Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 280945 11 pageshttpdxdoiorg1011552013280945
Research ArticleFilippov Ratio-Dependent Prey-Predator Model withThreshold Policy Control
Xianghong Zhang and Sanyi Tang
College of Mathematics and Information Science Shaanxi Normal University Xirsquoan Shaanxi 710062 China
Correspondence should be addressed to Sanyi Tang sytangsnnueducn
Received 10 June 2013 Revised 23 August 2013 Accepted 2 September 2013
Academic Editor Hamid Reza Karimi
Copyright copy 2013 X Zhang and S Tang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
The Filippov ratio-dependent prey-predator model with economic threshold is proposed and studied In particular the slidingmode domain sliding mode dynamics and the existence of four types of equilibria and tangent points are investigated firstlyFurther the stability of pseudoequilibrium is addressed by using theoretical and numerical methods and also the local slidingbifurcations including regularvirtual equilibrium bifurcations and boundary node bifurcations are studied Finally some globalsliding bifurcations are addressed numerically The globally stable touching cycle indicates that the density of pest population canbe successfully maintained below the economic threshold level by designing suitable threshold policy strategies
1 Introduction
Ordinary differential equation models (ODE models) arewidely used to describe the dynamics between predators andtheir prey which has long been and will continue to be oneof significant fields in mathematical ecology owing to itsuniversal existence and importance [1 2] The simplest prey-predator dynamic model is the Lotka-Volterra model [3]which has been modified in many ways since its original andrealism formulation in the 1920s
One important component of the prey-predator relationis predatorrsquos functional response which refers to the changein the density of prey attached per unit time per predatoras the prey density changes and makes the prey-predatorsystem more realistic There are several famous functionalresponse types in previous work which are monotonicallyincreasing and uniformly bounded functions in the firstquadrant Another functional response is the Michaelis-Menten (or Holling-type II) functional response which is themost common type of functional response among arthropodpredators [4 5] It takes the form 119901(119909 119910) = 119886119909(119887+119909) where119886 and 119887 are positive constants that stand for capturing rateand half capturing saturation constant respectively
Considering predators having to search for food a moresuitable general prey-predator theory based on the so-called
ratio-dependent theory is involved It can be roughly com-prehended as the per capita predator growth rate should bea function of the ratio of prey to predator abundance Andit is also strongly supported by numerous fields laboratoryexperiments and observations [6ndash8]
Therefore we can write the ratio-dependent prey-preda-tor model with Michaelis-Menten functional response as fol-lows
where 119909 and 119910 represent the density of prey (pest) andpredator (natural enemy) respectively The prey is assumedto grow logistically and 119896 is the carrying capacity of preyThepositive constants 119903 and 120575 stand for intrinsic growth rate ofprey and mortality rate of predator respectively 120573 denotesthe conversion rate of prey captured by predator
In population both ecologist and mathematicians areinterested in the ratio-dependent prey-predator model withMichaelis-Menten functional response [2 7ndash9] Hsu et al [7]resolved a complete classification of the asymptotic behaviorof the solutions of ratio-dependentmodel with theMichaelis-Menten functional response They also studied the global
2 Abstract and Applied Analysis
stability of all equilibria in various cases and reconsidered theuniqueness of limit cycle
It is well known that pests have been one of the principalthreats to crops important plants animals and humans allover the world Therefore it is necessary to apply acceptableand effective strategies to control pest outbreak In practice itis impossible to eradicate the pests completely nor is it biolog-ically or economically desirable Integrated pest management(IPM) is a long term management strategy [10ndash12] whichuses a combination of biological cultural and chemicaltactics so as to lower cost to the growers minimize effecton the environment and maintain pest population below theeconomic injury level (EIL) On the basis of IPM biologicalstrategy is useful and effective to suppress pest populationsuch as releasing beneficial natural enemies culture strategymakes the environment less favorable to pests such ascatching or harvesting artificially In most cropping systemswhen the above two tactics are unable to keep pest populationbelow the ET chemical strategy (ie insecticide) is still aprincipal means to control pests and prevent economic lossThus in order to control pest outbreak we should carryout control strategies when the number of pests reachesor exceeds the ET which is lower than the EIL and thecontrol strategies should be suspended once the density ofpest population falls below the ET which is the so-calledthreshold policy control (TPC) Considering IPM strategieseither fixed moment or state-dependent impulsive modelswith the ratio-dependent orMichaelis-Menten-type responsefunction of prey-predatormodel have been studied in [13ndash16]
However Zhao et al [17] have stated some disadvantagesof the impulsive differential equation models mentionedabove First in the fixed moment impulsive model withoutconsideration whether the density of pest reaches the ETor not control strategies are invariably implemented whichleads to consumption of vast resources Second in realityall kinds of control strategies need some time and cannot befinished instantaneously but in the state-dependent impul-sive differential models control strategies are carried outinstantaneously which is not reasonable
Therefore we use Filippov system which is a vectordifferential equation with discontinuous right-hand side todescribe prey-predator model with both noninstantaneousinterventions and the threshold policy Recently althoughFilippov systems have been widely utilized in scienceand engineering including harvesting thresholds oil welldrilling and liquid-gas reaction [18ndash23] However verylittle is involved that they are used to investigate theratio-dependent-type predator-prey model with Michaelis-Menten-type functional response We assume that a propor-tion of preys are caught or transferred (culture strategy) orkilled (chemical strategy) denoted by 119902
1 a proportion of
predators are released (biological strategy) denoted by 1199022 So
we have the following control model for 119909 gt ET
In this paper we aim to give a detailed analysis of Filippovratio-dependent prey-predator model with threshold policycontrol which describes that control measures are imple-mented only when the density of pest in a population exceedsthe ET We investigate the sliding mode domain slidingmode dynamics the existence of four types of equilibria andtangent point of Filippov system regularvirtual equilibriumbifurcation and boundary node bifurcations In addition thelocal stability of pseudoequilibrium implies global stabilityin our numerical simulations Globally touching bifurcationespecially indicates that the density of pest can be successfullymaintained below the ET by designing suitable threshold pol-icy strategy Therefore our control objective can be achievedin the above two cases which are desired situations in croplivestock sectors and forestry
The organization of this paper is as follows in Section 2we give some basic results and preliminaries for ODE systemand Filippov system In Section 3 the existence of slidingsegments and sliding mode dynamics for the Filippov system(3) are addressed In Section 4 we give the null-isoclines andequilibria Based on those results in Section 5 we considerthe bifurcation sets of equilibria and sliding bifurcationanalyses Then in the last section we give some discussions
2 The ODE System and Filippov System
21 The Basic Preliminaries and Results for ODE SystemThe ODE model (1) has been well studied in [2 7] and acomplete classification of the asymptotic behavior of the solu-tions of the ratio-dependent model with Michaelis-Mentenfunctional response has been proposed In the following wepresent some primary results in the following Lemma whichare useful for this study
Lemma 1 System (1) includes three equilibria (0 0) (119896 0)and a unique positive equilibrium 119864
lowast(119909lowast 119910lowast) if and only if the
following two conditions are true 120573119886 minus 120575 gt 0 and 119903 gt (120573119886 minus
120575)120573119887 where 119864lowast = (119909lowast 119910lowast) = (119896(120573119887119903 + 120575 minus 120573119886)120573119887119903 119896(120573119886 minus
120575)(120573119887119903 + 120575 minus 120573119886)1205731198872119903120575) If 120573119886 minus 120575 gt 0 and 120573119886 minus 120575 lt 119903 le
(119886 minus 119887120575)119887 (0 0) is globally asymptotically stable if 120573119886minus 120575 lt 0
and 119903 ge (119886 minus 119887120575)119887 (119896 0) is globally asymptotically stable if120573119886 minus 120575 gt 0 and (120573119886 minus 120575)120573119887 lt 119903 le 120573119886 minus 120575 or 120573119886 minus 120575 gt 0119903 gt 120573119886 minus 120575 and 119903 ge (119886 minus 119887120575)119887 the positive equilibriumof 119864lowast = (119909
lowast 119910lowast) is globally asymptotically stable However if
the positive equilibrium is locally asymptotically stable thenthe system (1) has no nontrivial positive periodic solutions If120573119886 minus 120575 gt 0 and max120573119886 minus 120575 (120573119886 minus 120575)120573119887 lt 119903 lt (119886 minus 119887120575)119887
hold true then the system (1) has at most one stable limit cycle
22 Filippov Ratio-Dependent Prey-Predator Model and Pre-liminaries By now based on IPM strategies and TPC themodels (1) and (2) can be incorporated and rewritten in thefollowing form
We first introduce some useful properties and definitionson Filippov system according to [24 25] so that we caninvestigate the model (3) in more detail Let119867(119885) = 119909 minus ETwith vector 119885 = (119909 119910)
Definition 4 A point 119885lowast is called a boundary equilibrium ofsystem (3) if 119865
1198781
(119885lowast) = 0 with 119867(119885lowast) = 0 or 119865
1198782
(119885lowast) = 0
with119867(119885lowast) = 0
Definition 5 A point 119885lowast is called a tangent point of sys-tem (3) if 119885lowast isin Σ
119878and ⟨119867
119885(119885lowast) 1198651198781
(119885lowast)⟩ = 0 or
⟨119867119885(119885lowast) 1198651198782
(119885lowast)⟩ = 0
The details and knowledge about the Filippov systemsuch as the concepts of Filippov solution sliding mode solu-tion and bifurcation can be found in reference [24]
3 Sliding Region and Sliding Mode Dynamics
A sliding mode exists if there are regions in the discontinuityboundary Σ where the vectors for the two subsystems of thesystem (3) are directed towards each other It is well knownthat two basic methods the so-called Filippov convexmethod[24] andUtkin equivalent control method [18] are developedfor the sliding mode and its domains which are shown in theappendix
31 Sliding Segment and Region Based on the appendix wehave
120582 (119885) =119903 (1 minus ET119896) minus 119886119910 (119887119910 + ET)
1199021
(11)
because the sliding mode regions can be determined by solv-ing the inequalities that is 120582(119885) ge 0 and 120582(119885) le 1 In orderto solve the above two inequalities with respect to 119910 we needto consider the following two algebraic equations
119903 (1 minusET119896) minus
119886119910
119887119910 + ET= 0
119903 (1 minusET119896) minus
119886119910
119887119910 + ETminus 1199021= 0
(12)
Solving the above two algebraic equations with respect to119910 yields two roots denoted by
1199101=
119903ET (119896 minus ET)119886119896 minus 119887119903 (119896 minus ET)
1199102=
(119903 (119896 minus ET) minus 1199021119896)ET
119886119896 minus 119887 (119903 (119896 minus ET) minus 1199021119896)
(13)
4 Abstract and Applied Analysis
Based on the relations between 1199101and 119910
2 there exist two
cases for the existence of sliding segments of Filippov system(3) By simply calculating and arranging we have the follow-ing results
(i) When 119903 lt 119886119896119887(119896 minus ET) the sliding segment can bedescribed as
Σ1
119878= (119909 119910) | max 0 119910
2 le 119910 le 119910
1 119909 = ET (14)
(ii) When max1199021119896(119896 minus ET) 119886119896119887(119896 minus ET) lt 119903 lt
min119896(119886 + 1198871199021)119887(119896 minus ET) 119886(119896 + 119887119902
1)119887(119896 minus ET) the
sliding segment can be described as
Σ2
119878= (119909 119910) | 0 le 119910 le 119910
2 119909 = ET (15)
32 Sliding Mode Dynamics Filippov system (3) only hasone piece of sliding segment and the solutions defined init can be obtained from the sliding mode dynamics whichcan be determined by employing theUtkin equivalent controlmethod (see the appendix)
From119867 = 0 we get that
119867119885= 1199091015840= 119903119909 (1 minus
119909
119896) minus
119886119909119910
119887119910 + 119909minus 1205761199021119909 = 0 119909 = ET
(16)
And solving the above equations with respect to 120576 yields
120576 =119903 (1 minus ET119896) minus 119886119910 (119887119910 + ET)
1199021
(17)
Hence the dynamics on the slidingmodeΣ119878can be deter-
mined by the following scalar differential equation
119910 (119905) = minus 120575119910 +120573119886119910ET119887119910 + ET
+119903 (1 minus ET119896) minus 119886119910 (119887119910 + ET)
1199021
1199022119910
= 1198751(119910) (119875
2119910 + 1198753) ≜ 120601 (119910)
(18)
where 119910 isin Σ1
119878or Σ2
1198781198751(119910) = 119910119902
1(119887119910 + ET) 119875
2= minus120575119887119902
1+
1199031198871199022(1 minus ET119896) minus 119886119902
2 and 119875
3= minus120575119902
1ET + 120573119886119902
1ET + 119903119902
2(1 minus
ET119896)ET
4 Null-Isoclines and Equilibria
41The Null-Isoclines of Filippov System (3) Null-isoclines ofboth systems 119878
1and 1198782are related to the existence of equilib-
ria and are useful for analysis of sliding dynamicNull-isoclines (119905) = 0 and 119910(119905) = 0 for both systems
1198781and 119878
2can be determined as follows For the system 119878
1
solving the equation of the null-isocline (119905) = 0 yeilds
1198911198781(119909) =
119903119909 (1 minus 119909119896)
119886 minus 119887119903 (1 minus 119909119896) (19)
and null-isocline 119910(119905) = 0 gives
1198921198781(119909) =
(120573119886 minus 120575) 119909
119887120575 (20)
For the system 1198782 solving the equation of the null-isocline
(119905) = 0 yields
1198911198782(119909) =
(119903119909 (1 minus 119909119896) minus 1199021) 119909
119886 + 1198871199021minus 119887119903 (1 minus 119909119896)
(21)
and null-isocline 119910(119905) = 0 gives
1198921198782(119909) =
(120573119886 minus (120575 minus 1199022)) 119909
119887 (120575 minus 1199022)
(22)
42 The Equilibria of Filippov System (3) Because the solu-tions of Filippov system (3) are composed of connectingstandard solutions in subsystems 119878
1 1198782and sliding mode
solutions on Σ1119878or Σ2119878The definitions of all types of equilibria
of Filippov system have been provided in Section 2 which areimportant to bifurcation analysis There may be several typesof equilibria for Filippov system (3) which include regularequilibrium (denoted by119864
119877) virtual equilibrium (denoted by
119864119881) pseudoequilibrium (denoted by 119864
119875) boundary equilib-
rium (denoted by119864119861) and one type of special point named as
tangent point (denoted by 119864119879) Detailed definitions of these
equilibria and the tangent point can be found in the literature[24]
From Lemma 1 we know that the subsystem 1198781with 119909 lt
ET has three possible equilibria (0 0) (119896 0) and interiorequilibrium
119864lowast
1198781
= (119909lowast
1198781
119910lowast
1198781
)
= (119896 (120573119887119903 + 120575 minus 120573119886)
120573119887119903119896 (120573119886 minus 120575) (120573119887119903 + 120575 minus 120573119886)
1205731198872119903120575)
(23)
If 120573119886 minus 120575 gt 0 and 119903 gt (120573119886 minus 120575)120573119887 then there is a positiveinterior equilibrium for the system 119878
1
Equilibria for the subsystem 1198782with 119909 gt ET satisfy
119903119909 (1 minus119909
119896) minus
119886119909119910
119887119910 + 119909minus 1199021119909 = 0
minus120575119910 +120573119886119909119910
119887119910 + 119909+ 1199022119910 = 0
(24)
Solving the above equations with respect to 119909 and 119910 yieldsthree possible equilibria that is (0 0) (119896(119903 minus 119902
1)119903 0) and
interior equilibrium
119864lowast
1198782
= (119909lowast
1198782
119910lowast
1198782
)
= (119896 (120573119887 (119903 minus 119902
1) + 120575 minus 119902
2minus 120573119886)
120573119887119903
119896 (120573119886 minus 120575 + 1199022) (120573119887 (119903 minus 119902
1) + 120575 minus 119902
2minus 120573119886)
1205731198872119903 (120575 minus 1199022)
)
(25)
Abstract and Applied Analysis 5
which indicates that if 119903 gt (120573119886 minus 120575 + 1199022)120573119887 + 119902
1and 119902
2lt
120575 lt 120573119886 + 1199022 then there is a positive interior equilibrium for
the system 1198782
421 Regular Equilibria For the subsystem 1198781with 119909 lt ET
(0 0) is a regular equilibrium while (119896 0) is a virtual equilib-rium In addition according to the coordinate of equilibrium119864lowast
1198781
we have the following results If 119903 gt (120573119886 minus 120575)120573119887 gt 0 and
119896 (120573119887119903 + 120575 minus 120573119886)
120573119887119903lt ET (26)
then it is a regular equilibrium for the system 1198781 denoted by
1198641
119877 If 119903 gt (120573119886 minus 120575)120573119887 gt 0 and
119896 (120573119887119903 + 120575 minus 120573119886)
120573119887119903gt ET (27)
then the equilibrium 119864lowast
1198781
becomes a virtual equilibriumdenoted by 1198641
119881
About regular equilibria for the subsystem 1198782with 119909 gt
ET (0 0) is a virtual equilibrium while (119896(119903 minus 1199011)119903 0) may
be a regular or virtual equilibrium which depends on theparameter space Moreover according to the coordinate ofequilibrium 119864
lowast
1198782
we have the following conclusions If 119903 gt
(120573119886 minus 120575 + 1199022)120573119887 + 119902
1 1199022lt 120575 lt 120573119886 + 119902
2and
119896 (120573119887 (119903 minus 1199021) + 120575 minus 119902
2minus 120573119886)
120573119887119903gt ET (28)
then it is a regular equilibrium for the system 1198782 denoted by
1198642
119877 Note that if 119903 gt (120573119886 minus 120575 + 119902
2)120573119887 + 119902
1 1199022lt 120575 lt 120573119886 + 119902
2
and119896 (120573119887 (119903 minus 119902
1) + 120575 minus 119902
2minus 120573119886)
120573119887119903lt ET (29)
then the equilibrium 119864lowast
1198782
becomes a virtual equilibriumdenoted by 1198642
119881 Based on the above analysis if
119896 (120573119886 minus 120575)
120573119887 (119896 minus ET)lt 119903 lt
119896 (120573119886 minus 120575 + 1199022+ 120573119887119902
1)
120573119887 (119896 minus ET) (30)
then the two virtual equilibria 1198641119881and 1198642
119881can coexist
422 Pseudoequilibrium For the existence of pseudoequilib-rium119864
119875= (ET 119910
119875) 119910119875component of the pseudoequilibrium
of sliding flow satisfies the following equation
1198751(119910) (119875
2119910 + 1198753) = 0 (31)
where 119910 isin Σ1
119878or Σ2
119878 Solving the above equation with
respect to 119910 yields two possible roots denoted by 1199101198751
= 0 and
1199101198752
=(1205751199021minus 120573119886119902
1minus 1199031199022(1 minus ET119896))ET
minus1205751198871199021+ 1199031198871199022(1 minus ET119896) minus 119886119902
Solving the above equations with respect to 119909 and 119910 yieldstwo tangent points including
1198641
119879= (ET 119903ET (1 minus ET119896)
119886 minus 119887119903 (1 minus ET119896))
1198642
119879= (
ET (119903 (1 minus ET119896) minus 1199021)
119886 + 1199021119887 minus 119887119903 (1 minus ET119896)
)
(40)
If we fix all parameters then the relations among null-isoclines regularvirtual equilibria pseudoequilibrium andsliding segment are provided in Figure 1 Note that virtualequilibria of both systems 119878
1and 119878
2imply the existence of
pseudoequilibrium andwewill prove this general result later
43 The Stability of Pseudoequilibrium In the process ofpest control we should apply all kinds of control strategiesso as to prevent multiple pest outbreaks or make sure thatthe total density of the pest stabilizes at a desired level ofET If the unique positive equilibrium of system 119878
1and
system 1198782is virtual simultaneously then the sliding flow
has a unique pseudoequilibrium In order to realize thisgoal we can choose a set of parameters such that all theequilibria of subsystems 119878
1and 1198782are virtual equilibria and
the pseudoequilibria are globally stable which have beenwidely used in pest control [26ndash28] For example if wefixed all parameter values as those in Figure 2 then bothvirtual equilibria coexist Therefore we address the stabilityof pseudoequilibrium119864
1198752
in the followingwhich is importantto control pest
Theorem 6 Either the inequalities (34) or (35) hold true orthe two virtual equilibria 1198641
119881and 1198642
119881can coexist then Filippov
system (3) contains a positive pseudoequilibrium 1198641198752
Regard-less of which cases would occur the pseudoequilibrium 119864
1198752
islocally stable with respect to sliding mode domain
Proof According to the conditions of Theorem 6 we seethat the inequalities (30) hold true which implies that theinequalities (34) are true Based on the discussions aboutthe existence of pseudoequilibrium in Section 42 we haveconcluded that if inequalities (34) or (35) hold true thenthe system (3) contains a positive pseudoequilibrium 119864
=(1198961199021(120575 minus 120573119886) minus 119903119902
2(119896 minus ET)) (120575119887119896119902
1minus 119903119887119902
2(119896 minus ET) + 119886119896119902
2)
11988611990211198962 (119887120573119902
1+ 1199022)
(41)
Therefore if the inequalities (33) hold true then 119889120601(119910)119889119910 lt 0 and 119910
1198752
is a positive root of (31) Note that theinequalities (34) and (35) indicate the inequalities (33) Thusthe positive pseudoequilibrium 119864
1198752
is locally stale if it exists
Abstract and Applied Analysis 7
It is difficult to directly prove the global stability ofpseudoequilibrium 119864
1198752
in this case Because we cannotemploy the classical Bendixson-Dulac theorem due to thediscontinuity of vector fields However if there is not crossingcycle surrounding sliding segment then pseudoequilibriumis globally stable [29] From the analysis of global bifurcationfor the system (3) in the following section the system just hastouching bifurcation and there is nonexistence of a slidingcycle which surrounds the 119864
1198752
By using the similar methodswe have that pseudoequilibrium 119864
1198752
is globally stable Thatis to say the local stability of pseudoequilibrium 119864
1198752
withrespect to sliding mode domain indicates its global stabilityin the first quadrant (shown in Figures 3(a) 3(b) 4(b) and4(c)) In practice in order to control pest outbreak we shouldchoose the desirable ET at first so that all equilibria of eachsystem such as system 119878
1and system 119878
2become virtual then
pseudoequilibrium not only exists but also is globally stableIn other words the density of pest can be stable at the ETWhen the density of pest reaches or exceeds the ET we shouldcarry out control strategies (eg releasing natural enemyetc) until it falls below the ET In this way our control goalcan be realized fully
5 Equilibria and Sliding Bifurcation Set
51 RegularVirtual Equilibrium Bifurcation According tothe above discussions it is obvious that 119903 and ET are primaryfactors in determining the existence of the above differenttypes of equilibria of the system (3) So we define four curvesabout parameters 119903 and ET as follows
1198711= (119903ET) | 119903 =
120573119886 minus 120575
120573119887
1198712= (119903ET) | 119903 =
120573119886 minus 120575 + 1199022
120573119887+ 1199021
1198713= (119903ET) | ET =
119896 (120573119887119903 + 120575 minus 120573119886)
120573119887119903
1198714= (119903ET) | ET =
119896 (120573119887 (119903 minus 1199021) + 120575 minus 119902
2minus 120573119886)
120573119887119903
(42)
The four curves (ie 1198711 1198712 1198713 and 119871
4) divide the 119903
and ET parameter space into six regions and the existenceor coexistence of regular or virtual equilibria is indicated ineach region The boundary equilibria 1198641
119861and 1198642
119861can appear
on the lines 1198713and 119871
4accordingly In particular it follows
from Figure 2 that the two virtual equilibria 1198641119881and 1198642
119881can
coexist which is very important to pest control However thetwo regular equilibria 1198641
119877and 1198642
119877cannot coexist
52 Boundary Node Bifurcations This type of bifurcationmay occur for Filippov system (3) once equilibria 119864
119875 119864119879 and
119864119877or 119864119879and 119864
119877collide together simultaneously when ET
passes through a critical value In this part we choose ETas bifurcation parameter and all other parameters are fixedas those in Figure 3 Note that once the parameter ET passes
through the first critical value ET1198881
= 93333 the regular equi-librium 119864
1
119877 tangent point 119864
119879 and pseudoequilibrium 119864
119875col-
lide together (see Figure 3(b)) where ET1198881
is determined by
ET1198881
=119896 (120573119887119903 + 120575 minus 120573119886)
120573119887119903 (43)
A virtual equilibrium 1198641
119881 an invisible tangent point 1198641
119879
and a pseudoequilibrium 119864119875coexist as shown in Figure 3(a)
when ET lt ET1198881
They collide at ET = ET1198881
and are substitut-ed by a visible tangent point1198641
119879 as shown in Figure 3(c) when
ET gt ET1198881
Similarly another boundary node bifurcation of Filippov
system (3) occurs at ET1198882
= 46667 (see Figure 4(b)) whereET1198882
is determined by
ET1198882
=119896 (120573119887 (119903 minus 119902
1) + 120575 minus 119902
2minus 120573119886)
120573119887119903 (44)
A stable regular equilibrium 1198642
119877and a visible tangent
point 1198642119879coexist as shown in Figure 4(a) when ET lt ET
1198882
They collide with a pseudoequilibrium 119864
119875at ET = ET
1198882
andare substituted by an invisible tangent point 1198642
119879as shown in
Figure 4(c) when ET gt ET1198882
53 Global Sliding Bifurcation Global sliding bifurcationsinvolve nonvanishing cycles which include sliding discon-nection touching (or grazing) bifurcation buckling bifurca-tion crossing bifurcation bifurcation of a sliding homoclinicorbit and heteroclinic orbit [22] Touching (or grazing) bifur-cation implies that a positive period solution can collide withthe sliding segments From the work of Kuang and BerettaandHsu et al [2 7] they concluded that when the value of theparameter 119886 passed slightly through the bifurcation value astable limit cycle bifurcates from the unstable positive interiorequilibrium for the system (1) According to numericalsimulations the system (3) just has touching bifurcation Forexample if we choose ET as bifurcation parameter and fixall other parameters as shown in Figure 5 when the valueof parameter ET varies touching bifurcation occurs at thecritical value ET
1198883
= 259 for the system (3) Note thatextensive numerical simulations indicate that nomatter whatthe value of ET is in touching bifurcation the whole periodicsolution lies in the region 119878
1(shown in Figure 5)This implies
that the density of pest can be successfully maintained belowET by designing suitable threshold policy strategies So ourcontrol objective can be fully realized which is a desiredsituation in crop livestock sectors and forestry
6 Discussion
Recently the threshold policy and IPM strategies haveattracted great attention in agriculture forestry animal hus-bandry and so on [10ndash12] In the process of pest controlIPM strategies would be used only when the density ofpest reaches or exceeds the ET In addition Filippov systemprovides a natural and rational framework for those realworld problems so it has been widely used in different fields
8 Abstract and Applied Analysis
0 5 10 150
1
2
3
4
5
6
7
8
9
10
11
x
y
E2V
E1V
EP
(a) ET = 83
0 5 10 150
1
2
3
4
5
6
7
8
9
10
11
x
y
E2V
E1B(EP)
(b) ET = 93333
0 5 10 150
1
2
3
4
5
6
7
8
9
10
11E2V
E1R
x
y
(c) ET = 98
Figure 3 Boundary node bifurcation for Filippov system (3) The parameter values are fixed as follows 119886 = 07 119887 = 06 119903 = 15 119896 = 12120573 = 05 120575 = 025 119902
1= 025 and 119902
2= 01
0 5 10
5
10
15
20
25
30
35
40
45
E1V
E2R
x
y
(a) ET = 4
0
5
10
15
20
25
30
35
0 5 10x
yE2B(EP)
E1V
(b) ET = 46667
0
5
10
15
20
25
0 5 10x
yE2V
E1V
EP
(c) ET = 58
Figure 4 Boundary node bifurcation for Filippov system (3) The parameter values are fixed as follows 119886 = 07 119887 = 06 119903 = 15 119896 = 12120573 = 05 120575 = 025 119902
1= 025 and 119902
2= 01
such as in science and engineering [19ndash21 23] In this paperwe employ the sliding analysis of Filippov system to describeand investigate the long term dynamical behavior of theratio-dependent-type prey-predator model with Michaelis-Menten-type functional response so that we can use theFilippov system to model intervention of pest control policyFirstly we investigate the sliding mode domain sliding modedynamics Secondly the null-isoclines and the existence of
four types of equilibria for Filippov system including regularvirtual boundary and pseudoequilibrium and the tangentpoints are discussed in detail Moreover the stability ofpseudoequilibrium is also studied Thirdly we have investi-gated the local sliding bifurcations including regularvirtualequilibrium bifurcation and boundary node bifurcationsFurther global touching bifurcation is also studied by numer-ical techniques
Abstract and Applied Analysis 9
0 1 2 30
5
10
15
x
y
E1R
(a) ET = 15
0 2 40
5
10
15
x
y
E1R
(b) ET = 259
0 2 4 60
5
10
15
x
y
E1R
(c) ET = 35
Figure 5 Globally touching bifurcation for Filippov system (3) The parameter values are fixed as follows 119886 = 051 119887 = 04 119903 = 12 119896 = 6120573 = 06863 120575 = 005 119902
1= 04 and 119902
2= 01
In the process of pest control we should apply all kindsof control strategies so as to prevent multiple pest outbreaksor make sure that the density of pest stabilizes at a desiredlevel of ET In order to realize these goals on the one handwe can determine a set of parameters such that not only all theequilibria of subsystems 119878
1and 1198782are virtual equilibria which
ensures that the pseudoequilibria exist but also the pseudoe-quilibrium is globally stable From Theorem 6 we providedthe conditions of the existence of pseudoequilibrium 119864
1198752
forthe system (3) and showed that it is locally stable if it existsFurther we showed that the local stability of pseudoequilib-rium 119864
1198752
implies global stability by numerical simulation Onthe other hand globally touching bifurcation indicates thatthe density of pest can be successfully maintained below theET by designing suitable threshold policy strategies (shownin Figure 5) Therefore our control objective can be achievedfully in the above two cases which can be used to pest controlin crop livestock sectors and forestry
Although impulsive prey-predator models with the ratio-dependent- or Michaelis-Menten-type response functionhave been studied in [13ndash16] the Filippov systems have manyadvantages in describing interventions including sprayingpesticides and releasing natural enemies compared withimpulsive models In this work the number of natural enemyto be released is proportional to its number It is interestingthat releasing number of natural enemy can be described byconstant independently of the existing numbers of pest andnatural enemy Moreover in practice considering environ-mental energy resource finiteness we should choose the totalnumber of both populations as a guide to switch the systemwhich is called the weighted escapement policy (WEP) [1930ndash32] Therefore in the future work we will focus on theabove two cases which could result in richer dynamics
Appendix
Methods for Analysis of the Sliding Solution
Filippov Convex Method The Filippov method associatesthe following convex combination 119865
119878(119885) of the two vectors
1198651198781
(119885) and 1198651198782
(119885) with each nonsingular sliding point 119885 isin
Σ2 that is
119865119878(119885) = (1 minus 120582 (119885)) 119865
1198781(119885) + 120582 (119885) 119865
1198782(119885) (A1)
where
120582 (119885) =⟨119867119885 1198651198781
⟩
⟨119867119885 1198651198781
minus F1198782
⟩ (A2)
119865119878(119885) is tangent to Σ
2and 0 le 120582(119885) le 1
Thus the sliding mode dynamics can be determined by
= 119865119878(119885) 119885 isin Σ
2 (A3)
which is smooth on a one-dimensional sliding interval of Σ2
The solution of the above equation is the sliding solutionThe equation 120582(119885) = 0 (or 120582(119885) = 1) indicates that the
flow is determined by 1198651198781
(or 1198651198782
) aloneTherefore the slidingmode domain can be defined as
Σ119878= 119885 = (119909 119910) isin Σ | 0 le 120582 (119885) le 1 (A4)
which is equivalent to Σ119878= 119885 isin Σ | 120590(119885) le 0 Denote
Σ+
119878= 119885 isin Σ | 120582 (119885) = 1 Σ
minus
119878= 119885 isin Σ | 120582 (119885) = 0
(A5)
10 Abstract and Applied Analysis
by the boundary of the sliding mode domain so the vectorfields are tangent to boundary Σ+
119878or Σminus119878
Utkin Equivalent Control Method It follows from Utkinrsquosworks [18] on sliding mode dynamics along the manifoldΣ we note that Filippov system (3) can be also rewritten asfollows
= 119865 (119885119880119867) (A6)
where the control 119880119867is defined as
119880119867=
0 119867 (119885) lt 0
1198801(119885 119905) 119867 (119885) gt 0
(A7)
where 1198801is a continuous function In the controlled system
(ie system 1198782) the control 119880
119867= 1198801is applied and in the
free system (ie system 1198781) 119880119867= 0 is applied Assume that
a sliding mode exists on manifold Σ that is Σ2is nonempty
Solving the following algebraic equation
=120597119867
120597119885119865 (119885U
119867) = 0 (A8)
with respect to 119880119867
on this manifold gives the solutiondenoted by 119880
lowast(119885 119905) which is referred to as equivalent
control Substituting for 119880119867in system (A6) yields
which determines the sliding mode dynamics of the Filippovsystem (3)
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China (NSFC 11171199) and by the FundamentalResearch Funds for the Central Universities (GK201003001)
References
[1] R M May ldquoLimit cycles in predator-prey communitiesrdquo Sci-ence vol 177 no 4052 pp 900ndash902 1972
[2] Y Kuang and E Beretta ldquoGlobal qualitative analysis of a ratio-dependent predator-prey systemrdquo Journal ofMathematical Biol-ogy vol 36 no 4 pp 389ndash406 1998
[3] W Murdoch C Briggs and R Nisbet Consumer-Resource Dy-namics Princeton University Press New York NY USA 2003
[4] H I Freedman Deterministic Mathematical Models in Popula-tion Ecology vol 57 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1980
[5] G T Skalski and J F Gilliam ldquoFunctional responseswith preda-tor interference viable alternatives to theHolling type IImodelrdquoEcology vol 82 no 11 pp 3083ndash3092 2001
[6] R Arditi and L R Ginzburg ldquoCoupling in predator-preydynamics ratio-dependencerdquo Journal ofTheoretical Biology vol139 no 3 pp 311ndash326 1989
[7] S-B Hsu T-W Hwang and Y Kuang ldquoGlobal analysis ofthe Michaelis-Menten-type ratio-dependent predator-prey sys-temrdquo Journal of Mathematical Biology vol 42 no 6 pp 489ndash506 2001
[8] C Jost O Arino and R Arditi ldquoAbout deterministic extinctionin ratio-dependent predator-prey modelsrdquo Bulletin of Mathe-matical Biology vol 61 no 1 pp 19ndash32 1999
[9] D Xiao and S Ruan ldquoGlobal dynamics of a ratio-dependentpredator-prey systemrdquo Journal of Mathematical Biology vol 43no 3 pp 268ndash290 2001
[10] Z Lu X Chi and L Chen ldquoImpulsive control strategies inbiological control of pesticiderdquo Theoretical Population Biologyvol 64 no 1 pp 39ndash47 2003
[11] S Tang and R A Cheke ldquoModels for integrated pest controland their biological implicationsrdquo Mathematical Biosciencesvol 215 no 1 pp 115ndash125 2008
[12] J C Van Lenteren and J Woets ldquoBiological and integrated pestcontrol in greenhousesrdquo Annual Review of Entomology vol 33pp 239ndash250 1988
[13] B Dai H Su and D Hu ldquoPeriodic solution of a delayed ratio-dependent predator-prey model with monotonic functionalresponse and impulserdquo Nonlinear Analysis Theory Methods ampApplications vol 70 no 1 pp 126ndash134 2009
[14] X Liu G Li and G Luo ldquoPositive periodic solution for a two-species ratio-dependent predator-prey system with time delayand impulserdquo Journal of Mathematical Analysis and Applica-tions vol 325 no 1 pp 715ndash723 2007
[15] G Jiang Q Lu and L Qian ldquoComplex dynamics of a Hollingtype II prey-predator system with state feedback controlrdquoChaos Solitons amp Fractals vol 31 no 2 pp 448ndash461 2007
[16] H Baek and Y Lim ldquoDynamics of an impulsively controlledMichaelis-Menten type predator-prey systemrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 16 no4 pp 2041ndash2053 2011
[17] T Zhao Y Xiao and R J Smith ldquoNon-smooth plant diseasemodels with economic thresholdsrdquo Mathematical Biosciencesvol 241 no 1 pp 34ndash48 2013
[18] V I Utkin Sliding Modes in Control and Optimization Com-munications and Control Engineering Series Springer BerlinGermany 1992
[19] M I S Costa E Kaszkurewicz A Bhaya and L Hsu ldquoAchiev-ing global convergence to an equilibrium population in pred-ator-prey systems by the use of a discontinuous harvestingpolicyrdquo Ecological Modelling vol 128 no 2-3 pp 89ndash99 2000
[20] B L Van De Vrande D H Van Campen and A De KrakerldquoApproximate analysis of dry-friction-induced stick-slip vibra-tions by a smoothing procedurerdquo Nonlinear Dynamics vol 19no 2 pp 157ndash169 1999
[21] S H Doole and S J Hogan ldquoA piecewise linear suspensionbridge model nonlinear dynamics and orbit continuationrdquoDynamics and Stability of Systems vol 11 no 1 pp 19ndash47 1996
[22] Yu A Kuznetsov S Rinaldi and A Gragnani ldquoOne-parameterbifurcations in planar Filippov systemsrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol13 no 8 pp 2157ndash2188 2003
[23] M di Bernardo C J Budd A R Champneys et al ldquoBifurca-tions in nonsmooth dynamical systemsrdquo SIAM Review vol 50no 4 pp 629ndash701 2008
[24] A F Filippov Differential Equations with Discontinuous Right-hand Sides vol 18 of Mathematics and Its Applications (SovietSeries) Kluwer Academic Dordrecht The Netherlands 1988
[25] S Tang J Liang Y Xiao and R A Cheke ldquoSliding bifurcationsof Filippov two stage pest control models with economicthresholdsrdquo SIAM Journal on Applied Mathematics vol 72 no4 pp 1061ndash1080 2012
Abstract and Applied Analysis 11
[26] M I da Silveira Costa and M E M Meza ldquoApplication of athreshold policy in the management of multispecies fisheriesand predator cullingrdquoMathematical Medicine and Biology vol23 no 1 pp 63ndash75 2006
[27] M E Mendoza Meza A Bhaya E Kaszkurewicz and M IDa Silveira Costa ldquoThreshold policies control for predator-preysystems using a control Liapunov function approachrdquoTheoreti-cal Population Biology vol 67 no 4 pp 273ndash284 2005
[28] F Dercole A Gragnani and S Rinaldi ldquoBifurcation analysisof piecewise smooth ecological modelsrdquoTheoretical PopulationBiology vol 72 no 2 pp 197ndash213 2007
[29] W Wang ldquoBackward bifurcation of an epidemic model withtreatmentrdquo Mathematical Biosciences vol 201 no 1-2 pp 58ndash71 2006
[30] I Noy-Meir ldquoStability of grazing systems an application ofpredator-prey graphsrdquo The Journal of Animal Ecology vol 63no 2 pp 459ndash481 1975
[31] R M Colombo and V Krivan ldquoSelective strategies in foodwebsrdquo Mathematical Medicine and Biology vol 10 no 4 pp281ndash291 1993
[32] V Krivan ldquoOptimal foraging and predator-prey dynamicsrdquoTheoretical Population Biology vol 49 no 3 pp 265ndash290 1996
stability of all equilibria in various cases and reconsidered theuniqueness of limit cycle
It is well known that pests have been one of the principalthreats to crops important plants animals and humans allover the world Therefore it is necessary to apply acceptableand effective strategies to control pest outbreak In practice itis impossible to eradicate the pests completely nor is it biolog-ically or economically desirable Integrated pest management(IPM) is a long term management strategy [10ndash12] whichuses a combination of biological cultural and chemicaltactics so as to lower cost to the growers minimize effecton the environment and maintain pest population below theeconomic injury level (EIL) On the basis of IPM biologicalstrategy is useful and effective to suppress pest populationsuch as releasing beneficial natural enemies culture strategymakes the environment less favorable to pests such ascatching or harvesting artificially In most cropping systemswhen the above two tactics are unable to keep pest populationbelow the ET chemical strategy (ie insecticide) is still aprincipal means to control pests and prevent economic lossThus in order to control pest outbreak we should carryout control strategies when the number of pests reachesor exceeds the ET which is lower than the EIL and thecontrol strategies should be suspended once the density ofpest population falls below the ET which is the so-calledthreshold policy control (TPC) Considering IPM strategieseither fixed moment or state-dependent impulsive modelswith the ratio-dependent orMichaelis-Menten-type responsefunction of prey-predatormodel have been studied in [13ndash16]
However Zhao et al [17] have stated some disadvantagesof the impulsive differential equation models mentionedabove First in the fixed moment impulsive model withoutconsideration whether the density of pest reaches the ETor not control strategies are invariably implemented whichleads to consumption of vast resources Second in realityall kinds of control strategies need some time and cannot befinished instantaneously but in the state-dependent impul-sive differential models control strategies are carried outinstantaneously which is not reasonable
Therefore we use Filippov system which is a vectordifferential equation with discontinuous right-hand side todescribe prey-predator model with both noninstantaneousinterventions and the threshold policy Recently althoughFilippov systems have been widely utilized in scienceand engineering including harvesting thresholds oil welldrilling and liquid-gas reaction [18ndash23] However verylittle is involved that they are used to investigate theratio-dependent-type predator-prey model with Michaelis-Menten-type functional response We assume that a propor-tion of preys are caught or transferred (culture strategy) orkilled (chemical strategy) denoted by 119902
1 a proportion of
predators are released (biological strategy) denoted by 1199022 So
we have the following control model for 119909 gt ET
In this paper we aim to give a detailed analysis of Filippovratio-dependent prey-predator model with threshold policycontrol which describes that control measures are imple-mented only when the density of pest in a population exceedsthe ET We investigate the sliding mode domain slidingmode dynamics the existence of four types of equilibria andtangent point of Filippov system regularvirtual equilibriumbifurcation and boundary node bifurcations In addition thelocal stability of pseudoequilibrium implies global stabilityin our numerical simulations Globally touching bifurcationespecially indicates that the density of pest can be successfullymaintained below the ET by designing suitable threshold pol-icy strategy Therefore our control objective can be achievedin the above two cases which are desired situations in croplivestock sectors and forestry
The organization of this paper is as follows in Section 2we give some basic results and preliminaries for ODE systemand Filippov system In Section 3 the existence of slidingsegments and sliding mode dynamics for the Filippov system(3) are addressed In Section 4 we give the null-isoclines andequilibria Based on those results in Section 5 we considerthe bifurcation sets of equilibria and sliding bifurcationanalyses Then in the last section we give some discussions
2 The ODE System and Filippov System
21 The Basic Preliminaries and Results for ODE SystemThe ODE model (1) has been well studied in [2 7] and acomplete classification of the asymptotic behavior of the solu-tions of the ratio-dependent model with Michaelis-Mentenfunctional response has been proposed In the following wepresent some primary results in the following Lemma whichare useful for this study
Lemma 1 System (1) includes three equilibria (0 0) (119896 0)and a unique positive equilibrium 119864
lowast(119909lowast 119910lowast) if and only if the
following two conditions are true 120573119886 minus 120575 gt 0 and 119903 gt (120573119886 minus
120575)120573119887 where 119864lowast = (119909lowast 119910lowast) = (119896(120573119887119903 + 120575 minus 120573119886)120573119887119903 119896(120573119886 minus
120575)(120573119887119903 + 120575 minus 120573119886)1205731198872119903120575) If 120573119886 minus 120575 gt 0 and 120573119886 minus 120575 lt 119903 le
(119886 minus 119887120575)119887 (0 0) is globally asymptotically stable if 120573119886minus 120575 lt 0
and 119903 ge (119886 minus 119887120575)119887 (119896 0) is globally asymptotically stable if120573119886 minus 120575 gt 0 and (120573119886 minus 120575)120573119887 lt 119903 le 120573119886 minus 120575 or 120573119886 minus 120575 gt 0119903 gt 120573119886 minus 120575 and 119903 ge (119886 minus 119887120575)119887 the positive equilibriumof 119864lowast = (119909
lowast 119910lowast) is globally asymptotically stable However if
the positive equilibrium is locally asymptotically stable thenthe system (1) has no nontrivial positive periodic solutions If120573119886 minus 120575 gt 0 and max120573119886 minus 120575 (120573119886 minus 120575)120573119887 lt 119903 lt (119886 minus 119887120575)119887
hold true then the system (1) has at most one stable limit cycle
22 Filippov Ratio-Dependent Prey-Predator Model and Pre-liminaries By now based on IPM strategies and TPC themodels (1) and (2) can be incorporated and rewritten in thefollowing form
We first introduce some useful properties and definitionson Filippov system according to [24 25] so that we caninvestigate the model (3) in more detail Let119867(119885) = 119909 minus ETwith vector 119885 = (119909 119910)
Definition 4 A point 119885lowast is called a boundary equilibrium ofsystem (3) if 119865
1198781
(119885lowast) = 0 with 119867(119885lowast) = 0 or 119865
1198782
(119885lowast) = 0
with119867(119885lowast) = 0
Definition 5 A point 119885lowast is called a tangent point of sys-tem (3) if 119885lowast isin Σ
119878and ⟨119867
119885(119885lowast) 1198651198781
(119885lowast)⟩ = 0 or
⟨119867119885(119885lowast) 1198651198782
(119885lowast)⟩ = 0
The details and knowledge about the Filippov systemsuch as the concepts of Filippov solution sliding mode solu-tion and bifurcation can be found in reference [24]
3 Sliding Region and Sliding Mode Dynamics
A sliding mode exists if there are regions in the discontinuityboundary Σ where the vectors for the two subsystems of thesystem (3) are directed towards each other It is well knownthat two basic methods the so-called Filippov convexmethod[24] andUtkin equivalent control method [18] are developedfor the sliding mode and its domains which are shown in theappendix
31 Sliding Segment and Region Based on the appendix wehave
120582 (119885) =119903 (1 minus ET119896) minus 119886119910 (119887119910 + ET)
1199021
(11)
because the sliding mode regions can be determined by solv-ing the inequalities that is 120582(119885) ge 0 and 120582(119885) le 1 In orderto solve the above two inequalities with respect to 119910 we needto consider the following two algebraic equations
119903 (1 minusET119896) minus
119886119910
119887119910 + ET= 0
119903 (1 minusET119896) minus
119886119910
119887119910 + ETminus 1199021= 0
(12)
Solving the above two algebraic equations with respect to119910 yields two roots denoted by
1199101=
119903ET (119896 minus ET)119886119896 minus 119887119903 (119896 minus ET)
1199102=
(119903 (119896 minus ET) minus 1199021119896)ET
119886119896 minus 119887 (119903 (119896 minus ET) minus 1199021119896)
(13)
4 Abstract and Applied Analysis
Based on the relations between 1199101and 119910
2 there exist two
cases for the existence of sliding segments of Filippov system(3) By simply calculating and arranging we have the follow-ing results
(i) When 119903 lt 119886119896119887(119896 minus ET) the sliding segment can bedescribed as
Σ1
119878= (119909 119910) | max 0 119910
2 le 119910 le 119910
1 119909 = ET (14)
(ii) When max1199021119896(119896 minus ET) 119886119896119887(119896 minus ET) lt 119903 lt
min119896(119886 + 1198871199021)119887(119896 minus ET) 119886(119896 + 119887119902
1)119887(119896 minus ET) the
sliding segment can be described as
Σ2
119878= (119909 119910) | 0 le 119910 le 119910
2 119909 = ET (15)
32 Sliding Mode Dynamics Filippov system (3) only hasone piece of sliding segment and the solutions defined init can be obtained from the sliding mode dynamics whichcan be determined by employing theUtkin equivalent controlmethod (see the appendix)
From119867 = 0 we get that
119867119885= 1199091015840= 119903119909 (1 minus
119909
119896) minus
119886119909119910
119887119910 + 119909minus 1205761199021119909 = 0 119909 = ET
(16)
And solving the above equations with respect to 120576 yields
120576 =119903 (1 minus ET119896) minus 119886119910 (119887119910 + ET)
1199021
(17)
Hence the dynamics on the slidingmodeΣ119878can be deter-
mined by the following scalar differential equation
119910 (119905) = minus 120575119910 +120573119886119910ET119887119910 + ET
+119903 (1 minus ET119896) minus 119886119910 (119887119910 + ET)
1199021
1199022119910
= 1198751(119910) (119875
2119910 + 1198753) ≜ 120601 (119910)
(18)
where 119910 isin Σ1
119878or Σ2
1198781198751(119910) = 119910119902
1(119887119910 + ET) 119875
2= minus120575119887119902
1+
1199031198871199022(1 minus ET119896) minus 119886119902
2 and 119875
3= minus120575119902
1ET + 120573119886119902
1ET + 119903119902
2(1 minus
ET119896)ET
4 Null-Isoclines and Equilibria
41The Null-Isoclines of Filippov System (3) Null-isoclines ofboth systems 119878
1and 1198782are related to the existence of equilib-
ria and are useful for analysis of sliding dynamicNull-isoclines (119905) = 0 and 119910(119905) = 0 for both systems
1198781and 119878
2can be determined as follows For the system 119878
1
solving the equation of the null-isocline (119905) = 0 yeilds
1198911198781(119909) =
119903119909 (1 minus 119909119896)
119886 minus 119887119903 (1 minus 119909119896) (19)
and null-isocline 119910(119905) = 0 gives
1198921198781(119909) =
(120573119886 minus 120575) 119909
119887120575 (20)
For the system 1198782 solving the equation of the null-isocline
(119905) = 0 yields
1198911198782(119909) =
(119903119909 (1 minus 119909119896) minus 1199021) 119909
119886 + 1198871199021minus 119887119903 (1 minus 119909119896)
(21)
and null-isocline 119910(119905) = 0 gives
1198921198782(119909) =
(120573119886 minus (120575 minus 1199022)) 119909
119887 (120575 minus 1199022)
(22)
42 The Equilibria of Filippov System (3) Because the solu-tions of Filippov system (3) are composed of connectingstandard solutions in subsystems 119878
1 1198782and sliding mode
solutions on Σ1119878or Σ2119878The definitions of all types of equilibria
of Filippov system have been provided in Section 2 which areimportant to bifurcation analysis There may be several typesof equilibria for Filippov system (3) which include regularequilibrium (denoted by119864
119877) virtual equilibrium (denoted by
119864119881) pseudoequilibrium (denoted by 119864
119875) boundary equilib-
rium (denoted by119864119861) and one type of special point named as
tangent point (denoted by 119864119879) Detailed definitions of these
equilibria and the tangent point can be found in the literature[24]
From Lemma 1 we know that the subsystem 1198781with 119909 lt
ET has three possible equilibria (0 0) (119896 0) and interiorequilibrium
119864lowast
1198781
= (119909lowast
1198781
119910lowast
1198781
)
= (119896 (120573119887119903 + 120575 minus 120573119886)
120573119887119903119896 (120573119886 minus 120575) (120573119887119903 + 120575 minus 120573119886)
1205731198872119903120575)
(23)
If 120573119886 minus 120575 gt 0 and 119903 gt (120573119886 minus 120575)120573119887 then there is a positiveinterior equilibrium for the system 119878
1
Equilibria for the subsystem 1198782with 119909 gt ET satisfy
119903119909 (1 minus119909
119896) minus
119886119909119910
119887119910 + 119909minus 1199021119909 = 0
minus120575119910 +120573119886119909119910
119887119910 + 119909+ 1199022119910 = 0
(24)
Solving the above equations with respect to 119909 and 119910 yieldsthree possible equilibria that is (0 0) (119896(119903 minus 119902
1)119903 0) and
interior equilibrium
119864lowast
1198782
= (119909lowast
1198782
119910lowast
1198782
)
= (119896 (120573119887 (119903 minus 119902
1) + 120575 minus 119902
2minus 120573119886)
120573119887119903
119896 (120573119886 minus 120575 + 1199022) (120573119887 (119903 minus 119902
1) + 120575 minus 119902
2minus 120573119886)
1205731198872119903 (120575 minus 1199022)
)
(25)
Abstract and Applied Analysis 5
which indicates that if 119903 gt (120573119886 minus 120575 + 1199022)120573119887 + 119902
1and 119902
2lt
120575 lt 120573119886 + 1199022 then there is a positive interior equilibrium for
the system 1198782
421 Regular Equilibria For the subsystem 1198781with 119909 lt ET
(0 0) is a regular equilibrium while (119896 0) is a virtual equilib-rium In addition according to the coordinate of equilibrium119864lowast
1198781
we have the following results If 119903 gt (120573119886 minus 120575)120573119887 gt 0 and
119896 (120573119887119903 + 120575 minus 120573119886)
120573119887119903lt ET (26)
then it is a regular equilibrium for the system 1198781 denoted by
1198641
119877 If 119903 gt (120573119886 minus 120575)120573119887 gt 0 and
119896 (120573119887119903 + 120575 minus 120573119886)
120573119887119903gt ET (27)
then the equilibrium 119864lowast
1198781
becomes a virtual equilibriumdenoted by 1198641
119881
About regular equilibria for the subsystem 1198782with 119909 gt
ET (0 0) is a virtual equilibrium while (119896(119903 minus 1199011)119903 0) may
be a regular or virtual equilibrium which depends on theparameter space Moreover according to the coordinate ofequilibrium 119864
lowast
1198782
we have the following conclusions If 119903 gt
(120573119886 minus 120575 + 1199022)120573119887 + 119902
1 1199022lt 120575 lt 120573119886 + 119902
2and
119896 (120573119887 (119903 minus 1199021) + 120575 minus 119902
2minus 120573119886)
120573119887119903gt ET (28)
then it is a regular equilibrium for the system 1198782 denoted by
1198642
119877 Note that if 119903 gt (120573119886 minus 120575 + 119902
2)120573119887 + 119902
1 1199022lt 120575 lt 120573119886 + 119902
2
and119896 (120573119887 (119903 minus 119902
1) + 120575 minus 119902
2minus 120573119886)
120573119887119903lt ET (29)
then the equilibrium 119864lowast
1198782
becomes a virtual equilibriumdenoted by 1198642
119881 Based on the above analysis if
119896 (120573119886 minus 120575)
120573119887 (119896 minus ET)lt 119903 lt
119896 (120573119886 minus 120575 + 1199022+ 120573119887119902
1)
120573119887 (119896 minus ET) (30)
then the two virtual equilibria 1198641119881and 1198642
119881can coexist
422 Pseudoequilibrium For the existence of pseudoequilib-rium119864
119875= (ET 119910
119875) 119910119875component of the pseudoequilibrium
of sliding flow satisfies the following equation
1198751(119910) (119875
2119910 + 1198753) = 0 (31)
where 119910 isin Σ1
119878or Σ2
119878 Solving the above equation with
respect to 119910 yields two possible roots denoted by 1199101198751
= 0 and
1199101198752
=(1205751199021minus 120573119886119902
1minus 1199031199022(1 minus ET119896))ET
minus1205751198871199021+ 1199031198871199022(1 minus ET119896) minus 119886119902
Solving the above equations with respect to 119909 and 119910 yieldstwo tangent points including
1198641
119879= (ET 119903ET (1 minus ET119896)
119886 minus 119887119903 (1 minus ET119896))
1198642
119879= (
ET (119903 (1 minus ET119896) minus 1199021)
119886 + 1199021119887 minus 119887119903 (1 minus ET119896)
)
(40)
If we fix all parameters then the relations among null-isoclines regularvirtual equilibria pseudoequilibrium andsliding segment are provided in Figure 1 Note that virtualequilibria of both systems 119878
1and 119878
2imply the existence of
pseudoequilibrium andwewill prove this general result later
43 The Stability of Pseudoequilibrium In the process ofpest control we should apply all kinds of control strategiesso as to prevent multiple pest outbreaks or make sure thatthe total density of the pest stabilizes at a desired level ofET If the unique positive equilibrium of system 119878
1and
system 1198782is virtual simultaneously then the sliding flow
has a unique pseudoequilibrium In order to realize thisgoal we can choose a set of parameters such that all theequilibria of subsystems 119878
1and 1198782are virtual equilibria and
the pseudoequilibria are globally stable which have beenwidely used in pest control [26ndash28] For example if wefixed all parameter values as those in Figure 2 then bothvirtual equilibria coexist Therefore we address the stabilityof pseudoequilibrium119864
1198752
in the followingwhich is importantto control pest
Theorem 6 Either the inequalities (34) or (35) hold true orthe two virtual equilibria 1198641
119881and 1198642
119881can coexist then Filippov
system (3) contains a positive pseudoequilibrium 1198641198752
Regard-less of which cases would occur the pseudoequilibrium 119864
1198752
islocally stable with respect to sliding mode domain
Proof According to the conditions of Theorem 6 we seethat the inequalities (30) hold true which implies that theinequalities (34) are true Based on the discussions aboutthe existence of pseudoequilibrium in Section 42 we haveconcluded that if inequalities (34) or (35) hold true thenthe system (3) contains a positive pseudoequilibrium 119864
=(1198961199021(120575 minus 120573119886) minus 119903119902
2(119896 minus ET)) (120575119887119896119902
1minus 119903119887119902
2(119896 minus ET) + 119886119896119902
2)
11988611990211198962 (119887120573119902
1+ 1199022)
(41)
Therefore if the inequalities (33) hold true then 119889120601(119910)119889119910 lt 0 and 119910
1198752
is a positive root of (31) Note that theinequalities (34) and (35) indicate the inequalities (33) Thusthe positive pseudoequilibrium 119864
1198752
is locally stale if it exists
Abstract and Applied Analysis 7
It is difficult to directly prove the global stability ofpseudoequilibrium 119864
1198752
in this case Because we cannotemploy the classical Bendixson-Dulac theorem due to thediscontinuity of vector fields However if there is not crossingcycle surrounding sliding segment then pseudoequilibriumis globally stable [29] From the analysis of global bifurcationfor the system (3) in the following section the system just hastouching bifurcation and there is nonexistence of a slidingcycle which surrounds the 119864
1198752
By using the similar methodswe have that pseudoequilibrium 119864
1198752
is globally stable Thatis to say the local stability of pseudoequilibrium 119864
1198752
withrespect to sliding mode domain indicates its global stabilityin the first quadrant (shown in Figures 3(a) 3(b) 4(b) and4(c)) In practice in order to control pest outbreak we shouldchoose the desirable ET at first so that all equilibria of eachsystem such as system 119878
1and system 119878
2become virtual then
pseudoequilibrium not only exists but also is globally stableIn other words the density of pest can be stable at the ETWhen the density of pest reaches or exceeds the ET we shouldcarry out control strategies (eg releasing natural enemyetc) until it falls below the ET In this way our control goalcan be realized fully
5 Equilibria and Sliding Bifurcation Set
51 RegularVirtual Equilibrium Bifurcation According tothe above discussions it is obvious that 119903 and ET are primaryfactors in determining the existence of the above differenttypes of equilibria of the system (3) So we define four curvesabout parameters 119903 and ET as follows
1198711= (119903ET) | 119903 =
120573119886 minus 120575
120573119887
1198712= (119903ET) | 119903 =
120573119886 minus 120575 + 1199022
120573119887+ 1199021
1198713= (119903ET) | ET =
119896 (120573119887119903 + 120575 minus 120573119886)
120573119887119903
1198714= (119903ET) | ET =
119896 (120573119887 (119903 minus 1199021) + 120575 minus 119902
2minus 120573119886)
120573119887119903
(42)
The four curves (ie 1198711 1198712 1198713 and 119871
4) divide the 119903
and ET parameter space into six regions and the existenceor coexistence of regular or virtual equilibria is indicated ineach region The boundary equilibria 1198641
119861and 1198642
119861can appear
on the lines 1198713and 119871
4accordingly In particular it follows
from Figure 2 that the two virtual equilibria 1198641119881and 1198642
119881can
coexist which is very important to pest control However thetwo regular equilibria 1198641
119877and 1198642
119877cannot coexist
52 Boundary Node Bifurcations This type of bifurcationmay occur for Filippov system (3) once equilibria 119864
119875 119864119879 and
119864119877or 119864119879and 119864
119877collide together simultaneously when ET
passes through a critical value In this part we choose ETas bifurcation parameter and all other parameters are fixedas those in Figure 3 Note that once the parameter ET passes
through the first critical value ET1198881
= 93333 the regular equi-librium 119864
1
119877 tangent point 119864
119879 and pseudoequilibrium 119864
119875col-
lide together (see Figure 3(b)) where ET1198881
is determined by
ET1198881
=119896 (120573119887119903 + 120575 minus 120573119886)
120573119887119903 (43)
A virtual equilibrium 1198641
119881 an invisible tangent point 1198641
119879
and a pseudoequilibrium 119864119875coexist as shown in Figure 3(a)
when ET lt ET1198881
They collide at ET = ET1198881
and are substitut-ed by a visible tangent point1198641
119879 as shown in Figure 3(c) when
ET gt ET1198881
Similarly another boundary node bifurcation of Filippov
system (3) occurs at ET1198882
= 46667 (see Figure 4(b)) whereET1198882
is determined by
ET1198882
=119896 (120573119887 (119903 minus 119902
1) + 120575 minus 119902
2minus 120573119886)
120573119887119903 (44)
A stable regular equilibrium 1198642
119877and a visible tangent
point 1198642119879coexist as shown in Figure 4(a) when ET lt ET
1198882
They collide with a pseudoequilibrium 119864
119875at ET = ET
1198882
andare substituted by an invisible tangent point 1198642
119879as shown in
Figure 4(c) when ET gt ET1198882
53 Global Sliding Bifurcation Global sliding bifurcationsinvolve nonvanishing cycles which include sliding discon-nection touching (or grazing) bifurcation buckling bifurca-tion crossing bifurcation bifurcation of a sliding homoclinicorbit and heteroclinic orbit [22] Touching (or grazing) bifur-cation implies that a positive period solution can collide withthe sliding segments From the work of Kuang and BerettaandHsu et al [2 7] they concluded that when the value of theparameter 119886 passed slightly through the bifurcation value astable limit cycle bifurcates from the unstable positive interiorequilibrium for the system (1) According to numericalsimulations the system (3) just has touching bifurcation Forexample if we choose ET as bifurcation parameter and fixall other parameters as shown in Figure 5 when the valueof parameter ET varies touching bifurcation occurs at thecritical value ET
1198883
= 259 for the system (3) Note thatextensive numerical simulations indicate that nomatter whatthe value of ET is in touching bifurcation the whole periodicsolution lies in the region 119878
1(shown in Figure 5)This implies
that the density of pest can be successfully maintained belowET by designing suitable threshold policy strategies So ourcontrol objective can be fully realized which is a desiredsituation in crop livestock sectors and forestry
6 Discussion
Recently the threshold policy and IPM strategies haveattracted great attention in agriculture forestry animal hus-bandry and so on [10ndash12] In the process of pest controlIPM strategies would be used only when the density ofpest reaches or exceeds the ET In addition Filippov systemprovides a natural and rational framework for those realworld problems so it has been widely used in different fields
8 Abstract and Applied Analysis
0 5 10 150
1
2
3
4
5
6
7
8
9
10
11
x
y
E2V
E1V
EP
(a) ET = 83
0 5 10 150
1
2
3
4
5
6
7
8
9
10
11
x
y
E2V
E1B(EP)
(b) ET = 93333
0 5 10 150
1
2
3
4
5
6
7
8
9
10
11E2V
E1R
x
y
(c) ET = 98
Figure 3 Boundary node bifurcation for Filippov system (3) The parameter values are fixed as follows 119886 = 07 119887 = 06 119903 = 15 119896 = 12120573 = 05 120575 = 025 119902
1= 025 and 119902
2= 01
0 5 10
5
10
15
20
25
30
35
40
45
E1V
E2R
x
y
(a) ET = 4
0
5
10
15
20
25
30
35
0 5 10x
yE2B(EP)
E1V
(b) ET = 46667
0
5
10
15
20
25
0 5 10x
yE2V
E1V
EP
(c) ET = 58
Figure 4 Boundary node bifurcation for Filippov system (3) The parameter values are fixed as follows 119886 = 07 119887 = 06 119903 = 15 119896 = 12120573 = 05 120575 = 025 119902
1= 025 and 119902
2= 01
such as in science and engineering [19ndash21 23] In this paperwe employ the sliding analysis of Filippov system to describeand investigate the long term dynamical behavior of theratio-dependent-type prey-predator model with Michaelis-Menten-type functional response so that we can use theFilippov system to model intervention of pest control policyFirstly we investigate the sliding mode domain sliding modedynamics Secondly the null-isoclines and the existence of
four types of equilibria for Filippov system including regularvirtual boundary and pseudoequilibrium and the tangentpoints are discussed in detail Moreover the stability ofpseudoequilibrium is also studied Thirdly we have investi-gated the local sliding bifurcations including regularvirtualequilibrium bifurcation and boundary node bifurcationsFurther global touching bifurcation is also studied by numer-ical techniques
Abstract and Applied Analysis 9
0 1 2 30
5
10
15
x
y
E1R
(a) ET = 15
0 2 40
5
10
15
x
y
E1R
(b) ET = 259
0 2 4 60
5
10
15
x
y
E1R
(c) ET = 35
Figure 5 Globally touching bifurcation for Filippov system (3) The parameter values are fixed as follows 119886 = 051 119887 = 04 119903 = 12 119896 = 6120573 = 06863 120575 = 005 119902
1= 04 and 119902
2= 01
In the process of pest control we should apply all kindsof control strategies so as to prevent multiple pest outbreaksor make sure that the density of pest stabilizes at a desiredlevel of ET In order to realize these goals on the one handwe can determine a set of parameters such that not only all theequilibria of subsystems 119878
1and 1198782are virtual equilibria which
ensures that the pseudoequilibria exist but also the pseudoe-quilibrium is globally stable From Theorem 6 we providedthe conditions of the existence of pseudoequilibrium 119864
1198752
forthe system (3) and showed that it is locally stable if it existsFurther we showed that the local stability of pseudoequilib-rium 119864
1198752
implies global stability by numerical simulation Onthe other hand globally touching bifurcation indicates thatthe density of pest can be successfully maintained below theET by designing suitable threshold policy strategies (shownin Figure 5) Therefore our control objective can be achievedfully in the above two cases which can be used to pest controlin crop livestock sectors and forestry
Although impulsive prey-predator models with the ratio-dependent- or Michaelis-Menten-type response functionhave been studied in [13ndash16] the Filippov systems have manyadvantages in describing interventions including sprayingpesticides and releasing natural enemies compared withimpulsive models In this work the number of natural enemyto be released is proportional to its number It is interestingthat releasing number of natural enemy can be described byconstant independently of the existing numbers of pest andnatural enemy Moreover in practice considering environ-mental energy resource finiteness we should choose the totalnumber of both populations as a guide to switch the systemwhich is called the weighted escapement policy (WEP) [1930ndash32] Therefore in the future work we will focus on theabove two cases which could result in richer dynamics
Appendix
Methods for Analysis of the Sliding Solution
Filippov Convex Method The Filippov method associatesthe following convex combination 119865
119878(119885) of the two vectors
1198651198781
(119885) and 1198651198782
(119885) with each nonsingular sliding point 119885 isin
Σ2 that is
119865119878(119885) = (1 minus 120582 (119885)) 119865
1198781(119885) + 120582 (119885) 119865
1198782(119885) (A1)
where
120582 (119885) =⟨119867119885 1198651198781
⟩
⟨119867119885 1198651198781
minus F1198782
⟩ (A2)
119865119878(119885) is tangent to Σ
2and 0 le 120582(119885) le 1
Thus the sliding mode dynamics can be determined by
= 119865119878(119885) 119885 isin Σ
2 (A3)
which is smooth on a one-dimensional sliding interval of Σ2
The solution of the above equation is the sliding solutionThe equation 120582(119885) = 0 (or 120582(119885) = 1) indicates that the
flow is determined by 1198651198781
(or 1198651198782
) aloneTherefore the slidingmode domain can be defined as
Σ119878= 119885 = (119909 119910) isin Σ | 0 le 120582 (119885) le 1 (A4)
which is equivalent to Σ119878= 119885 isin Σ | 120590(119885) le 0 Denote
Σ+
119878= 119885 isin Σ | 120582 (119885) = 1 Σ
minus
119878= 119885 isin Σ | 120582 (119885) = 0
(A5)
10 Abstract and Applied Analysis
by the boundary of the sliding mode domain so the vectorfields are tangent to boundary Σ+
119878or Σminus119878
Utkin Equivalent Control Method It follows from Utkinrsquosworks [18] on sliding mode dynamics along the manifoldΣ we note that Filippov system (3) can be also rewritten asfollows
= 119865 (119885119880119867) (A6)
where the control 119880119867is defined as
119880119867=
0 119867 (119885) lt 0
1198801(119885 119905) 119867 (119885) gt 0
(A7)
where 1198801is a continuous function In the controlled system
(ie system 1198782) the control 119880
119867= 1198801is applied and in the
free system (ie system 1198781) 119880119867= 0 is applied Assume that
a sliding mode exists on manifold Σ that is Σ2is nonempty
Solving the following algebraic equation
=120597119867
120597119885119865 (119885U
119867) = 0 (A8)
with respect to 119880119867
on this manifold gives the solutiondenoted by 119880
lowast(119885 119905) which is referred to as equivalent
control Substituting for 119880119867in system (A6) yields
which determines the sliding mode dynamics of the Filippovsystem (3)
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China (NSFC 11171199) and by the FundamentalResearch Funds for the Central Universities (GK201003001)
References
[1] R M May ldquoLimit cycles in predator-prey communitiesrdquo Sci-ence vol 177 no 4052 pp 900ndash902 1972
[2] Y Kuang and E Beretta ldquoGlobal qualitative analysis of a ratio-dependent predator-prey systemrdquo Journal ofMathematical Biol-ogy vol 36 no 4 pp 389ndash406 1998
[3] W Murdoch C Briggs and R Nisbet Consumer-Resource Dy-namics Princeton University Press New York NY USA 2003
[4] H I Freedman Deterministic Mathematical Models in Popula-tion Ecology vol 57 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1980
[5] G T Skalski and J F Gilliam ldquoFunctional responseswith preda-tor interference viable alternatives to theHolling type IImodelrdquoEcology vol 82 no 11 pp 3083ndash3092 2001
[6] R Arditi and L R Ginzburg ldquoCoupling in predator-preydynamics ratio-dependencerdquo Journal ofTheoretical Biology vol139 no 3 pp 311ndash326 1989
[7] S-B Hsu T-W Hwang and Y Kuang ldquoGlobal analysis ofthe Michaelis-Menten-type ratio-dependent predator-prey sys-temrdquo Journal of Mathematical Biology vol 42 no 6 pp 489ndash506 2001
[8] C Jost O Arino and R Arditi ldquoAbout deterministic extinctionin ratio-dependent predator-prey modelsrdquo Bulletin of Mathe-matical Biology vol 61 no 1 pp 19ndash32 1999
[9] D Xiao and S Ruan ldquoGlobal dynamics of a ratio-dependentpredator-prey systemrdquo Journal of Mathematical Biology vol 43no 3 pp 268ndash290 2001
[10] Z Lu X Chi and L Chen ldquoImpulsive control strategies inbiological control of pesticiderdquo Theoretical Population Biologyvol 64 no 1 pp 39ndash47 2003
[11] S Tang and R A Cheke ldquoModels for integrated pest controland their biological implicationsrdquo Mathematical Biosciencesvol 215 no 1 pp 115ndash125 2008
[12] J C Van Lenteren and J Woets ldquoBiological and integrated pestcontrol in greenhousesrdquo Annual Review of Entomology vol 33pp 239ndash250 1988
[13] B Dai H Su and D Hu ldquoPeriodic solution of a delayed ratio-dependent predator-prey model with monotonic functionalresponse and impulserdquo Nonlinear Analysis Theory Methods ampApplications vol 70 no 1 pp 126ndash134 2009
[14] X Liu G Li and G Luo ldquoPositive periodic solution for a two-species ratio-dependent predator-prey system with time delayand impulserdquo Journal of Mathematical Analysis and Applica-tions vol 325 no 1 pp 715ndash723 2007
[15] G Jiang Q Lu and L Qian ldquoComplex dynamics of a Hollingtype II prey-predator system with state feedback controlrdquoChaos Solitons amp Fractals vol 31 no 2 pp 448ndash461 2007
[16] H Baek and Y Lim ldquoDynamics of an impulsively controlledMichaelis-Menten type predator-prey systemrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 16 no4 pp 2041ndash2053 2011
[17] T Zhao Y Xiao and R J Smith ldquoNon-smooth plant diseasemodels with economic thresholdsrdquo Mathematical Biosciencesvol 241 no 1 pp 34ndash48 2013
[18] V I Utkin Sliding Modes in Control and Optimization Com-munications and Control Engineering Series Springer BerlinGermany 1992
[19] M I S Costa E Kaszkurewicz A Bhaya and L Hsu ldquoAchiev-ing global convergence to an equilibrium population in pred-ator-prey systems by the use of a discontinuous harvestingpolicyrdquo Ecological Modelling vol 128 no 2-3 pp 89ndash99 2000
[20] B L Van De Vrande D H Van Campen and A De KrakerldquoApproximate analysis of dry-friction-induced stick-slip vibra-tions by a smoothing procedurerdquo Nonlinear Dynamics vol 19no 2 pp 157ndash169 1999
[21] S H Doole and S J Hogan ldquoA piecewise linear suspensionbridge model nonlinear dynamics and orbit continuationrdquoDynamics and Stability of Systems vol 11 no 1 pp 19ndash47 1996
[22] Yu A Kuznetsov S Rinaldi and A Gragnani ldquoOne-parameterbifurcations in planar Filippov systemsrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol13 no 8 pp 2157ndash2188 2003
[23] M di Bernardo C J Budd A R Champneys et al ldquoBifurca-tions in nonsmooth dynamical systemsrdquo SIAM Review vol 50no 4 pp 629ndash701 2008
[24] A F Filippov Differential Equations with Discontinuous Right-hand Sides vol 18 of Mathematics and Its Applications (SovietSeries) Kluwer Academic Dordrecht The Netherlands 1988
[25] S Tang J Liang Y Xiao and R A Cheke ldquoSliding bifurcationsof Filippov two stage pest control models with economicthresholdsrdquo SIAM Journal on Applied Mathematics vol 72 no4 pp 1061ndash1080 2012
Abstract and Applied Analysis 11
[26] M I da Silveira Costa and M E M Meza ldquoApplication of athreshold policy in the management of multispecies fisheriesand predator cullingrdquoMathematical Medicine and Biology vol23 no 1 pp 63ndash75 2006
[27] M E Mendoza Meza A Bhaya E Kaszkurewicz and M IDa Silveira Costa ldquoThreshold policies control for predator-preysystems using a control Liapunov function approachrdquoTheoreti-cal Population Biology vol 67 no 4 pp 273ndash284 2005
[28] F Dercole A Gragnani and S Rinaldi ldquoBifurcation analysisof piecewise smooth ecological modelsrdquoTheoretical PopulationBiology vol 72 no 2 pp 197ndash213 2007
[29] W Wang ldquoBackward bifurcation of an epidemic model withtreatmentrdquo Mathematical Biosciences vol 201 no 1-2 pp 58ndash71 2006
[30] I Noy-Meir ldquoStability of grazing systems an application ofpredator-prey graphsrdquo The Journal of Animal Ecology vol 63no 2 pp 459ndash481 1975
[31] R M Colombo and V Krivan ldquoSelective strategies in foodwebsrdquo Mathematical Medicine and Biology vol 10 no 4 pp281ndash291 1993
[32] V Krivan ldquoOptimal foraging and predator-prey dynamicsrdquoTheoretical Population Biology vol 49 no 3 pp 265ndash290 1996
We first introduce some useful properties and definitionson Filippov system according to [24 25] so that we caninvestigate the model (3) in more detail Let119867(119885) = 119909 minus ETwith vector 119885 = (119909 119910)
Definition 4 A point 119885lowast is called a boundary equilibrium ofsystem (3) if 119865
1198781
(119885lowast) = 0 with 119867(119885lowast) = 0 or 119865
1198782
(119885lowast) = 0
with119867(119885lowast) = 0
Definition 5 A point 119885lowast is called a tangent point of sys-tem (3) if 119885lowast isin Σ
119878and ⟨119867
119885(119885lowast) 1198651198781
(119885lowast)⟩ = 0 or
⟨119867119885(119885lowast) 1198651198782
(119885lowast)⟩ = 0
The details and knowledge about the Filippov systemsuch as the concepts of Filippov solution sliding mode solu-tion and bifurcation can be found in reference [24]
3 Sliding Region and Sliding Mode Dynamics
A sliding mode exists if there are regions in the discontinuityboundary Σ where the vectors for the two subsystems of thesystem (3) are directed towards each other It is well knownthat two basic methods the so-called Filippov convexmethod[24] andUtkin equivalent control method [18] are developedfor the sliding mode and its domains which are shown in theappendix
31 Sliding Segment and Region Based on the appendix wehave
120582 (119885) =119903 (1 minus ET119896) minus 119886119910 (119887119910 + ET)
1199021
(11)
because the sliding mode regions can be determined by solv-ing the inequalities that is 120582(119885) ge 0 and 120582(119885) le 1 In orderto solve the above two inequalities with respect to 119910 we needto consider the following two algebraic equations
119903 (1 minusET119896) minus
119886119910
119887119910 + ET= 0
119903 (1 minusET119896) minus
119886119910
119887119910 + ETminus 1199021= 0
(12)
Solving the above two algebraic equations with respect to119910 yields two roots denoted by
1199101=
119903ET (119896 minus ET)119886119896 minus 119887119903 (119896 minus ET)
1199102=
(119903 (119896 minus ET) minus 1199021119896)ET
119886119896 minus 119887 (119903 (119896 minus ET) minus 1199021119896)
(13)
4 Abstract and Applied Analysis
Based on the relations between 1199101and 119910
2 there exist two
cases for the existence of sliding segments of Filippov system(3) By simply calculating and arranging we have the follow-ing results
(i) When 119903 lt 119886119896119887(119896 minus ET) the sliding segment can bedescribed as
Σ1
119878= (119909 119910) | max 0 119910
2 le 119910 le 119910
1 119909 = ET (14)
(ii) When max1199021119896(119896 minus ET) 119886119896119887(119896 minus ET) lt 119903 lt
min119896(119886 + 1198871199021)119887(119896 minus ET) 119886(119896 + 119887119902
1)119887(119896 minus ET) the
sliding segment can be described as
Σ2
119878= (119909 119910) | 0 le 119910 le 119910
2 119909 = ET (15)
32 Sliding Mode Dynamics Filippov system (3) only hasone piece of sliding segment and the solutions defined init can be obtained from the sliding mode dynamics whichcan be determined by employing theUtkin equivalent controlmethod (see the appendix)
From119867 = 0 we get that
119867119885= 1199091015840= 119903119909 (1 minus
119909
119896) minus
119886119909119910
119887119910 + 119909minus 1205761199021119909 = 0 119909 = ET
(16)
And solving the above equations with respect to 120576 yields
120576 =119903 (1 minus ET119896) minus 119886119910 (119887119910 + ET)
1199021
(17)
Hence the dynamics on the slidingmodeΣ119878can be deter-
mined by the following scalar differential equation
119910 (119905) = minus 120575119910 +120573119886119910ET119887119910 + ET
+119903 (1 minus ET119896) minus 119886119910 (119887119910 + ET)
1199021
1199022119910
= 1198751(119910) (119875
2119910 + 1198753) ≜ 120601 (119910)
(18)
where 119910 isin Σ1
119878or Σ2
1198781198751(119910) = 119910119902
1(119887119910 + ET) 119875
2= minus120575119887119902
1+
1199031198871199022(1 minus ET119896) minus 119886119902
2 and 119875
3= minus120575119902
1ET + 120573119886119902
1ET + 119903119902
2(1 minus
ET119896)ET
4 Null-Isoclines and Equilibria
41The Null-Isoclines of Filippov System (3) Null-isoclines ofboth systems 119878
1and 1198782are related to the existence of equilib-
ria and are useful for analysis of sliding dynamicNull-isoclines (119905) = 0 and 119910(119905) = 0 for both systems
1198781and 119878
2can be determined as follows For the system 119878
1
solving the equation of the null-isocline (119905) = 0 yeilds
1198911198781(119909) =
119903119909 (1 minus 119909119896)
119886 minus 119887119903 (1 minus 119909119896) (19)
and null-isocline 119910(119905) = 0 gives
1198921198781(119909) =
(120573119886 minus 120575) 119909
119887120575 (20)
For the system 1198782 solving the equation of the null-isocline
(119905) = 0 yields
1198911198782(119909) =
(119903119909 (1 minus 119909119896) minus 1199021) 119909
119886 + 1198871199021minus 119887119903 (1 minus 119909119896)
(21)
and null-isocline 119910(119905) = 0 gives
1198921198782(119909) =
(120573119886 minus (120575 minus 1199022)) 119909
119887 (120575 minus 1199022)
(22)
42 The Equilibria of Filippov System (3) Because the solu-tions of Filippov system (3) are composed of connectingstandard solutions in subsystems 119878
1 1198782and sliding mode
solutions on Σ1119878or Σ2119878The definitions of all types of equilibria
of Filippov system have been provided in Section 2 which areimportant to bifurcation analysis There may be several typesof equilibria for Filippov system (3) which include regularequilibrium (denoted by119864
119877) virtual equilibrium (denoted by
119864119881) pseudoequilibrium (denoted by 119864
119875) boundary equilib-
rium (denoted by119864119861) and one type of special point named as
tangent point (denoted by 119864119879) Detailed definitions of these
equilibria and the tangent point can be found in the literature[24]
From Lemma 1 we know that the subsystem 1198781with 119909 lt
ET has three possible equilibria (0 0) (119896 0) and interiorequilibrium
119864lowast
1198781
= (119909lowast
1198781
119910lowast
1198781
)
= (119896 (120573119887119903 + 120575 minus 120573119886)
120573119887119903119896 (120573119886 minus 120575) (120573119887119903 + 120575 minus 120573119886)
1205731198872119903120575)
(23)
If 120573119886 minus 120575 gt 0 and 119903 gt (120573119886 minus 120575)120573119887 then there is a positiveinterior equilibrium for the system 119878
1
Equilibria for the subsystem 1198782with 119909 gt ET satisfy
119903119909 (1 minus119909
119896) minus
119886119909119910
119887119910 + 119909minus 1199021119909 = 0
minus120575119910 +120573119886119909119910
119887119910 + 119909+ 1199022119910 = 0
(24)
Solving the above equations with respect to 119909 and 119910 yieldsthree possible equilibria that is (0 0) (119896(119903 minus 119902
1)119903 0) and
interior equilibrium
119864lowast
1198782
= (119909lowast
1198782
119910lowast
1198782
)
= (119896 (120573119887 (119903 minus 119902
1) + 120575 minus 119902
2minus 120573119886)
120573119887119903
119896 (120573119886 minus 120575 + 1199022) (120573119887 (119903 minus 119902
1) + 120575 minus 119902
2minus 120573119886)
1205731198872119903 (120575 minus 1199022)
)
(25)
Abstract and Applied Analysis 5
which indicates that if 119903 gt (120573119886 minus 120575 + 1199022)120573119887 + 119902
1and 119902
2lt
120575 lt 120573119886 + 1199022 then there is a positive interior equilibrium for
the system 1198782
421 Regular Equilibria For the subsystem 1198781with 119909 lt ET
(0 0) is a regular equilibrium while (119896 0) is a virtual equilib-rium In addition according to the coordinate of equilibrium119864lowast
1198781
we have the following results If 119903 gt (120573119886 minus 120575)120573119887 gt 0 and
119896 (120573119887119903 + 120575 minus 120573119886)
120573119887119903lt ET (26)
then it is a regular equilibrium for the system 1198781 denoted by
1198641
119877 If 119903 gt (120573119886 minus 120575)120573119887 gt 0 and
119896 (120573119887119903 + 120575 minus 120573119886)
120573119887119903gt ET (27)
then the equilibrium 119864lowast
1198781
becomes a virtual equilibriumdenoted by 1198641
119881
About regular equilibria for the subsystem 1198782with 119909 gt
ET (0 0) is a virtual equilibrium while (119896(119903 minus 1199011)119903 0) may
be a regular or virtual equilibrium which depends on theparameter space Moreover according to the coordinate ofequilibrium 119864
lowast
1198782
we have the following conclusions If 119903 gt
(120573119886 minus 120575 + 1199022)120573119887 + 119902
1 1199022lt 120575 lt 120573119886 + 119902
2and
119896 (120573119887 (119903 minus 1199021) + 120575 minus 119902
2minus 120573119886)
120573119887119903gt ET (28)
then it is a regular equilibrium for the system 1198782 denoted by
1198642
119877 Note that if 119903 gt (120573119886 minus 120575 + 119902
2)120573119887 + 119902
1 1199022lt 120575 lt 120573119886 + 119902
2
and119896 (120573119887 (119903 minus 119902
1) + 120575 minus 119902
2minus 120573119886)
120573119887119903lt ET (29)
then the equilibrium 119864lowast
1198782
becomes a virtual equilibriumdenoted by 1198642
119881 Based on the above analysis if
119896 (120573119886 minus 120575)
120573119887 (119896 minus ET)lt 119903 lt
119896 (120573119886 minus 120575 + 1199022+ 120573119887119902
1)
120573119887 (119896 minus ET) (30)
then the two virtual equilibria 1198641119881and 1198642
119881can coexist
422 Pseudoequilibrium For the existence of pseudoequilib-rium119864
119875= (ET 119910
119875) 119910119875component of the pseudoequilibrium
of sliding flow satisfies the following equation
1198751(119910) (119875
2119910 + 1198753) = 0 (31)
where 119910 isin Σ1
119878or Σ2
119878 Solving the above equation with
respect to 119910 yields two possible roots denoted by 1199101198751
= 0 and
1199101198752
=(1205751199021minus 120573119886119902
1minus 1199031199022(1 minus ET119896))ET
minus1205751198871199021+ 1199031198871199022(1 minus ET119896) minus 119886119902
Solving the above equations with respect to 119909 and 119910 yieldstwo tangent points including
1198641
119879= (ET 119903ET (1 minus ET119896)
119886 minus 119887119903 (1 minus ET119896))
1198642
119879= (
ET (119903 (1 minus ET119896) minus 1199021)
119886 + 1199021119887 minus 119887119903 (1 minus ET119896)
)
(40)
If we fix all parameters then the relations among null-isoclines regularvirtual equilibria pseudoequilibrium andsliding segment are provided in Figure 1 Note that virtualequilibria of both systems 119878
1and 119878
2imply the existence of
pseudoequilibrium andwewill prove this general result later
43 The Stability of Pseudoequilibrium In the process ofpest control we should apply all kinds of control strategiesso as to prevent multiple pest outbreaks or make sure thatthe total density of the pest stabilizes at a desired level ofET If the unique positive equilibrium of system 119878
1and
system 1198782is virtual simultaneously then the sliding flow
has a unique pseudoequilibrium In order to realize thisgoal we can choose a set of parameters such that all theequilibria of subsystems 119878
1and 1198782are virtual equilibria and
the pseudoequilibria are globally stable which have beenwidely used in pest control [26ndash28] For example if wefixed all parameter values as those in Figure 2 then bothvirtual equilibria coexist Therefore we address the stabilityof pseudoequilibrium119864
1198752
in the followingwhich is importantto control pest
Theorem 6 Either the inequalities (34) or (35) hold true orthe two virtual equilibria 1198641
119881and 1198642
119881can coexist then Filippov
system (3) contains a positive pseudoequilibrium 1198641198752
Regard-less of which cases would occur the pseudoequilibrium 119864
1198752
islocally stable with respect to sliding mode domain
Proof According to the conditions of Theorem 6 we seethat the inequalities (30) hold true which implies that theinequalities (34) are true Based on the discussions aboutthe existence of pseudoequilibrium in Section 42 we haveconcluded that if inequalities (34) or (35) hold true thenthe system (3) contains a positive pseudoequilibrium 119864
=(1198961199021(120575 minus 120573119886) minus 119903119902
2(119896 minus ET)) (120575119887119896119902
1minus 119903119887119902
2(119896 minus ET) + 119886119896119902
2)
11988611990211198962 (119887120573119902
1+ 1199022)
(41)
Therefore if the inequalities (33) hold true then 119889120601(119910)119889119910 lt 0 and 119910
1198752
is a positive root of (31) Note that theinequalities (34) and (35) indicate the inequalities (33) Thusthe positive pseudoequilibrium 119864
1198752
is locally stale if it exists
Abstract and Applied Analysis 7
It is difficult to directly prove the global stability ofpseudoequilibrium 119864
1198752
in this case Because we cannotemploy the classical Bendixson-Dulac theorem due to thediscontinuity of vector fields However if there is not crossingcycle surrounding sliding segment then pseudoequilibriumis globally stable [29] From the analysis of global bifurcationfor the system (3) in the following section the system just hastouching bifurcation and there is nonexistence of a slidingcycle which surrounds the 119864
1198752
By using the similar methodswe have that pseudoequilibrium 119864
1198752
is globally stable Thatis to say the local stability of pseudoequilibrium 119864
1198752
withrespect to sliding mode domain indicates its global stabilityin the first quadrant (shown in Figures 3(a) 3(b) 4(b) and4(c)) In practice in order to control pest outbreak we shouldchoose the desirable ET at first so that all equilibria of eachsystem such as system 119878
1and system 119878
2become virtual then
pseudoequilibrium not only exists but also is globally stableIn other words the density of pest can be stable at the ETWhen the density of pest reaches or exceeds the ET we shouldcarry out control strategies (eg releasing natural enemyetc) until it falls below the ET In this way our control goalcan be realized fully
5 Equilibria and Sliding Bifurcation Set
51 RegularVirtual Equilibrium Bifurcation According tothe above discussions it is obvious that 119903 and ET are primaryfactors in determining the existence of the above differenttypes of equilibria of the system (3) So we define four curvesabout parameters 119903 and ET as follows
1198711= (119903ET) | 119903 =
120573119886 minus 120575
120573119887
1198712= (119903ET) | 119903 =
120573119886 minus 120575 + 1199022
120573119887+ 1199021
1198713= (119903ET) | ET =
119896 (120573119887119903 + 120575 minus 120573119886)
120573119887119903
1198714= (119903ET) | ET =
119896 (120573119887 (119903 minus 1199021) + 120575 minus 119902
2minus 120573119886)
120573119887119903
(42)
The four curves (ie 1198711 1198712 1198713 and 119871
4) divide the 119903
and ET parameter space into six regions and the existenceor coexistence of regular or virtual equilibria is indicated ineach region The boundary equilibria 1198641
119861and 1198642
119861can appear
on the lines 1198713and 119871
4accordingly In particular it follows
from Figure 2 that the two virtual equilibria 1198641119881and 1198642
119881can
coexist which is very important to pest control However thetwo regular equilibria 1198641
119877and 1198642
119877cannot coexist
52 Boundary Node Bifurcations This type of bifurcationmay occur for Filippov system (3) once equilibria 119864
119875 119864119879 and
119864119877or 119864119879and 119864
119877collide together simultaneously when ET
passes through a critical value In this part we choose ETas bifurcation parameter and all other parameters are fixedas those in Figure 3 Note that once the parameter ET passes
through the first critical value ET1198881
= 93333 the regular equi-librium 119864
1
119877 tangent point 119864
119879 and pseudoequilibrium 119864
119875col-
lide together (see Figure 3(b)) where ET1198881
is determined by
ET1198881
=119896 (120573119887119903 + 120575 minus 120573119886)
120573119887119903 (43)
A virtual equilibrium 1198641
119881 an invisible tangent point 1198641
119879
and a pseudoequilibrium 119864119875coexist as shown in Figure 3(a)
when ET lt ET1198881
They collide at ET = ET1198881
and are substitut-ed by a visible tangent point1198641
119879 as shown in Figure 3(c) when
ET gt ET1198881
Similarly another boundary node bifurcation of Filippov
system (3) occurs at ET1198882
= 46667 (see Figure 4(b)) whereET1198882
is determined by
ET1198882
=119896 (120573119887 (119903 minus 119902
1) + 120575 minus 119902
2minus 120573119886)
120573119887119903 (44)
A stable regular equilibrium 1198642
119877and a visible tangent
point 1198642119879coexist as shown in Figure 4(a) when ET lt ET
1198882
They collide with a pseudoequilibrium 119864
119875at ET = ET
1198882
andare substituted by an invisible tangent point 1198642
119879as shown in
Figure 4(c) when ET gt ET1198882
53 Global Sliding Bifurcation Global sliding bifurcationsinvolve nonvanishing cycles which include sliding discon-nection touching (or grazing) bifurcation buckling bifurca-tion crossing bifurcation bifurcation of a sliding homoclinicorbit and heteroclinic orbit [22] Touching (or grazing) bifur-cation implies that a positive period solution can collide withthe sliding segments From the work of Kuang and BerettaandHsu et al [2 7] they concluded that when the value of theparameter 119886 passed slightly through the bifurcation value astable limit cycle bifurcates from the unstable positive interiorequilibrium for the system (1) According to numericalsimulations the system (3) just has touching bifurcation Forexample if we choose ET as bifurcation parameter and fixall other parameters as shown in Figure 5 when the valueof parameter ET varies touching bifurcation occurs at thecritical value ET
1198883
= 259 for the system (3) Note thatextensive numerical simulations indicate that nomatter whatthe value of ET is in touching bifurcation the whole periodicsolution lies in the region 119878
1(shown in Figure 5)This implies
that the density of pest can be successfully maintained belowET by designing suitable threshold policy strategies So ourcontrol objective can be fully realized which is a desiredsituation in crop livestock sectors and forestry
6 Discussion
Recently the threshold policy and IPM strategies haveattracted great attention in agriculture forestry animal hus-bandry and so on [10ndash12] In the process of pest controlIPM strategies would be used only when the density ofpest reaches or exceeds the ET In addition Filippov systemprovides a natural and rational framework for those realworld problems so it has been widely used in different fields
8 Abstract and Applied Analysis
0 5 10 150
1
2
3
4
5
6
7
8
9
10
11
x
y
E2V
E1V
EP
(a) ET = 83
0 5 10 150
1
2
3
4
5
6
7
8
9
10
11
x
y
E2V
E1B(EP)
(b) ET = 93333
0 5 10 150
1
2
3
4
5
6
7
8
9
10
11E2V
E1R
x
y
(c) ET = 98
Figure 3 Boundary node bifurcation for Filippov system (3) The parameter values are fixed as follows 119886 = 07 119887 = 06 119903 = 15 119896 = 12120573 = 05 120575 = 025 119902
1= 025 and 119902
2= 01
0 5 10
5
10
15
20
25
30
35
40
45
E1V
E2R
x
y
(a) ET = 4
0
5
10
15
20
25
30
35
0 5 10x
yE2B(EP)
E1V
(b) ET = 46667
0
5
10
15
20
25
0 5 10x
yE2V
E1V
EP
(c) ET = 58
Figure 4 Boundary node bifurcation for Filippov system (3) The parameter values are fixed as follows 119886 = 07 119887 = 06 119903 = 15 119896 = 12120573 = 05 120575 = 025 119902
1= 025 and 119902
2= 01
such as in science and engineering [19ndash21 23] In this paperwe employ the sliding analysis of Filippov system to describeand investigate the long term dynamical behavior of theratio-dependent-type prey-predator model with Michaelis-Menten-type functional response so that we can use theFilippov system to model intervention of pest control policyFirstly we investigate the sliding mode domain sliding modedynamics Secondly the null-isoclines and the existence of
four types of equilibria for Filippov system including regularvirtual boundary and pseudoequilibrium and the tangentpoints are discussed in detail Moreover the stability ofpseudoequilibrium is also studied Thirdly we have investi-gated the local sliding bifurcations including regularvirtualequilibrium bifurcation and boundary node bifurcationsFurther global touching bifurcation is also studied by numer-ical techniques
Abstract and Applied Analysis 9
0 1 2 30
5
10
15
x
y
E1R
(a) ET = 15
0 2 40
5
10
15
x
y
E1R
(b) ET = 259
0 2 4 60
5
10
15
x
y
E1R
(c) ET = 35
Figure 5 Globally touching bifurcation for Filippov system (3) The parameter values are fixed as follows 119886 = 051 119887 = 04 119903 = 12 119896 = 6120573 = 06863 120575 = 005 119902
1= 04 and 119902
2= 01
In the process of pest control we should apply all kindsof control strategies so as to prevent multiple pest outbreaksor make sure that the density of pest stabilizes at a desiredlevel of ET In order to realize these goals on the one handwe can determine a set of parameters such that not only all theequilibria of subsystems 119878
1and 1198782are virtual equilibria which
ensures that the pseudoequilibria exist but also the pseudoe-quilibrium is globally stable From Theorem 6 we providedthe conditions of the existence of pseudoequilibrium 119864
1198752
forthe system (3) and showed that it is locally stable if it existsFurther we showed that the local stability of pseudoequilib-rium 119864
1198752
implies global stability by numerical simulation Onthe other hand globally touching bifurcation indicates thatthe density of pest can be successfully maintained below theET by designing suitable threshold policy strategies (shownin Figure 5) Therefore our control objective can be achievedfully in the above two cases which can be used to pest controlin crop livestock sectors and forestry
Although impulsive prey-predator models with the ratio-dependent- or Michaelis-Menten-type response functionhave been studied in [13ndash16] the Filippov systems have manyadvantages in describing interventions including sprayingpesticides and releasing natural enemies compared withimpulsive models In this work the number of natural enemyto be released is proportional to its number It is interestingthat releasing number of natural enemy can be described byconstant independently of the existing numbers of pest andnatural enemy Moreover in practice considering environ-mental energy resource finiteness we should choose the totalnumber of both populations as a guide to switch the systemwhich is called the weighted escapement policy (WEP) [1930ndash32] Therefore in the future work we will focus on theabove two cases which could result in richer dynamics
Appendix
Methods for Analysis of the Sliding Solution
Filippov Convex Method The Filippov method associatesthe following convex combination 119865
119878(119885) of the two vectors
1198651198781
(119885) and 1198651198782
(119885) with each nonsingular sliding point 119885 isin
Σ2 that is
119865119878(119885) = (1 minus 120582 (119885)) 119865
1198781(119885) + 120582 (119885) 119865
1198782(119885) (A1)
where
120582 (119885) =⟨119867119885 1198651198781
⟩
⟨119867119885 1198651198781
minus F1198782
⟩ (A2)
119865119878(119885) is tangent to Σ
2and 0 le 120582(119885) le 1
Thus the sliding mode dynamics can be determined by
= 119865119878(119885) 119885 isin Σ
2 (A3)
which is smooth on a one-dimensional sliding interval of Σ2
The solution of the above equation is the sliding solutionThe equation 120582(119885) = 0 (or 120582(119885) = 1) indicates that the
flow is determined by 1198651198781
(or 1198651198782
) aloneTherefore the slidingmode domain can be defined as
Σ119878= 119885 = (119909 119910) isin Σ | 0 le 120582 (119885) le 1 (A4)
which is equivalent to Σ119878= 119885 isin Σ | 120590(119885) le 0 Denote
Σ+
119878= 119885 isin Σ | 120582 (119885) = 1 Σ
minus
119878= 119885 isin Σ | 120582 (119885) = 0
(A5)
10 Abstract and Applied Analysis
by the boundary of the sliding mode domain so the vectorfields are tangent to boundary Σ+
119878or Σminus119878
Utkin Equivalent Control Method It follows from Utkinrsquosworks [18] on sliding mode dynamics along the manifoldΣ we note that Filippov system (3) can be also rewritten asfollows
= 119865 (119885119880119867) (A6)
where the control 119880119867is defined as
119880119867=
0 119867 (119885) lt 0
1198801(119885 119905) 119867 (119885) gt 0
(A7)
where 1198801is a continuous function In the controlled system
(ie system 1198782) the control 119880
119867= 1198801is applied and in the
free system (ie system 1198781) 119880119867= 0 is applied Assume that
a sliding mode exists on manifold Σ that is Σ2is nonempty
Solving the following algebraic equation
=120597119867
120597119885119865 (119885U
119867) = 0 (A8)
with respect to 119880119867
on this manifold gives the solutiondenoted by 119880
lowast(119885 119905) which is referred to as equivalent
control Substituting for 119880119867in system (A6) yields
which determines the sliding mode dynamics of the Filippovsystem (3)
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China (NSFC 11171199) and by the FundamentalResearch Funds for the Central Universities (GK201003001)
References
[1] R M May ldquoLimit cycles in predator-prey communitiesrdquo Sci-ence vol 177 no 4052 pp 900ndash902 1972
[2] Y Kuang and E Beretta ldquoGlobal qualitative analysis of a ratio-dependent predator-prey systemrdquo Journal ofMathematical Biol-ogy vol 36 no 4 pp 389ndash406 1998
[3] W Murdoch C Briggs and R Nisbet Consumer-Resource Dy-namics Princeton University Press New York NY USA 2003
[4] H I Freedman Deterministic Mathematical Models in Popula-tion Ecology vol 57 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1980
[5] G T Skalski and J F Gilliam ldquoFunctional responseswith preda-tor interference viable alternatives to theHolling type IImodelrdquoEcology vol 82 no 11 pp 3083ndash3092 2001
[6] R Arditi and L R Ginzburg ldquoCoupling in predator-preydynamics ratio-dependencerdquo Journal ofTheoretical Biology vol139 no 3 pp 311ndash326 1989
[7] S-B Hsu T-W Hwang and Y Kuang ldquoGlobal analysis ofthe Michaelis-Menten-type ratio-dependent predator-prey sys-temrdquo Journal of Mathematical Biology vol 42 no 6 pp 489ndash506 2001
[8] C Jost O Arino and R Arditi ldquoAbout deterministic extinctionin ratio-dependent predator-prey modelsrdquo Bulletin of Mathe-matical Biology vol 61 no 1 pp 19ndash32 1999
[9] D Xiao and S Ruan ldquoGlobal dynamics of a ratio-dependentpredator-prey systemrdquo Journal of Mathematical Biology vol 43no 3 pp 268ndash290 2001
[10] Z Lu X Chi and L Chen ldquoImpulsive control strategies inbiological control of pesticiderdquo Theoretical Population Biologyvol 64 no 1 pp 39ndash47 2003
[11] S Tang and R A Cheke ldquoModels for integrated pest controland their biological implicationsrdquo Mathematical Biosciencesvol 215 no 1 pp 115ndash125 2008
[12] J C Van Lenteren and J Woets ldquoBiological and integrated pestcontrol in greenhousesrdquo Annual Review of Entomology vol 33pp 239ndash250 1988
[13] B Dai H Su and D Hu ldquoPeriodic solution of a delayed ratio-dependent predator-prey model with monotonic functionalresponse and impulserdquo Nonlinear Analysis Theory Methods ampApplications vol 70 no 1 pp 126ndash134 2009
[14] X Liu G Li and G Luo ldquoPositive periodic solution for a two-species ratio-dependent predator-prey system with time delayand impulserdquo Journal of Mathematical Analysis and Applica-tions vol 325 no 1 pp 715ndash723 2007
[15] G Jiang Q Lu and L Qian ldquoComplex dynamics of a Hollingtype II prey-predator system with state feedback controlrdquoChaos Solitons amp Fractals vol 31 no 2 pp 448ndash461 2007
[16] H Baek and Y Lim ldquoDynamics of an impulsively controlledMichaelis-Menten type predator-prey systemrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 16 no4 pp 2041ndash2053 2011
[17] T Zhao Y Xiao and R J Smith ldquoNon-smooth plant diseasemodels with economic thresholdsrdquo Mathematical Biosciencesvol 241 no 1 pp 34ndash48 2013
[18] V I Utkin Sliding Modes in Control and Optimization Com-munications and Control Engineering Series Springer BerlinGermany 1992
[19] M I S Costa E Kaszkurewicz A Bhaya and L Hsu ldquoAchiev-ing global convergence to an equilibrium population in pred-ator-prey systems by the use of a discontinuous harvestingpolicyrdquo Ecological Modelling vol 128 no 2-3 pp 89ndash99 2000
[20] B L Van De Vrande D H Van Campen and A De KrakerldquoApproximate analysis of dry-friction-induced stick-slip vibra-tions by a smoothing procedurerdquo Nonlinear Dynamics vol 19no 2 pp 157ndash169 1999
[21] S H Doole and S J Hogan ldquoA piecewise linear suspensionbridge model nonlinear dynamics and orbit continuationrdquoDynamics and Stability of Systems vol 11 no 1 pp 19ndash47 1996
[22] Yu A Kuznetsov S Rinaldi and A Gragnani ldquoOne-parameterbifurcations in planar Filippov systemsrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol13 no 8 pp 2157ndash2188 2003
[23] M di Bernardo C J Budd A R Champneys et al ldquoBifurca-tions in nonsmooth dynamical systemsrdquo SIAM Review vol 50no 4 pp 629ndash701 2008
[24] A F Filippov Differential Equations with Discontinuous Right-hand Sides vol 18 of Mathematics and Its Applications (SovietSeries) Kluwer Academic Dordrecht The Netherlands 1988
[25] S Tang J Liang Y Xiao and R A Cheke ldquoSliding bifurcationsof Filippov two stage pest control models with economicthresholdsrdquo SIAM Journal on Applied Mathematics vol 72 no4 pp 1061ndash1080 2012
Abstract and Applied Analysis 11
[26] M I da Silveira Costa and M E M Meza ldquoApplication of athreshold policy in the management of multispecies fisheriesand predator cullingrdquoMathematical Medicine and Biology vol23 no 1 pp 63ndash75 2006
[27] M E Mendoza Meza A Bhaya E Kaszkurewicz and M IDa Silveira Costa ldquoThreshold policies control for predator-preysystems using a control Liapunov function approachrdquoTheoreti-cal Population Biology vol 67 no 4 pp 273ndash284 2005
[28] F Dercole A Gragnani and S Rinaldi ldquoBifurcation analysisof piecewise smooth ecological modelsrdquoTheoretical PopulationBiology vol 72 no 2 pp 197ndash213 2007
[29] W Wang ldquoBackward bifurcation of an epidemic model withtreatmentrdquo Mathematical Biosciences vol 201 no 1-2 pp 58ndash71 2006
[30] I Noy-Meir ldquoStability of grazing systems an application ofpredator-prey graphsrdquo The Journal of Animal Ecology vol 63no 2 pp 459ndash481 1975
[31] R M Colombo and V Krivan ldquoSelective strategies in foodwebsrdquo Mathematical Medicine and Biology vol 10 no 4 pp281ndash291 1993
[32] V Krivan ldquoOptimal foraging and predator-prey dynamicsrdquoTheoretical Population Biology vol 49 no 3 pp 265ndash290 1996
cases for the existence of sliding segments of Filippov system(3) By simply calculating and arranging we have the follow-ing results
(i) When 119903 lt 119886119896119887(119896 minus ET) the sliding segment can bedescribed as
Σ1
119878= (119909 119910) | max 0 119910
2 le 119910 le 119910
1 119909 = ET (14)
(ii) When max1199021119896(119896 minus ET) 119886119896119887(119896 minus ET) lt 119903 lt
min119896(119886 + 1198871199021)119887(119896 minus ET) 119886(119896 + 119887119902
1)119887(119896 minus ET) the
sliding segment can be described as
Σ2
119878= (119909 119910) | 0 le 119910 le 119910
2 119909 = ET (15)
32 Sliding Mode Dynamics Filippov system (3) only hasone piece of sliding segment and the solutions defined init can be obtained from the sliding mode dynamics whichcan be determined by employing theUtkin equivalent controlmethod (see the appendix)
From119867 = 0 we get that
119867119885= 1199091015840= 119903119909 (1 minus
119909
119896) minus
119886119909119910
119887119910 + 119909minus 1205761199021119909 = 0 119909 = ET
(16)
And solving the above equations with respect to 120576 yields
120576 =119903 (1 minus ET119896) minus 119886119910 (119887119910 + ET)
1199021
(17)
Hence the dynamics on the slidingmodeΣ119878can be deter-
mined by the following scalar differential equation
119910 (119905) = minus 120575119910 +120573119886119910ET119887119910 + ET
+119903 (1 minus ET119896) minus 119886119910 (119887119910 + ET)
1199021
1199022119910
= 1198751(119910) (119875
2119910 + 1198753) ≜ 120601 (119910)
(18)
where 119910 isin Σ1
119878or Σ2
1198781198751(119910) = 119910119902
1(119887119910 + ET) 119875
2= minus120575119887119902
1+
1199031198871199022(1 minus ET119896) minus 119886119902
2 and 119875
3= minus120575119902
1ET + 120573119886119902
1ET + 119903119902
2(1 minus
ET119896)ET
4 Null-Isoclines and Equilibria
41The Null-Isoclines of Filippov System (3) Null-isoclines ofboth systems 119878
1and 1198782are related to the existence of equilib-
ria and are useful for analysis of sliding dynamicNull-isoclines (119905) = 0 and 119910(119905) = 0 for both systems
1198781and 119878
2can be determined as follows For the system 119878
1
solving the equation of the null-isocline (119905) = 0 yeilds
1198911198781(119909) =
119903119909 (1 minus 119909119896)
119886 minus 119887119903 (1 minus 119909119896) (19)
and null-isocline 119910(119905) = 0 gives
1198921198781(119909) =
(120573119886 minus 120575) 119909
119887120575 (20)
For the system 1198782 solving the equation of the null-isocline
(119905) = 0 yields
1198911198782(119909) =
(119903119909 (1 minus 119909119896) minus 1199021) 119909
119886 + 1198871199021minus 119887119903 (1 minus 119909119896)
(21)
and null-isocline 119910(119905) = 0 gives
1198921198782(119909) =
(120573119886 minus (120575 minus 1199022)) 119909
119887 (120575 minus 1199022)
(22)
42 The Equilibria of Filippov System (3) Because the solu-tions of Filippov system (3) are composed of connectingstandard solutions in subsystems 119878
1 1198782and sliding mode
solutions on Σ1119878or Σ2119878The definitions of all types of equilibria
of Filippov system have been provided in Section 2 which areimportant to bifurcation analysis There may be several typesof equilibria for Filippov system (3) which include regularequilibrium (denoted by119864
119877) virtual equilibrium (denoted by
119864119881) pseudoequilibrium (denoted by 119864
119875) boundary equilib-
rium (denoted by119864119861) and one type of special point named as
tangent point (denoted by 119864119879) Detailed definitions of these
equilibria and the tangent point can be found in the literature[24]
From Lemma 1 we know that the subsystem 1198781with 119909 lt
ET has three possible equilibria (0 0) (119896 0) and interiorequilibrium
119864lowast
1198781
= (119909lowast
1198781
119910lowast
1198781
)
= (119896 (120573119887119903 + 120575 minus 120573119886)
120573119887119903119896 (120573119886 minus 120575) (120573119887119903 + 120575 minus 120573119886)
1205731198872119903120575)
(23)
If 120573119886 minus 120575 gt 0 and 119903 gt (120573119886 minus 120575)120573119887 then there is a positiveinterior equilibrium for the system 119878
1
Equilibria for the subsystem 1198782with 119909 gt ET satisfy
119903119909 (1 minus119909
119896) minus
119886119909119910
119887119910 + 119909minus 1199021119909 = 0
minus120575119910 +120573119886119909119910
119887119910 + 119909+ 1199022119910 = 0
(24)
Solving the above equations with respect to 119909 and 119910 yieldsthree possible equilibria that is (0 0) (119896(119903 minus 119902
1)119903 0) and
interior equilibrium
119864lowast
1198782
= (119909lowast
1198782
119910lowast
1198782
)
= (119896 (120573119887 (119903 minus 119902
1) + 120575 minus 119902
2minus 120573119886)
120573119887119903
119896 (120573119886 minus 120575 + 1199022) (120573119887 (119903 minus 119902
1) + 120575 minus 119902
2minus 120573119886)
1205731198872119903 (120575 minus 1199022)
)
(25)
Abstract and Applied Analysis 5
which indicates that if 119903 gt (120573119886 minus 120575 + 1199022)120573119887 + 119902
1and 119902
2lt
120575 lt 120573119886 + 1199022 then there is a positive interior equilibrium for
the system 1198782
421 Regular Equilibria For the subsystem 1198781with 119909 lt ET
(0 0) is a regular equilibrium while (119896 0) is a virtual equilib-rium In addition according to the coordinate of equilibrium119864lowast
1198781
we have the following results If 119903 gt (120573119886 minus 120575)120573119887 gt 0 and
119896 (120573119887119903 + 120575 minus 120573119886)
120573119887119903lt ET (26)
then it is a regular equilibrium for the system 1198781 denoted by
1198641
119877 If 119903 gt (120573119886 minus 120575)120573119887 gt 0 and
119896 (120573119887119903 + 120575 minus 120573119886)
120573119887119903gt ET (27)
then the equilibrium 119864lowast
1198781
becomes a virtual equilibriumdenoted by 1198641
119881
About regular equilibria for the subsystem 1198782with 119909 gt
ET (0 0) is a virtual equilibrium while (119896(119903 minus 1199011)119903 0) may
be a regular or virtual equilibrium which depends on theparameter space Moreover according to the coordinate ofequilibrium 119864
lowast
1198782
we have the following conclusions If 119903 gt
(120573119886 minus 120575 + 1199022)120573119887 + 119902
1 1199022lt 120575 lt 120573119886 + 119902
2and
119896 (120573119887 (119903 minus 1199021) + 120575 minus 119902
2minus 120573119886)
120573119887119903gt ET (28)
then it is a regular equilibrium for the system 1198782 denoted by
1198642
119877 Note that if 119903 gt (120573119886 minus 120575 + 119902
2)120573119887 + 119902
1 1199022lt 120575 lt 120573119886 + 119902
2
and119896 (120573119887 (119903 minus 119902
1) + 120575 minus 119902
2minus 120573119886)
120573119887119903lt ET (29)
then the equilibrium 119864lowast
1198782
becomes a virtual equilibriumdenoted by 1198642
119881 Based on the above analysis if
119896 (120573119886 minus 120575)
120573119887 (119896 minus ET)lt 119903 lt
119896 (120573119886 minus 120575 + 1199022+ 120573119887119902
1)
120573119887 (119896 minus ET) (30)
then the two virtual equilibria 1198641119881and 1198642
119881can coexist
422 Pseudoequilibrium For the existence of pseudoequilib-rium119864
119875= (ET 119910
119875) 119910119875component of the pseudoequilibrium
of sliding flow satisfies the following equation
1198751(119910) (119875
2119910 + 1198753) = 0 (31)
where 119910 isin Σ1
119878or Σ2
119878 Solving the above equation with
respect to 119910 yields two possible roots denoted by 1199101198751
= 0 and
1199101198752
=(1205751199021minus 120573119886119902
1minus 1199031199022(1 minus ET119896))ET
minus1205751198871199021+ 1199031198871199022(1 minus ET119896) minus 119886119902
Solving the above equations with respect to 119909 and 119910 yieldstwo tangent points including
1198641
119879= (ET 119903ET (1 minus ET119896)
119886 minus 119887119903 (1 minus ET119896))
1198642
119879= (
ET (119903 (1 minus ET119896) minus 1199021)
119886 + 1199021119887 minus 119887119903 (1 minus ET119896)
)
(40)
If we fix all parameters then the relations among null-isoclines regularvirtual equilibria pseudoequilibrium andsliding segment are provided in Figure 1 Note that virtualequilibria of both systems 119878
1and 119878
2imply the existence of
pseudoequilibrium andwewill prove this general result later
43 The Stability of Pseudoequilibrium In the process ofpest control we should apply all kinds of control strategiesso as to prevent multiple pest outbreaks or make sure thatthe total density of the pest stabilizes at a desired level ofET If the unique positive equilibrium of system 119878
1and
system 1198782is virtual simultaneously then the sliding flow
has a unique pseudoequilibrium In order to realize thisgoal we can choose a set of parameters such that all theequilibria of subsystems 119878
1and 1198782are virtual equilibria and
the pseudoequilibria are globally stable which have beenwidely used in pest control [26ndash28] For example if wefixed all parameter values as those in Figure 2 then bothvirtual equilibria coexist Therefore we address the stabilityof pseudoequilibrium119864
1198752
in the followingwhich is importantto control pest
Theorem 6 Either the inequalities (34) or (35) hold true orthe two virtual equilibria 1198641
119881and 1198642
119881can coexist then Filippov
system (3) contains a positive pseudoequilibrium 1198641198752
Regard-less of which cases would occur the pseudoequilibrium 119864
1198752
islocally stable with respect to sliding mode domain
Proof According to the conditions of Theorem 6 we seethat the inequalities (30) hold true which implies that theinequalities (34) are true Based on the discussions aboutthe existence of pseudoequilibrium in Section 42 we haveconcluded that if inequalities (34) or (35) hold true thenthe system (3) contains a positive pseudoequilibrium 119864
=(1198961199021(120575 minus 120573119886) minus 119903119902
2(119896 minus ET)) (120575119887119896119902
1minus 119903119887119902
2(119896 minus ET) + 119886119896119902
2)
11988611990211198962 (119887120573119902
1+ 1199022)
(41)
Therefore if the inequalities (33) hold true then 119889120601(119910)119889119910 lt 0 and 119910
1198752
is a positive root of (31) Note that theinequalities (34) and (35) indicate the inequalities (33) Thusthe positive pseudoequilibrium 119864
1198752
is locally stale if it exists
Abstract and Applied Analysis 7
It is difficult to directly prove the global stability ofpseudoequilibrium 119864
1198752
in this case Because we cannotemploy the classical Bendixson-Dulac theorem due to thediscontinuity of vector fields However if there is not crossingcycle surrounding sliding segment then pseudoequilibriumis globally stable [29] From the analysis of global bifurcationfor the system (3) in the following section the system just hastouching bifurcation and there is nonexistence of a slidingcycle which surrounds the 119864
1198752
By using the similar methodswe have that pseudoequilibrium 119864
1198752
is globally stable Thatis to say the local stability of pseudoequilibrium 119864
1198752
withrespect to sliding mode domain indicates its global stabilityin the first quadrant (shown in Figures 3(a) 3(b) 4(b) and4(c)) In practice in order to control pest outbreak we shouldchoose the desirable ET at first so that all equilibria of eachsystem such as system 119878
1and system 119878
2become virtual then
pseudoequilibrium not only exists but also is globally stableIn other words the density of pest can be stable at the ETWhen the density of pest reaches or exceeds the ET we shouldcarry out control strategies (eg releasing natural enemyetc) until it falls below the ET In this way our control goalcan be realized fully
5 Equilibria and Sliding Bifurcation Set
51 RegularVirtual Equilibrium Bifurcation According tothe above discussions it is obvious that 119903 and ET are primaryfactors in determining the existence of the above differenttypes of equilibria of the system (3) So we define four curvesabout parameters 119903 and ET as follows
1198711= (119903ET) | 119903 =
120573119886 minus 120575
120573119887
1198712= (119903ET) | 119903 =
120573119886 minus 120575 + 1199022
120573119887+ 1199021
1198713= (119903ET) | ET =
119896 (120573119887119903 + 120575 minus 120573119886)
120573119887119903
1198714= (119903ET) | ET =
119896 (120573119887 (119903 minus 1199021) + 120575 minus 119902
2minus 120573119886)
120573119887119903
(42)
The four curves (ie 1198711 1198712 1198713 and 119871
4) divide the 119903
and ET parameter space into six regions and the existenceor coexistence of regular or virtual equilibria is indicated ineach region The boundary equilibria 1198641
119861and 1198642
119861can appear
on the lines 1198713and 119871
4accordingly In particular it follows
from Figure 2 that the two virtual equilibria 1198641119881and 1198642
119881can
coexist which is very important to pest control However thetwo regular equilibria 1198641
119877and 1198642
119877cannot coexist
52 Boundary Node Bifurcations This type of bifurcationmay occur for Filippov system (3) once equilibria 119864
119875 119864119879 and
119864119877or 119864119879and 119864
119877collide together simultaneously when ET
passes through a critical value In this part we choose ETas bifurcation parameter and all other parameters are fixedas those in Figure 3 Note that once the parameter ET passes
through the first critical value ET1198881
= 93333 the regular equi-librium 119864
1
119877 tangent point 119864
119879 and pseudoequilibrium 119864
119875col-
lide together (see Figure 3(b)) where ET1198881
is determined by
ET1198881
=119896 (120573119887119903 + 120575 minus 120573119886)
120573119887119903 (43)
A virtual equilibrium 1198641
119881 an invisible tangent point 1198641
119879
and a pseudoequilibrium 119864119875coexist as shown in Figure 3(a)
when ET lt ET1198881
They collide at ET = ET1198881
and are substitut-ed by a visible tangent point1198641
119879 as shown in Figure 3(c) when
ET gt ET1198881
Similarly another boundary node bifurcation of Filippov
system (3) occurs at ET1198882
= 46667 (see Figure 4(b)) whereET1198882
is determined by
ET1198882
=119896 (120573119887 (119903 minus 119902
1) + 120575 minus 119902
2minus 120573119886)
120573119887119903 (44)
A stable regular equilibrium 1198642
119877and a visible tangent
point 1198642119879coexist as shown in Figure 4(a) when ET lt ET
1198882
They collide with a pseudoequilibrium 119864
119875at ET = ET
1198882
andare substituted by an invisible tangent point 1198642
119879as shown in
Figure 4(c) when ET gt ET1198882
53 Global Sliding Bifurcation Global sliding bifurcationsinvolve nonvanishing cycles which include sliding discon-nection touching (or grazing) bifurcation buckling bifurca-tion crossing bifurcation bifurcation of a sliding homoclinicorbit and heteroclinic orbit [22] Touching (or grazing) bifur-cation implies that a positive period solution can collide withthe sliding segments From the work of Kuang and BerettaandHsu et al [2 7] they concluded that when the value of theparameter 119886 passed slightly through the bifurcation value astable limit cycle bifurcates from the unstable positive interiorequilibrium for the system (1) According to numericalsimulations the system (3) just has touching bifurcation Forexample if we choose ET as bifurcation parameter and fixall other parameters as shown in Figure 5 when the valueof parameter ET varies touching bifurcation occurs at thecritical value ET
1198883
= 259 for the system (3) Note thatextensive numerical simulations indicate that nomatter whatthe value of ET is in touching bifurcation the whole periodicsolution lies in the region 119878
1(shown in Figure 5)This implies
that the density of pest can be successfully maintained belowET by designing suitable threshold policy strategies So ourcontrol objective can be fully realized which is a desiredsituation in crop livestock sectors and forestry
6 Discussion
Recently the threshold policy and IPM strategies haveattracted great attention in agriculture forestry animal hus-bandry and so on [10ndash12] In the process of pest controlIPM strategies would be used only when the density ofpest reaches or exceeds the ET In addition Filippov systemprovides a natural and rational framework for those realworld problems so it has been widely used in different fields
8 Abstract and Applied Analysis
0 5 10 150
1
2
3
4
5
6
7
8
9
10
11
x
y
E2V
E1V
EP
(a) ET = 83
0 5 10 150
1
2
3
4
5
6
7
8
9
10
11
x
y
E2V
E1B(EP)
(b) ET = 93333
0 5 10 150
1
2
3
4
5
6
7
8
9
10
11E2V
E1R
x
y
(c) ET = 98
Figure 3 Boundary node bifurcation for Filippov system (3) The parameter values are fixed as follows 119886 = 07 119887 = 06 119903 = 15 119896 = 12120573 = 05 120575 = 025 119902
1= 025 and 119902
2= 01
0 5 10
5
10
15
20
25
30
35
40
45
E1V
E2R
x
y
(a) ET = 4
0
5
10
15
20
25
30
35
0 5 10x
yE2B(EP)
E1V
(b) ET = 46667
0
5
10
15
20
25
0 5 10x
yE2V
E1V
EP
(c) ET = 58
Figure 4 Boundary node bifurcation for Filippov system (3) The parameter values are fixed as follows 119886 = 07 119887 = 06 119903 = 15 119896 = 12120573 = 05 120575 = 025 119902
1= 025 and 119902
2= 01
such as in science and engineering [19ndash21 23] In this paperwe employ the sliding analysis of Filippov system to describeand investigate the long term dynamical behavior of theratio-dependent-type prey-predator model with Michaelis-Menten-type functional response so that we can use theFilippov system to model intervention of pest control policyFirstly we investigate the sliding mode domain sliding modedynamics Secondly the null-isoclines and the existence of
four types of equilibria for Filippov system including regularvirtual boundary and pseudoequilibrium and the tangentpoints are discussed in detail Moreover the stability ofpseudoequilibrium is also studied Thirdly we have investi-gated the local sliding bifurcations including regularvirtualequilibrium bifurcation and boundary node bifurcationsFurther global touching bifurcation is also studied by numer-ical techniques
Abstract and Applied Analysis 9
0 1 2 30
5
10
15
x
y
E1R
(a) ET = 15
0 2 40
5
10
15
x
y
E1R
(b) ET = 259
0 2 4 60
5
10
15
x
y
E1R
(c) ET = 35
Figure 5 Globally touching bifurcation for Filippov system (3) The parameter values are fixed as follows 119886 = 051 119887 = 04 119903 = 12 119896 = 6120573 = 06863 120575 = 005 119902
1= 04 and 119902
2= 01
In the process of pest control we should apply all kindsof control strategies so as to prevent multiple pest outbreaksor make sure that the density of pest stabilizes at a desiredlevel of ET In order to realize these goals on the one handwe can determine a set of parameters such that not only all theequilibria of subsystems 119878
1and 1198782are virtual equilibria which
ensures that the pseudoequilibria exist but also the pseudoe-quilibrium is globally stable From Theorem 6 we providedthe conditions of the existence of pseudoequilibrium 119864
1198752
forthe system (3) and showed that it is locally stable if it existsFurther we showed that the local stability of pseudoequilib-rium 119864
1198752
implies global stability by numerical simulation Onthe other hand globally touching bifurcation indicates thatthe density of pest can be successfully maintained below theET by designing suitable threshold policy strategies (shownin Figure 5) Therefore our control objective can be achievedfully in the above two cases which can be used to pest controlin crop livestock sectors and forestry
Although impulsive prey-predator models with the ratio-dependent- or Michaelis-Menten-type response functionhave been studied in [13ndash16] the Filippov systems have manyadvantages in describing interventions including sprayingpesticides and releasing natural enemies compared withimpulsive models In this work the number of natural enemyto be released is proportional to its number It is interestingthat releasing number of natural enemy can be described byconstant independently of the existing numbers of pest andnatural enemy Moreover in practice considering environ-mental energy resource finiteness we should choose the totalnumber of both populations as a guide to switch the systemwhich is called the weighted escapement policy (WEP) [1930ndash32] Therefore in the future work we will focus on theabove two cases which could result in richer dynamics
Appendix
Methods for Analysis of the Sliding Solution
Filippov Convex Method The Filippov method associatesthe following convex combination 119865
119878(119885) of the two vectors
1198651198781
(119885) and 1198651198782
(119885) with each nonsingular sliding point 119885 isin
Σ2 that is
119865119878(119885) = (1 minus 120582 (119885)) 119865
1198781(119885) + 120582 (119885) 119865
1198782(119885) (A1)
where
120582 (119885) =⟨119867119885 1198651198781
⟩
⟨119867119885 1198651198781
minus F1198782
⟩ (A2)
119865119878(119885) is tangent to Σ
2and 0 le 120582(119885) le 1
Thus the sliding mode dynamics can be determined by
= 119865119878(119885) 119885 isin Σ
2 (A3)
which is smooth on a one-dimensional sliding interval of Σ2
The solution of the above equation is the sliding solutionThe equation 120582(119885) = 0 (or 120582(119885) = 1) indicates that the
flow is determined by 1198651198781
(or 1198651198782
) aloneTherefore the slidingmode domain can be defined as
Σ119878= 119885 = (119909 119910) isin Σ | 0 le 120582 (119885) le 1 (A4)
which is equivalent to Σ119878= 119885 isin Σ | 120590(119885) le 0 Denote
Σ+
119878= 119885 isin Σ | 120582 (119885) = 1 Σ
minus
119878= 119885 isin Σ | 120582 (119885) = 0
(A5)
10 Abstract and Applied Analysis
by the boundary of the sliding mode domain so the vectorfields are tangent to boundary Σ+
119878or Σminus119878
Utkin Equivalent Control Method It follows from Utkinrsquosworks [18] on sliding mode dynamics along the manifoldΣ we note that Filippov system (3) can be also rewritten asfollows
= 119865 (119885119880119867) (A6)
where the control 119880119867is defined as
119880119867=
0 119867 (119885) lt 0
1198801(119885 119905) 119867 (119885) gt 0
(A7)
where 1198801is a continuous function In the controlled system
(ie system 1198782) the control 119880
119867= 1198801is applied and in the
free system (ie system 1198781) 119880119867= 0 is applied Assume that
a sliding mode exists on manifold Σ that is Σ2is nonempty
Solving the following algebraic equation
=120597119867
120597119885119865 (119885U
119867) = 0 (A8)
with respect to 119880119867
on this manifold gives the solutiondenoted by 119880
lowast(119885 119905) which is referred to as equivalent
control Substituting for 119880119867in system (A6) yields
which determines the sliding mode dynamics of the Filippovsystem (3)
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China (NSFC 11171199) and by the FundamentalResearch Funds for the Central Universities (GK201003001)
References
[1] R M May ldquoLimit cycles in predator-prey communitiesrdquo Sci-ence vol 177 no 4052 pp 900ndash902 1972
[2] Y Kuang and E Beretta ldquoGlobal qualitative analysis of a ratio-dependent predator-prey systemrdquo Journal ofMathematical Biol-ogy vol 36 no 4 pp 389ndash406 1998
[3] W Murdoch C Briggs and R Nisbet Consumer-Resource Dy-namics Princeton University Press New York NY USA 2003
[4] H I Freedman Deterministic Mathematical Models in Popula-tion Ecology vol 57 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1980
[5] G T Skalski and J F Gilliam ldquoFunctional responseswith preda-tor interference viable alternatives to theHolling type IImodelrdquoEcology vol 82 no 11 pp 3083ndash3092 2001
[6] R Arditi and L R Ginzburg ldquoCoupling in predator-preydynamics ratio-dependencerdquo Journal ofTheoretical Biology vol139 no 3 pp 311ndash326 1989
[7] S-B Hsu T-W Hwang and Y Kuang ldquoGlobal analysis ofthe Michaelis-Menten-type ratio-dependent predator-prey sys-temrdquo Journal of Mathematical Biology vol 42 no 6 pp 489ndash506 2001
[8] C Jost O Arino and R Arditi ldquoAbout deterministic extinctionin ratio-dependent predator-prey modelsrdquo Bulletin of Mathe-matical Biology vol 61 no 1 pp 19ndash32 1999
[9] D Xiao and S Ruan ldquoGlobal dynamics of a ratio-dependentpredator-prey systemrdquo Journal of Mathematical Biology vol 43no 3 pp 268ndash290 2001
[10] Z Lu X Chi and L Chen ldquoImpulsive control strategies inbiological control of pesticiderdquo Theoretical Population Biologyvol 64 no 1 pp 39ndash47 2003
[11] S Tang and R A Cheke ldquoModels for integrated pest controland their biological implicationsrdquo Mathematical Biosciencesvol 215 no 1 pp 115ndash125 2008
[12] J C Van Lenteren and J Woets ldquoBiological and integrated pestcontrol in greenhousesrdquo Annual Review of Entomology vol 33pp 239ndash250 1988
[13] B Dai H Su and D Hu ldquoPeriodic solution of a delayed ratio-dependent predator-prey model with monotonic functionalresponse and impulserdquo Nonlinear Analysis Theory Methods ampApplications vol 70 no 1 pp 126ndash134 2009
[14] X Liu G Li and G Luo ldquoPositive periodic solution for a two-species ratio-dependent predator-prey system with time delayand impulserdquo Journal of Mathematical Analysis and Applica-tions vol 325 no 1 pp 715ndash723 2007
[15] G Jiang Q Lu and L Qian ldquoComplex dynamics of a Hollingtype II prey-predator system with state feedback controlrdquoChaos Solitons amp Fractals vol 31 no 2 pp 448ndash461 2007
[16] H Baek and Y Lim ldquoDynamics of an impulsively controlledMichaelis-Menten type predator-prey systemrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 16 no4 pp 2041ndash2053 2011
[17] T Zhao Y Xiao and R J Smith ldquoNon-smooth plant diseasemodels with economic thresholdsrdquo Mathematical Biosciencesvol 241 no 1 pp 34ndash48 2013
[18] V I Utkin Sliding Modes in Control and Optimization Com-munications and Control Engineering Series Springer BerlinGermany 1992
[19] M I S Costa E Kaszkurewicz A Bhaya and L Hsu ldquoAchiev-ing global convergence to an equilibrium population in pred-ator-prey systems by the use of a discontinuous harvestingpolicyrdquo Ecological Modelling vol 128 no 2-3 pp 89ndash99 2000
[20] B L Van De Vrande D H Van Campen and A De KrakerldquoApproximate analysis of dry-friction-induced stick-slip vibra-tions by a smoothing procedurerdquo Nonlinear Dynamics vol 19no 2 pp 157ndash169 1999
[21] S H Doole and S J Hogan ldquoA piecewise linear suspensionbridge model nonlinear dynamics and orbit continuationrdquoDynamics and Stability of Systems vol 11 no 1 pp 19ndash47 1996
[22] Yu A Kuznetsov S Rinaldi and A Gragnani ldquoOne-parameterbifurcations in planar Filippov systemsrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol13 no 8 pp 2157ndash2188 2003
[23] M di Bernardo C J Budd A R Champneys et al ldquoBifurca-tions in nonsmooth dynamical systemsrdquo SIAM Review vol 50no 4 pp 629ndash701 2008
[24] A F Filippov Differential Equations with Discontinuous Right-hand Sides vol 18 of Mathematics and Its Applications (SovietSeries) Kluwer Academic Dordrecht The Netherlands 1988
[25] S Tang J Liang Y Xiao and R A Cheke ldquoSliding bifurcationsof Filippov two stage pest control models with economicthresholdsrdquo SIAM Journal on Applied Mathematics vol 72 no4 pp 1061ndash1080 2012
Abstract and Applied Analysis 11
[26] M I da Silveira Costa and M E M Meza ldquoApplication of athreshold policy in the management of multispecies fisheriesand predator cullingrdquoMathematical Medicine and Biology vol23 no 1 pp 63ndash75 2006
[27] M E Mendoza Meza A Bhaya E Kaszkurewicz and M IDa Silveira Costa ldquoThreshold policies control for predator-preysystems using a control Liapunov function approachrdquoTheoreti-cal Population Biology vol 67 no 4 pp 273ndash284 2005
[28] F Dercole A Gragnani and S Rinaldi ldquoBifurcation analysisof piecewise smooth ecological modelsrdquoTheoretical PopulationBiology vol 72 no 2 pp 197ndash213 2007
[29] W Wang ldquoBackward bifurcation of an epidemic model withtreatmentrdquo Mathematical Biosciences vol 201 no 1-2 pp 58ndash71 2006
[30] I Noy-Meir ldquoStability of grazing systems an application ofpredator-prey graphsrdquo The Journal of Animal Ecology vol 63no 2 pp 459ndash481 1975
[31] R M Colombo and V Krivan ldquoSelective strategies in foodwebsrdquo Mathematical Medicine and Biology vol 10 no 4 pp281ndash291 1993
[32] V Krivan ldquoOptimal foraging and predator-prey dynamicsrdquoTheoretical Population Biology vol 49 no 3 pp 265ndash290 1996
Solving the above equations with respect to 119909 and 119910 yieldstwo tangent points including
1198641
119879= (ET 119903ET (1 minus ET119896)
119886 minus 119887119903 (1 minus ET119896))
1198642
119879= (
ET (119903 (1 minus ET119896) minus 1199021)
119886 + 1199021119887 minus 119887119903 (1 minus ET119896)
)
(40)
If we fix all parameters then the relations among null-isoclines regularvirtual equilibria pseudoequilibrium andsliding segment are provided in Figure 1 Note that virtualequilibria of both systems 119878
1and 119878
2imply the existence of
pseudoequilibrium andwewill prove this general result later
43 The Stability of Pseudoequilibrium In the process ofpest control we should apply all kinds of control strategiesso as to prevent multiple pest outbreaks or make sure thatthe total density of the pest stabilizes at a desired level ofET If the unique positive equilibrium of system 119878
1and
system 1198782is virtual simultaneously then the sliding flow
has a unique pseudoequilibrium In order to realize thisgoal we can choose a set of parameters such that all theequilibria of subsystems 119878
1and 1198782are virtual equilibria and
the pseudoequilibria are globally stable which have beenwidely used in pest control [26ndash28] For example if wefixed all parameter values as those in Figure 2 then bothvirtual equilibria coexist Therefore we address the stabilityof pseudoequilibrium119864
1198752
in the followingwhich is importantto control pest
Theorem 6 Either the inequalities (34) or (35) hold true orthe two virtual equilibria 1198641
119881and 1198642
119881can coexist then Filippov
system (3) contains a positive pseudoequilibrium 1198641198752
Regard-less of which cases would occur the pseudoequilibrium 119864
1198752
islocally stable with respect to sliding mode domain
Proof According to the conditions of Theorem 6 we seethat the inequalities (30) hold true which implies that theinequalities (34) are true Based on the discussions aboutthe existence of pseudoequilibrium in Section 42 we haveconcluded that if inequalities (34) or (35) hold true thenthe system (3) contains a positive pseudoequilibrium 119864
=(1198961199021(120575 minus 120573119886) minus 119903119902
2(119896 minus ET)) (120575119887119896119902
1minus 119903119887119902
2(119896 minus ET) + 119886119896119902
2)
11988611990211198962 (119887120573119902
1+ 1199022)
(41)
Therefore if the inequalities (33) hold true then 119889120601(119910)119889119910 lt 0 and 119910
1198752
is a positive root of (31) Note that theinequalities (34) and (35) indicate the inequalities (33) Thusthe positive pseudoequilibrium 119864
1198752
is locally stale if it exists
Abstract and Applied Analysis 7
It is difficult to directly prove the global stability ofpseudoequilibrium 119864
1198752
in this case Because we cannotemploy the classical Bendixson-Dulac theorem due to thediscontinuity of vector fields However if there is not crossingcycle surrounding sliding segment then pseudoequilibriumis globally stable [29] From the analysis of global bifurcationfor the system (3) in the following section the system just hastouching bifurcation and there is nonexistence of a slidingcycle which surrounds the 119864
1198752
By using the similar methodswe have that pseudoequilibrium 119864
1198752
is globally stable Thatis to say the local stability of pseudoequilibrium 119864
1198752
withrespect to sliding mode domain indicates its global stabilityin the first quadrant (shown in Figures 3(a) 3(b) 4(b) and4(c)) In practice in order to control pest outbreak we shouldchoose the desirable ET at first so that all equilibria of eachsystem such as system 119878
1and system 119878
2become virtual then
pseudoequilibrium not only exists but also is globally stableIn other words the density of pest can be stable at the ETWhen the density of pest reaches or exceeds the ET we shouldcarry out control strategies (eg releasing natural enemyetc) until it falls below the ET In this way our control goalcan be realized fully
5 Equilibria and Sliding Bifurcation Set
51 RegularVirtual Equilibrium Bifurcation According tothe above discussions it is obvious that 119903 and ET are primaryfactors in determining the existence of the above differenttypes of equilibria of the system (3) So we define four curvesabout parameters 119903 and ET as follows
1198711= (119903ET) | 119903 =
120573119886 minus 120575
120573119887
1198712= (119903ET) | 119903 =
120573119886 minus 120575 + 1199022
120573119887+ 1199021
1198713= (119903ET) | ET =
119896 (120573119887119903 + 120575 minus 120573119886)
120573119887119903
1198714= (119903ET) | ET =
119896 (120573119887 (119903 minus 1199021) + 120575 minus 119902
2minus 120573119886)
120573119887119903
(42)
The four curves (ie 1198711 1198712 1198713 and 119871
4) divide the 119903
and ET parameter space into six regions and the existenceor coexistence of regular or virtual equilibria is indicated ineach region The boundary equilibria 1198641
119861and 1198642
119861can appear
on the lines 1198713and 119871
4accordingly In particular it follows
from Figure 2 that the two virtual equilibria 1198641119881and 1198642
119881can
coexist which is very important to pest control However thetwo regular equilibria 1198641
119877and 1198642
119877cannot coexist
52 Boundary Node Bifurcations This type of bifurcationmay occur for Filippov system (3) once equilibria 119864
119875 119864119879 and
119864119877or 119864119879and 119864
119877collide together simultaneously when ET
passes through a critical value In this part we choose ETas bifurcation parameter and all other parameters are fixedas those in Figure 3 Note that once the parameter ET passes
through the first critical value ET1198881
= 93333 the regular equi-librium 119864
1
119877 tangent point 119864
119879 and pseudoequilibrium 119864
119875col-
lide together (see Figure 3(b)) where ET1198881
is determined by
ET1198881
=119896 (120573119887119903 + 120575 minus 120573119886)
120573119887119903 (43)
A virtual equilibrium 1198641
119881 an invisible tangent point 1198641
119879
and a pseudoequilibrium 119864119875coexist as shown in Figure 3(a)
when ET lt ET1198881
They collide at ET = ET1198881
and are substitut-ed by a visible tangent point1198641
119879 as shown in Figure 3(c) when
ET gt ET1198881
Similarly another boundary node bifurcation of Filippov
system (3) occurs at ET1198882
= 46667 (see Figure 4(b)) whereET1198882
is determined by
ET1198882
=119896 (120573119887 (119903 minus 119902
1) + 120575 minus 119902
2minus 120573119886)
120573119887119903 (44)
A stable regular equilibrium 1198642
119877and a visible tangent
point 1198642119879coexist as shown in Figure 4(a) when ET lt ET
1198882
They collide with a pseudoequilibrium 119864
119875at ET = ET
1198882
andare substituted by an invisible tangent point 1198642
119879as shown in
Figure 4(c) when ET gt ET1198882
53 Global Sliding Bifurcation Global sliding bifurcationsinvolve nonvanishing cycles which include sliding discon-nection touching (or grazing) bifurcation buckling bifurca-tion crossing bifurcation bifurcation of a sliding homoclinicorbit and heteroclinic orbit [22] Touching (or grazing) bifur-cation implies that a positive period solution can collide withthe sliding segments From the work of Kuang and BerettaandHsu et al [2 7] they concluded that when the value of theparameter 119886 passed slightly through the bifurcation value astable limit cycle bifurcates from the unstable positive interiorequilibrium for the system (1) According to numericalsimulations the system (3) just has touching bifurcation Forexample if we choose ET as bifurcation parameter and fixall other parameters as shown in Figure 5 when the valueof parameter ET varies touching bifurcation occurs at thecritical value ET
1198883
= 259 for the system (3) Note thatextensive numerical simulations indicate that nomatter whatthe value of ET is in touching bifurcation the whole periodicsolution lies in the region 119878
1(shown in Figure 5)This implies
that the density of pest can be successfully maintained belowET by designing suitable threshold policy strategies So ourcontrol objective can be fully realized which is a desiredsituation in crop livestock sectors and forestry
6 Discussion
Recently the threshold policy and IPM strategies haveattracted great attention in agriculture forestry animal hus-bandry and so on [10ndash12] In the process of pest controlIPM strategies would be used only when the density ofpest reaches or exceeds the ET In addition Filippov systemprovides a natural and rational framework for those realworld problems so it has been widely used in different fields
8 Abstract and Applied Analysis
0 5 10 150
1
2
3
4
5
6
7
8
9
10
11
x
y
E2V
E1V
EP
(a) ET = 83
0 5 10 150
1
2
3
4
5
6
7
8
9
10
11
x
y
E2V
E1B(EP)
(b) ET = 93333
0 5 10 150
1
2
3
4
5
6
7
8
9
10
11E2V
E1R
x
y
(c) ET = 98
Figure 3 Boundary node bifurcation for Filippov system (3) The parameter values are fixed as follows 119886 = 07 119887 = 06 119903 = 15 119896 = 12120573 = 05 120575 = 025 119902
1= 025 and 119902
2= 01
0 5 10
5
10
15
20
25
30
35
40
45
E1V
E2R
x
y
(a) ET = 4
0
5
10
15
20
25
30
35
0 5 10x
yE2B(EP)
E1V
(b) ET = 46667
0
5
10
15
20
25
0 5 10x
yE2V
E1V
EP
(c) ET = 58
Figure 4 Boundary node bifurcation for Filippov system (3) The parameter values are fixed as follows 119886 = 07 119887 = 06 119903 = 15 119896 = 12120573 = 05 120575 = 025 119902
1= 025 and 119902
2= 01
such as in science and engineering [19ndash21 23] In this paperwe employ the sliding analysis of Filippov system to describeand investigate the long term dynamical behavior of theratio-dependent-type prey-predator model with Michaelis-Menten-type functional response so that we can use theFilippov system to model intervention of pest control policyFirstly we investigate the sliding mode domain sliding modedynamics Secondly the null-isoclines and the existence of
four types of equilibria for Filippov system including regularvirtual boundary and pseudoequilibrium and the tangentpoints are discussed in detail Moreover the stability ofpseudoequilibrium is also studied Thirdly we have investi-gated the local sliding bifurcations including regularvirtualequilibrium bifurcation and boundary node bifurcationsFurther global touching bifurcation is also studied by numer-ical techniques
Abstract and Applied Analysis 9
0 1 2 30
5
10
15
x
y
E1R
(a) ET = 15
0 2 40
5
10
15
x
y
E1R
(b) ET = 259
0 2 4 60
5
10
15
x
y
E1R
(c) ET = 35
Figure 5 Globally touching bifurcation for Filippov system (3) The parameter values are fixed as follows 119886 = 051 119887 = 04 119903 = 12 119896 = 6120573 = 06863 120575 = 005 119902
1= 04 and 119902
2= 01
In the process of pest control we should apply all kindsof control strategies so as to prevent multiple pest outbreaksor make sure that the density of pest stabilizes at a desiredlevel of ET In order to realize these goals on the one handwe can determine a set of parameters such that not only all theequilibria of subsystems 119878
1and 1198782are virtual equilibria which
ensures that the pseudoequilibria exist but also the pseudoe-quilibrium is globally stable From Theorem 6 we providedthe conditions of the existence of pseudoequilibrium 119864
1198752
forthe system (3) and showed that it is locally stable if it existsFurther we showed that the local stability of pseudoequilib-rium 119864
1198752
implies global stability by numerical simulation Onthe other hand globally touching bifurcation indicates thatthe density of pest can be successfully maintained below theET by designing suitable threshold policy strategies (shownin Figure 5) Therefore our control objective can be achievedfully in the above two cases which can be used to pest controlin crop livestock sectors and forestry
Although impulsive prey-predator models with the ratio-dependent- or Michaelis-Menten-type response functionhave been studied in [13ndash16] the Filippov systems have manyadvantages in describing interventions including sprayingpesticides and releasing natural enemies compared withimpulsive models In this work the number of natural enemyto be released is proportional to its number It is interestingthat releasing number of natural enemy can be described byconstant independently of the existing numbers of pest andnatural enemy Moreover in practice considering environ-mental energy resource finiteness we should choose the totalnumber of both populations as a guide to switch the systemwhich is called the weighted escapement policy (WEP) [1930ndash32] Therefore in the future work we will focus on theabove two cases which could result in richer dynamics
Appendix
Methods for Analysis of the Sliding Solution
Filippov Convex Method The Filippov method associatesthe following convex combination 119865
119878(119885) of the two vectors
1198651198781
(119885) and 1198651198782
(119885) with each nonsingular sliding point 119885 isin
Σ2 that is
119865119878(119885) = (1 minus 120582 (119885)) 119865
1198781(119885) + 120582 (119885) 119865
1198782(119885) (A1)
where
120582 (119885) =⟨119867119885 1198651198781
⟩
⟨119867119885 1198651198781
minus F1198782
⟩ (A2)
119865119878(119885) is tangent to Σ
2and 0 le 120582(119885) le 1
Thus the sliding mode dynamics can be determined by
= 119865119878(119885) 119885 isin Σ
2 (A3)
which is smooth on a one-dimensional sliding interval of Σ2
The solution of the above equation is the sliding solutionThe equation 120582(119885) = 0 (or 120582(119885) = 1) indicates that the
flow is determined by 1198651198781
(or 1198651198782
) aloneTherefore the slidingmode domain can be defined as
Σ119878= 119885 = (119909 119910) isin Σ | 0 le 120582 (119885) le 1 (A4)
which is equivalent to Σ119878= 119885 isin Σ | 120590(119885) le 0 Denote
Σ+
119878= 119885 isin Σ | 120582 (119885) = 1 Σ
minus
119878= 119885 isin Σ | 120582 (119885) = 0
(A5)
10 Abstract and Applied Analysis
by the boundary of the sliding mode domain so the vectorfields are tangent to boundary Σ+
119878or Σminus119878
Utkin Equivalent Control Method It follows from Utkinrsquosworks [18] on sliding mode dynamics along the manifoldΣ we note that Filippov system (3) can be also rewritten asfollows
= 119865 (119885119880119867) (A6)
where the control 119880119867is defined as
119880119867=
0 119867 (119885) lt 0
1198801(119885 119905) 119867 (119885) gt 0
(A7)
where 1198801is a continuous function In the controlled system
(ie system 1198782) the control 119880
119867= 1198801is applied and in the
free system (ie system 1198781) 119880119867= 0 is applied Assume that
a sliding mode exists on manifold Σ that is Σ2is nonempty
Solving the following algebraic equation
=120597119867
120597119885119865 (119885U
119867) = 0 (A8)
with respect to 119880119867
on this manifold gives the solutiondenoted by 119880
lowast(119885 119905) which is referred to as equivalent
control Substituting for 119880119867in system (A6) yields
which determines the sliding mode dynamics of the Filippovsystem (3)
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China (NSFC 11171199) and by the FundamentalResearch Funds for the Central Universities (GK201003001)
References
[1] R M May ldquoLimit cycles in predator-prey communitiesrdquo Sci-ence vol 177 no 4052 pp 900ndash902 1972
[2] Y Kuang and E Beretta ldquoGlobal qualitative analysis of a ratio-dependent predator-prey systemrdquo Journal ofMathematical Biol-ogy vol 36 no 4 pp 389ndash406 1998
[3] W Murdoch C Briggs and R Nisbet Consumer-Resource Dy-namics Princeton University Press New York NY USA 2003
[4] H I Freedman Deterministic Mathematical Models in Popula-tion Ecology vol 57 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1980
[5] G T Skalski and J F Gilliam ldquoFunctional responseswith preda-tor interference viable alternatives to theHolling type IImodelrdquoEcology vol 82 no 11 pp 3083ndash3092 2001
[6] R Arditi and L R Ginzburg ldquoCoupling in predator-preydynamics ratio-dependencerdquo Journal ofTheoretical Biology vol139 no 3 pp 311ndash326 1989
[7] S-B Hsu T-W Hwang and Y Kuang ldquoGlobal analysis ofthe Michaelis-Menten-type ratio-dependent predator-prey sys-temrdquo Journal of Mathematical Biology vol 42 no 6 pp 489ndash506 2001
[8] C Jost O Arino and R Arditi ldquoAbout deterministic extinctionin ratio-dependent predator-prey modelsrdquo Bulletin of Mathe-matical Biology vol 61 no 1 pp 19ndash32 1999
[9] D Xiao and S Ruan ldquoGlobal dynamics of a ratio-dependentpredator-prey systemrdquo Journal of Mathematical Biology vol 43no 3 pp 268ndash290 2001
[10] Z Lu X Chi and L Chen ldquoImpulsive control strategies inbiological control of pesticiderdquo Theoretical Population Biologyvol 64 no 1 pp 39ndash47 2003
[11] S Tang and R A Cheke ldquoModels for integrated pest controland their biological implicationsrdquo Mathematical Biosciencesvol 215 no 1 pp 115ndash125 2008
[12] J C Van Lenteren and J Woets ldquoBiological and integrated pestcontrol in greenhousesrdquo Annual Review of Entomology vol 33pp 239ndash250 1988
[13] B Dai H Su and D Hu ldquoPeriodic solution of a delayed ratio-dependent predator-prey model with monotonic functionalresponse and impulserdquo Nonlinear Analysis Theory Methods ampApplications vol 70 no 1 pp 126ndash134 2009
[14] X Liu G Li and G Luo ldquoPositive periodic solution for a two-species ratio-dependent predator-prey system with time delayand impulserdquo Journal of Mathematical Analysis and Applica-tions vol 325 no 1 pp 715ndash723 2007
[15] G Jiang Q Lu and L Qian ldquoComplex dynamics of a Hollingtype II prey-predator system with state feedback controlrdquoChaos Solitons amp Fractals vol 31 no 2 pp 448ndash461 2007
[16] H Baek and Y Lim ldquoDynamics of an impulsively controlledMichaelis-Menten type predator-prey systemrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 16 no4 pp 2041ndash2053 2011
[17] T Zhao Y Xiao and R J Smith ldquoNon-smooth plant diseasemodels with economic thresholdsrdquo Mathematical Biosciencesvol 241 no 1 pp 34ndash48 2013
[18] V I Utkin Sliding Modes in Control and Optimization Com-munications and Control Engineering Series Springer BerlinGermany 1992
[19] M I S Costa E Kaszkurewicz A Bhaya and L Hsu ldquoAchiev-ing global convergence to an equilibrium population in pred-ator-prey systems by the use of a discontinuous harvestingpolicyrdquo Ecological Modelling vol 128 no 2-3 pp 89ndash99 2000
[20] B L Van De Vrande D H Van Campen and A De KrakerldquoApproximate analysis of dry-friction-induced stick-slip vibra-tions by a smoothing procedurerdquo Nonlinear Dynamics vol 19no 2 pp 157ndash169 1999
[21] S H Doole and S J Hogan ldquoA piecewise linear suspensionbridge model nonlinear dynamics and orbit continuationrdquoDynamics and Stability of Systems vol 11 no 1 pp 19ndash47 1996
[22] Yu A Kuznetsov S Rinaldi and A Gragnani ldquoOne-parameterbifurcations in planar Filippov systemsrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol13 no 8 pp 2157ndash2188 2003
[23] M di Bernardo C J Budd A R Champneys et al ldquoBifurca-tions in nonsmooth dynamical systemsrdquo SIAM Review vol 50no 4 pp 629ndash701 2008
[24] A F Filippov Differential Equations with Discontinuous Right-hand Sides vol 18 of Mathematics and Its Applications (SovietSeries) Kluwer Academic Dordrecht The Netherlands 1988
[25] S Tang J Liang Y Xiao and R A Cheke ldquoSliding bifurcationsof Filippov two stage pest control models with economicthresholdsrdquo SIAM Journal on Applied Mathematics vol 72 no4 pp 1061ndash1080 2012
Abstract and Applied Analysis 11
[26] M I da Silveira Costa and M E M Meza ldquoApplication of athreshold policy in the management of multispecies fisheriesand predator cullingrdquoMathematical Medicine and Biology vol23 no 1 pp 63ndash75 2006
[27] M E Mendoza Meza A Bhaya E Kaszkurewicz and M IDa Silveira Costa ldquoThreshold policies control for predator-preysystems using a control Liapunov function approachrdquoTheoreti-cal Population Biology vol 67 no 4 pp 273ndash284 2005
[28] F Dercole A Gragnani and S Rinaldi ldquoBifurcation analysisof piecewise smooth ecological modelsrdquoTheoretical PopulationBiology vol 72 no 2 pp 197ndash213 2007
[29] W Wang ldquoBackward bifurcation of an epidemic model withtreatmentrdquo Mathematical Biosciences vol 201 no 1-2 pp 58ndash71 2006
[30] I Noy-Meir ldquoStability of grazing systems an application ofpredator-prey graphsrdquo The Journal of Animal Ecology vol 63no 2 pp 459ndash481 1975
[31] R M Colombo and V Krivan ldquoSelective strategies in foodwebsrdquo Mathematical Medicine and Biology vol 10 no 4 pp281ndash291 1993
[32] V Krivan ldquoOptimal foraging and predator-prey dynamicsrdquoTheoretical Population Biology vol 49 no 3 pp 265ndash290 1996
Solving the above equations with respect to 119909 and 119910 yieldstwo tangent points including
1198641
119879= (ET 119903ET (1 minus ET119896)
119886 minus 119887119903 (1 minus ET119896))
1198642
119879= (
ET (119903 (1 minus ET119896) minus 1199021)
119886 + 1199021119887 minus 119887119903 (1 minus ET119896)
)
(40)
If we fix all parameters then the relations among null-isoclines regularvirtual equilibria pseudoequilibrium andsliding segment are provided in Figure 1 Note that virtualequilibria of both systems 119878
1and 119878
2imply the existence of
pseudoequilibrium andwewill prove this general result later
43 The Stability of Pseudoequilibrium In the process ofpest control we should apply all kinds of control strategiesso as to prevent multiple pest outbreaks or make sure thatthe total density of the pest stabilizes at a desired level ofET If the unique positive equilibrium of system 119878
1and
system 1198782is virtual simultaneously then the sliding flow
has a unique pseudoequilibrium In order to realize thisgoal we can choose a set of parameters such that all theequilibria of subsystems 119878
1and 1198782are virtual equilibria and
the pseudoequilibria are globally stable which have beenwidely used in pest control [26ndash28] For example if wefixed all parameter values as those in Figure 2 then bothvirtual equilibria coexist Therefore we address the stabilityof pseudoequilibrium119864
1198752
in the followingwhich is importantto control pest
Theorem 6 Either the inequalities (34) or (35) hold true orthe two virtual equilibria 1198641
119881and 1198642
119881can coexist then Filippov
system (3) contains a positive pseudoequilibrium 1198641198752
Regard-less of which cases would occur the pseudoequilibrium 119864
1198752
islocally stable with respect to sliding mode domain
Proof According to the conditions of Theorem 6 we seethat the inequalities (30) hold true which implies that theinequalities (34) are true Based on the discussions aboutthe existence of pseudoequilibrium in Section 42 we haveconcluded that if inequalities (34) or (35) hold true thenthe system (3) contains a positive pseudoequilibrium 119864
=(1198961199021(120575 minus 120573119886) minus 119903119902
2(119896 minus ET)) (120575119887119896119902
1minus 119903119887119902
2(119896 minus ET) + 119886119896119902
2)
11988611990211198962 (119887120573119902
1+ 1199022)
(41)
Therefore if the inequalities (33) hold true then 119889120601(119910)119889119910 lt 0 and 119910
1198752
is a positive root of (31) Note that theinequalities (34) and (35) indicate the inequalities (33) Thusthe positive pseudoequilibrium 119864
1198752
is locally stale if it exists
Abstract and Applied Analysis 7
It is difficult to directly prove the global stability ofpseudoequilibrium 119864
1198752
in this case Because we cannotemploy the classical Bendixson-Dulac theorem due to thediscontinuity of vector fields However if there is not crossingcycle surrounding sliding segment then pseudoequilibriumis globally stable [29] From the analysis of global bifurcationfor the system (3) in the following section the system just hastouching bifurcation and there is nonexistence of a slidingcycle which surrounds the 119864
1198752
By using the similar methodswe have that pseudoequilibrium 119864
1198752
is globally stable Thatis to say the local stability of pseudoequilibrium 119864
1198752
withrespect to sliding mode domain indicates its global stabilityin the first quadrant (shown in Figures 3(a) 3(b) 4(b) and4(c)) In practice in order to control pest outbreak we shouldchoose the desirable ET at first so that all equilibria of eachsystem such as system 119878
1and system 119878
2become virtual then
pseudoequilibrium not only exists but also is globally stableIn other words the density of pest can be stable at the ETWhen the density of pest reaches or exceeds the ET we shouldcarry out control strategies (eg releasing natural enemyetc) until it falls below the ET In this way our control goalcan be realized fully
5 Equilibria and Sliding Bifurcation Set
51 RegularVirtual Equilibrium Bifurcation According tothe above discussions it is obvious that 119903 and ET are primaryfactors in determining the existence of the above differenttypes of equilibria of the system (3) So we define four curvesabout parameters 119903 and ET as follows
1198711= (119903ET) | 119903 =
120573119886 minus 120575
120573119887
1198712= (119903ET) | 119903 =
120573119886 minus 120575 + 1199022
120573119887+ 1199021
1198713= (119903ET) | ET =
119896 (120573119887119903 + 120575 minus 120573119886)
120573119887119903
1198714= (119903ET) | ET =
119896 (120573119887 (119903 minus 1199021) + 120575 minus 119902
2minus 120573119886)
120573119887119903
(42)
The four curves (ie 1198711 1198712 1198713 and 119871
4) divide the 119903
and ET parameter space into six regions and the existenceor coexistence of regular or virtual equilibria is indicated ineach region The boundary equilibria 1198641
119861and 1198642
119861can appear
on the lines 1198713and 119871
4accordingly In particular it follows
from Figure 2 that the two virtual equilibria 1198641119881and 1198642
119881can
coexist which is very important to pest control However thetwo regular equilibria 1198641
119877and 1198642
119877cannot coexist
52 Boundary Node Bifurcations This type of bifurcationmay occur for Filippov system (3) once equilibria 119864
119875 119864119879 and
119864119877or 119864119879and 119864
119877collide together simultaneously when ET
passes through a critical value In this part we choose ETas bifurcation parameter and all other parameters are fixedas those in Figure 3 Note that once the parameter ET passes
through the first critical value ET1198881
= 93333 the regular equi-librium 119864
1
119877 tangent point 119864
119879 and pseudoequilibrium 119864
119875col-
lide together (see Figure 3(b)) where ET1198881
is determined by
ET1198881
=119896 (120573119887119903 + 120575 minus 120573119886)
120573119887119903 (43)
A virtual equilibrium 1198641
119881 an invisible tangent point 1198641
119879
and a pseudoequilibrium 119864119875coexist as shown in Figure 3(a)
when ET lt ET1198881
They collide at ET = ET1198881
and are substitut-ed by a visible tangent point1198641
119879 as shown in Figure 3(c) when
ET gt ET1198881
Similarly another boundary node bifurcation of Filippov
system (3) occurs at ET1198882
= 46667 (see Figure 4(b)) whereET1198882
is determined by
ET1198882
=119896 (120573119887 (119903 minus 119902
1) + 120575 minus 119902
2minus 120573119886)
120573119887119903 (44)
A stable regular equilibrium 1198642
119877and a visible tangent
point 1198642119879coexist as shown in Figure 4(a) when ET lt ET
1198882
They collide with a pseudoequilibrium 119864
119875at ET = ET
1198882
andare substituted by an invisible tangent point 1198642
119879as shown in
Figure 4(c) when ET gt ET1198882
53 Global Sliding Bifurcation Global sliding bifurcationsinvolve nonvanishing cycles which include sliding discon-nection touching (or grazing) bifurcation buckling bifurca-tion crossing bifurcation bifurcation of a sliding homoclinicorbit and heteroclinic orbit [22] Touching (or grazing) bifur-cation implies that a positive period solution can collide withthe sliding segments From the work of Kuang and BerettaandHsu et al [2 7] they concluded that when the value of theparameter 119886 passed slightly through the bifurcation value astable limit cycle bifurcates from the unstable positive interiorequilibrium for the system (1) According to numericalsimulations the system (3) just has touching bifurcation Forexample if we choose ET as bifurcation parameter and fixall other parameters as shown in Figure 5 when the valueof parameter ET varies touching bifurcation occurs at thecritical value ET
1198883
= 259 for the system (3) Note thatextensive numerical simulations indicate that nomatter whatthe value of ET is in touching bifurcation the whole periodicsolution lies in the region 119878
1(shown in Figure 5)This implies
that the density of pest can be successfully maintained belowET by designing suitable threshold policy strategies So ourcontrol objective can be fully realized which is a desiredsituation in crop livestock sectors and forestry
6 Discussion
Recently the threshold policy and IPM strategies haveattracted great attention in agriculture forestry animal hus-bandry and so on [10ndash12] In the process of pest controlIPM strategies would be used only when the density ofpest reaches or exceeds the ET In addition Filippov systemprovides a natural and rational framework for those realworld problems so it has been widely used in different fields
8 Abstract and Applied Analysis
0 5 10 150
1
2
3
4
5
6
7
8
9
10
11
x
y
E2V
E1V
EP
(a) ET = 83
0 5 10 150
1
2
3
4
5
6
7
8
9
10
11
x
y
E2V
E1B(EP)
(b) ET = 93333
0 5 10 150
1
2
3
4
5
6
7
8
9
10
11E2V
E1R
x
y
(c) ET = 98
Figure 3 Boundary node bifurcation for Filippov system (3) The parameter values are fixed as follows 119886 = 07 119887 = 06 119903 = 15 119896 = 12120573 = 05 120575 = 025 119902
1= 025 and 119902
2= 01
0 5 10
5
10
15
20
25
30
35
40
45
E1V
E2R
x
y
(a) ET = 4
0
5
10
15
20
25
30
35
0 5 10x
yE2B(EP)
E1V
(b) ET = 46667
0
5
10
15
20
25
0 5 10x
yE2V
E1V
EP
(c) ET = 58
Figure 4 Boundary node bifurcation for Filippov system (3) The parameter values are fixed as follows 119886 = 07 119887 = 06 119903 = 15 119896 = 12120573 = 05 120575 = 025 119902
1= 025 and 119902
2= 01
such as in science and engineering [19ndash21 23] In this paperwe employ the sliding analysis of Filippov system to describeand investigate the long term dynamical behavior of theratio-dependent-type prey-predator model with Michaelis-Menten-type functional response so that we can use theFilippov system to model intervention of pest control policyFirstly we investigate the sliding mode domain sliding modedynamics Secondly the null-isoclines and the existence of
four types of equilibria for Filippov system including regularvirtual boundary and pseudoequilibrium and the tangentpoints are discussed in detail Moreover the stability ofpseudoequilibrium is also studied Thirdly we have investi-gated the local sliding bifurcations including regularvirtualequilibrium bifurcation and boundary node bifurcationsFurther global touching bifurcation is also studied by numer-ical techniques
Abstract and Applied Analysis 9
0 1 2 30
5
10
15
x
y
E1R
(a) ET = 15
0 2 40
5
10
15
x
y
E1R
(b) ET = 259
0 2 4 60
5
10
15
x
y
E1R
(c) ET = 35
Figure 5 Globally touching bifurcation for Filippov system (3) The parameter values are fixed as follows 119886 = 051 119887 = 04 119903 = 12 119896 = 6120573 = 06863 120575 = 005 119902
1= 04 and 119902
2= 01
In the process of pest control we should apply all kindsof control strategies so as to prevent multiple pest outbreaksor make sure that the density of pest stabilizes at a desiredlevel of ET In order to realize these goals on the one handwe can determine a set of parameters such that not only all theequilibria of subsystems 119878
1and 1198782are virtual equilibria which
ensures that the pseudoequilibria exist but also the pseudoe-quilibrium is globally stable From Theorem 6 we providedthe conditions of the existence of pseudoequilibrium 119864
1198752
forthe system (3) and showed that it is locally stable if it existsFurther we showed that the local stability of pseudoequilib-rium 119864
1198752
implies global stability by numerical simulation Onthe other hand globally touching bifurcation indicates thatthe density of pest can be successfully maintained below theET by designing suitable threshold policy strategies (shownin Figure 5) Therefore our control objective can be achievedfully in the above two cases which can be used to pest controlin crop livestock sectors and forestry
Although impulsive prey-predator models with the ratio-dependent- or Michaelis-Menten-type response functionhave been studied in [13ndash16] the Filippov systems have manyadvantages in describing interventions including sprayingpesticides and releasing natural enemies compared withimpulsive models In this work the number of natural enemyto be released is proportional to its number It is interestingthat releasing number of natural enemy can be described byconstant independently of the existing numbers of pest andnatural enemy Moreover in practice considering environ-mental energy resource finiteness we should choose the totalnumber of both populations as a guide to switch the systemwhich is called the weighted escapement policy (WEP) [1930ndash32] Therefore in the future work we will focus on theabove two cases which could result in richer dynamics
Appendix
Methods for Analysis of the Sliding Solution
Filippov Convex Method The Filippov method associatesthe following convex combination 119865
119878(119885) of the two vectors
1198651198781
(119885) and 1198651198782
(119885) with each nonsingular sliding point 119885 isin
Σ2 that is
119865119878(119885) = (1 minus 120582 (119885)) 119865
1198781(119885) + 120582 (119885) 119865
1198782(119885) (A1)
where
120582 (119885) =⟨119867119885 1198651198781
⟩
⟨119867119885 1198651198781
minus F1198782
⟩ (A2)
119865119878(119885) is tangent to Σ
2and 0 le 120582(119885) le 1
Thus the sliding mode dynamics can be determined by
= 119865119878(119885) 119885 isin Σ
2 (A3)
which is smooth on a one-dimensional sliding interval of Σ2
The solution of the above equation is the sliding solutionThe equation 120582(119885) = 0 (or 120582(119885) = 1) indicates that the
flow is determined by 1198651198781
(or 1198651198782
) aloneTherefore the slidingmode domain can be defined as
Σ119878= 119885 = (119909 119910) isin Σ | 0 le 120582 (119885) le 1 (A4)
which is equivalent to Σ119878= 119885 isin Σ | 120590(119885) le 0 Denote
Σ+
119878= 119885 isin Σ | 120582 (119885) = 1 Σ
minus
119878= 119885 isin Σ | 120582 (119885) = 0
(A5)
10 Abstract and Applied Analysis
by the boundary of the sliding mode domain so the vectorfields are tangent to boundary Σ+
119878or Σminus119878
Utkin Equivalent Control Method It follows from Utkinrsquosworks [18] on sliding mode dynamics along the manifoldΣ we note that Filippov system (3) can be also rewritten asfollows
= 119865 (119885119880119867) (A6)
where the control 119880119867is defined as
119880119867=
0 119867 (119885) lt 0
1198801(119885 119905) 119867 (119885) gt 0
(A7)
where 1198801is a continuous function In the controlled system
(ie system 1198782) the control 119880
119867= 1198801is applied and in the
free system (ie system 1198781) 119880119867= 0 is applied Assume that
a sliding mode exists on manifold Σ that is Σ2is nonempty
Solving the following algebraic equation
=120597119867
120597119885119865 (119885U
119867) = 0 (A8)
with respect to 119880119867
on this manifold gives the solutiondenoted by 119880
lowast(119885 119905) which is referred to as equivalent
control Substituting for 119880119867in system (A6) yields
which determines the sliding mode dynamics of the Filippovsystem (3)
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China (NSFC 11171199) and by the FundamentalResearch Funds for the Central Universities (GK201003001)
References
[1] R M May ldquoLimit cycles in predator-prey communitiesrdquo Sci-ence vol 177 no 4052 pp 900ndash902 1972
[2] Y Kuang and E Beretta ldquoGlobal qualitative analysis of a ratio-dependent predator-prey systemrdquo Journal ofMathematical Biol-ogy vol 36 no 4 pp 389ndash406 1998
[3] W Murdoch C Briggs and R Nisbet Consumer-Resource Dy-namics Princeton University Press New York NY USA 2003
[4] H I Freedman Deterministic Mathematical Models in Popula-tion Ecology vol 57 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1980
[5] G T Skalski and J F Gilliam ldquoFunctional responseswith preda-tor interference viable alternatives to theHolling type IImodelrdquoEcology vol 82 no 11 pp 3083ndash3092 2001
[6] R Arditi and L R Ginzburg ldquoCoupling in predator-preydynamics ratio-dependencerdquo Journal ofTheoretical Biology vol139 no 3 pp 311ndash326 1989
[7] S-B Hsu T-W Hwang and Y Kuang ldquoGlobal analysis ofthe Michaelis-Menten-type ratio-dependent predator-prey sys-temrdquo Journal of Mathematical Biology vol 42 no 6 pp 489ndash506 2001
[8] C Jost O Arino and R Arditi ldquoAbout deterministic extinctionin ratio-dependent predator-prey modelsrdquo Bulletin of Mathe-matical Biology vol 61 no 1 pp 19ndash32 1999
[9] D Xiao and S Ruan ldquoGlobal dynamics of a ratio-dependentpredator-prey systemrdquo Journal of Mathematical Biology vol 43no 3 pp 268ndash290 2001
[10] Z Lu X Chi and L Chen ldquoImpulsive control strategies inbiological control of pesticiderdquo Theoretical Population Biologyvol 64 no 1 pp 39ndash47 2003
[11] S Tang and R A Cheke ldquoModels for integrated pest controland their biological implicationsrdquo Mathematical Biosciencesvol 215 no 1 pp 115ndash125 2008
[12] J C Van Lenteren and J Woets ldquoBiological and integrated pestcontrol in greenhousesrdquo Annual Review of Entomology vol 33pp 239ndash250 1988
[13] B Dai H Su and D Hu ldquoPeriodic solution of a delayed ratio-dependent predator-prey model with monotonic functionalresponse and impulserdquo Nonlinear Analysis Theory Methods ampApplications vol 70 no 1 pp 126ndash134 2009
[14] X Liu G Li and G Luo ldquoPositive periodic solution for a two-species ratio-dependent predator-prey system with time delayand impulserdquo Journal of Mathematical Analysis and Applica-tions vol 325 no 1 pp 715ndash723 2007
[15] G Jiang Q Lu and L Qian ldquoComplex dynamics of a Hollingtype II prey-predator system with state feedback controlrdquoChaos Solitons amp Fractals vol 31 no 2 pp 448ndash461 2007
[16] H Baek and Y Lim ldquoDynamics of an impulsively controlledMichaelis-Menten type predator-prey systemrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 16 no4 pp 2041ndash2053 2011
[17] T Zhao Y Xiao and R J Smith ldquoNon-smooth plant diseasemodels with economic thresholdsrdquo Mathematical Biosciencesvol 241 no 1 pp 34ndash48 2013
[18] V I Utkin Sliding Modes in Control and Optimization Com-munications and Control Engineering Series Springer BerlinGermany 1992
[19] M I S Costa E Kaszkurewicz A Bhaya and L Hsu ldquoAchiev-ing global convergence to an equilibrium population in pred-ator-prey systems by the use of a discontinuous harvestingpolicyrdquo Ecological Modelling vol 128 no 2-3 pp 89ndash99 2000
[20] B L Van De Vrande D H Van Campen and A De KrakerldquoApproximate analysis of dry-friction-induced stick-slip vibra-tions by a smoothing procedurerdquo Nonlinear Dynamics vol 19no 2 pp 157ndash169 1999
[21] S H Doole and S J Hogan ldquoA piecewise linear suspensionbridge model nonlinear dynamics and orbit continuationrdquoDynamics and Stability of Systems vol 11 no 1 pp 19ndash47 1996
[22] Yu A Kuznetsov S Rinaldi and A Gragnani ldquoOne-parameterbifurcations in planar Filippov systemsrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol13 no 8 pp 2157ndash2188 2003
[23] M di Bernardo C J Budd A R Champneys et al ldquoBifurca-tions in nonsmooth dynamical systemsrdquo SIAM Review vol 50no 4 pp 629ndash701 2008
[24] A F Filippov Differential Equations with Discontinuous Right-hand Sides vol 18 of Mathematics and Its Applications (SovietSeries) Kluwer Academic Dordrecht The Netherlands 1988
[25] S Tang J Liang Y Xiao and R A Cheke ldquoSliding bifurcationsof Filippov two stage pest control models with economicthresholdsrdquo SIAM Journal on Applied Mathematics vol 72 no4 pp 1061ndash1080 2012
Abstract and Applied Analysis 11
[26] M I da Silveira Costa and M E M Meza ldquoApplication of athreshold policy in the management of multispecies fisheriesand predator cullingrdquoMathematical Medicine and Biology vol23 no 1 pp 63ndash75 2006
[27] M E Mendoza Meza A Bhaya E Kaszkurewicz and M IDa Silveira Costa ldquoThreshold policies control for predator-preysystems using a control Liapunov function approachrdquoTheoreti-cal Population Biology vol 67 no 4 pp 273ndash284 2005
[28] F Dercole A Gragnani and S Rinaldi ldquoBifurcation analysisof piecewise smooth ecological modelsrdquoTheoretical PopulationBiology vol 72 no 2 pp 197ndash213 2007
[29] W Wang ldquoBackward bifurcation of an epidemic model withtreatmentrdquo Mathematical Biosciences vol 201 no 1-2 pp 58ndash71 2006
[30] I Noy-Meir ldquoStability of grazing systems an application ofpredator-prey graphsrdquo The Journal of Animal Ecology vol 63no 2 pp 459ndash481 1975
[31] R M Colombo and V Krivan ldquoSelective strategies in foodwebsrdquo Mathematical Medicine and Biology vol 10 no 4 pp281ndash291 1993
[32] V Krivan ldquoOptimal foraging and predator-prey dynamicsrdquoTheoretical Population Biology vol 49 no 3 pp 265ndash290 1996
It is difficult to directly prove the global stability ofpseudoequilibrium 119864
1198752
in this case Because we cannotemploy the classical Bendixson-Dulac theorem due to thediscontinuity of vector fields However if there is not crossingcycle surrounding sliding segment then pseudoequilibriumis globally stable [29] From the analysis of global bifurcationfor the system (3) in the following section the system just hastouching bifurcation and there is nonexistence of a slidingcycle which surrounds the 119864
1198752
By using the similar methodswe have that pseudoequilibrium 119864
1198752
is globally stable Thatis to say the local stability of pseudoequilibrium 119864
1198752
withrespect to sliding mode domain indicates its global stabilityin the first quadrant (shown in Figures 3(a) 3(b) 4(b) and4(c)) In practice in order to control pest outbreak we shouldchoose the desirable ET at first so that all equilibria of eachsystem such as system 119878
1and system 119878
2become virtual then
pseudoequilibrium not only exists but also is globally stableIn other words the density of pest can be stable at the ETWhen the density of pest reaches or exceeds the ET we shouldcarry out control strategies (eg releasing natural enemyetc) until it falls below the ET In this way our control goalcan be realized fully
5 Equilibria and Sliding Bifurcation Set
51 RegularVirtual Equilibrium Bifurcation According tothe above discussions it is obvious that 119903 and ET are primaryfactors in determining the existence of the above differenttypes of equilibria of the system (3) So we define four curvesabout parameters 119903 and ET as follows
1198711= (119903ET) | 119903 =
120573119886 minus 120575
120573119887
1198712= (119903ET) | 119903 =
120573119886 minus 120575 + 1199022
120573119887+ 1199021
1198713= (119903ET) | ET =
119896 (120573119887119903 + 120575 minus 120573119886)
120573119887119903
1198714= (119903ET) | ET =
119896 (120573119887 (119903 minus 1199021) + 120575 minus 119902
2minus 120573119886)
120573119887119903
(42)
The four curves (ie 1198711 1198712 1198713 and 119871
4) divide the 119903
and ET parameter space into six regions and the existenceor coexistence of regular or virtual equilibria is indicated ineach region The boundary equilibria 1198641
119861and 1198642
119861can appear
on the lines 1198713and 119871
4accordingly In particular it follows
from Figure 2 that the two virtual equilibria 1198641119881and 1198642
119881can
coexist which is very important to pest control However thetwo regular equilibria 1198641
119877and 1198642
119877cannot coexist
52 Boundary Node Bifurcations This type of bifurcationmay occur for Filippov system (3) once equilibria 119864
119875 119864119879 and
119864119877or 119864119879and 119864
119877collide together simultaneously when ET
passes through a critical value In this part we choose ETas bifurcation parameter and all other parameters are fixedas those in Figure 3 Note that once the parameter ET passes
through the first critical value ET1198881
= 93333 the regular equi-librium 119864
1
119877 tangent point 119864
119879 and pseudoequilibrium 119864
119875col-
lide together (see Figure 3(b)) where ET1198881
is determined by
ET1198881
=119896 (120573119887119903 + 120575 minus 120573119886)
120573119887119903 (43)
A virtual equilibrium 1198641
119881 an invisible tangent point 1198641
119879
and a pseudoequilibrium 119864119875coexist as shown in Figure 3(a)
when ET lt ET1198881
They collide at ET = ET1198881
and are substitut-ed by a visible tangent point1198641
119879 as shown in Figure 3(c) when
ET gt ET1198881
Similarly another boundary node bifurcation of Filippov
system (3) occurs at ET1198882
= 46667 (see Figure 4(b)) whereET1198882
is determined by
ET1198882
=119896 (120573119887 (119903 minus 119902
1) + 120575 minus 119902
2minus 120573119886)
120573119887119903 (44)
A stable regular equilibrium 1198642
119877and a visible tangent
point 1198642119879coexist as shown in Figure 4(a) when ET lt ET
1198882
They collide with a pseudoequilibrium 119864
119875at ET = ET
1198882
andare substituted by an invisible tangent point 1198642
119879as shown in
Figure 4(c) when ET gt ET1198882
53 Global Sliding Bifurcation Global sliding bifurcationsinvolve nonvanishing cycles which include sliding discon-nection touching (or grazing) bifurcation buckling bifurca-tion crossing bifurcation bifurcation of a sliding homoclinicorbit and heteroclinic orbit [22] Touching (or grazing) bifur-cation implies that a positive period solution can collide withthe sliding segments From the work of Kuang and BerettaandHsu et al [2 7] they concluded that when the value of theparameter 119886 passed slightly through the bifurcation value astable limit cycle bifurcates from the unstable positive interiorequilibrium for the system (1) According to numericalsimulations the system (3) just has touching bifurcation Forexample if we choose ET as bifurcation parameter and fixall other parameters as shown in Figure 5 when the valueof parameter ET varies touching bifurcation occurs at thecritical value ET
1198883
= 259 for the system (3) Note thatextensive numerical simulations indicate that nomatter whatthe value of ET is in touching bifurcation the whole periodicsolution lies in the region 119878
1(shown in Figure 5)This implies
that the density of pest can be successfully maintained belowET by designing suitable threshold policy strategies So ourcontrol objective can be fully realized which is a desiredsituation in crop livestock sectors and forestry
6 Discussion
Recently the threshold policy and IPM strategies haveattracted great attention in agriculture forestry animal hus-bandry and so on [10ndash12] In the process of pest controlIPM strategies would be used only when the density ofpest reaches or exceeds the ET In addition Filippov systemprovides a natural and rational framework for those realworld problems so it has been widely used in different fields
8 Abstract and Applied Analysis
0 5 10 150
1
2
3
4
5
6
7
8
9
10
11
x
y
E2V
E1V
EP
(a) ET = 83
0 5 10 150
1
2
3
4
5
6
7
8
9
10
11
x
y
E2V
E1B(EP)
(b) ET = 93333
0 5 10 150
1
2
3
4
5
6
7
8
9
10
11E2V
E1R
x
y
(c) ET = 98
Figure 3 Boundary node bifurcation for Filippov system (3) The parameter values are fixed as follows 119886 = 07 119887 = 06 119903 = 15 119896 = 12120573 = 05 120575 = 025 119902
1= 025 and 119902
2= 01
0 5 10
5
10
15
20
25
30
35
40
45
E1V
E2R
x
y
(a) ET = 4
0
5
10
15
20
25
30
35
0 5 10x
yE2B(EP)
E1V
(b) ET = 46667
0
5
10
15
20
25
0 5 10x
yE2V
E1V
EP
(c) ET = 58
Figure 4 Boundary node bifurcation for Filippov system (3) The parameter values are fixed as follows 119886 = 07 119887 = 06 119903 = 15 119896 = 12120573 = 05 120575 = 025 119902
1= 025 and 119902
2= 01
such as in science and engineering [19ndash21 23] In this paperwe employ the sliding analysis of Filippov system to describeand investigate the long term dynamical behavior of theratio-dependent-type prey-predator model with Michaelis-Menten-type functional response so that we can use theFilippov system to model intervention of pest control policyFirstly we investigate the sliding mode domain sliding modedynamics Secondly the null-isoclines and the existence of
four types of equilibria for Filippov system including regularvirtual boundary and pseudoequilibrium and the tangentpoints are discussed in detail Moreover the stability ofpseudoequilibrium is also studied Thirdly we have investi-gated the local sliding bifurcations including regularvirtualequilibrium bifurcation and boundary node bifurcationsFurther global touching bifurcation is also studied by numer-ical techniques
Abstract and Applied Analysis 9
0 1 2 30
5
10
15
x
y
E1R
(a) ET = 15
0 2 40
5
10
15
x
y
E1R
(b) ET = 259
0 2 4 60
5
10
15
x
y
E1R
(c) ET = 35
Figure 5 Globally touching bifurcation for Filippov system (3) The parameter values are fixed as follows 119886 = 051 119887 = 04 119903 = 12 119896 = 6120573 = 06863 120575 = 005 119902
1= 04 and 119902
2= 01
In the process of pest control we should apply all kindsof control strategies so as to prevent multiple pest outbreaksor make sure that the density of pest stabilizes at a desiredlevel of ET In order to realize these goals on the one handwe can determine a set of parameters such that not only all theequilibria of subsystems 119878
1and 1198782are virtual equilibria which
ensures that the pseudoequilibria exist but also the pseudoe-quilibrium is globally stable From Theorem 6 we providedthe conditions of the existence of pseudoequilibrium 119864
1198752
forthe system (3) and showed that it is locally stable if it existsFurther we showed that the local stability of pseudoequilib-rium 119864
1198752
implies global stability by numerical simulation Onthe other hand globally touching bifurcation indicates thatthe density of pest can be successfully maintained below theET by designing suitable threshold policy strategies (shownin Figure 5) Therefore our control objective can be achievedfully in the above two cases which can be used to pest controlin crop livestock sectors and forestry
Although impulsive prey-predator models with the ratio-dependent- or Michaelis-Menten-type response functionhave been studied in [13ndash16] the Filippov systems have manyadvantages in describing interventions including sprayingpesticides and releasing natural enemies compared withimpulsive models In this work the number of natural enemyto be released is proportional to its number It is interestingthat releasing number of natural enemy can be described byconstant independently of the existing numbers of pest andnatural enemy Moreover in practice considering environ-mental energy resource finiteness we should choose the totalnumber of both populations as a guide to switch the systemwhich is called the weighted escapement policy (WEP) [1930ndash32] Therefore in the future work we will focus on theabove two cases which could result in richer dynamics
Appendix
Methods for Analysis of the Sliding Solution
Filippov Convex Method The Filippov method associatesthe following convex combination 119865
119878(119885) of the two vectors
1198651198781
(119885) and 1198651198782
(119885) with each nonsingular sliding point 119885 isin
Σ2 that is
119865119878(119885) = (1 minus 120582 (119885)) 119865
1198781(119885) + 120582 (119885) 119865
1198782(119885) (A1)
where
120582 (119885) =⟨119867119885 1198651198781
⟩
⟨119867119885 1198651198781
minus F1198782
⟩ (A2)
119865119878(119885) is tangent to Σ
2and 0 le 120582(119885) le 1
Thus the sliding mode dynamics can be determined by
= 119865119878(119885) 119885 isin Σ
2 (A3)
which is smooth on a one-dimensional sliding interval of Σ2
The solution of the above equation is the sliding solutionThe equation 120582(119885) = 0 (or 120582(119885) = 1) indicates that the
flow is determined by 1198651198781
(or 1198651198782
) aloneTherefore the slidingmode domain can be defined as
Σ119878= 119885 = (119909 119910) isin Σ | 0 le 120582 (119885) le 1 (A4)
which is equivalent to Σ119878= 119885 isin Σ | 120590(119885) le 0 Denote
Σ+
119878= 119885 isin Σ | 120582 (119885) = 1 Σ
minus
119878= 119885 isin Σ | 120582 (119885) = 0
(A5)
10 Abstract and Applied Analysis
by the boundary of the sliding mode domain so the vectorfields are tangent to boundary Σ+
119878or Σminus119878
Utkin Equivalent Control Method It follows from Utkinrsquosworks [18] on sliding mode dynamics along the manifoldΣ we note that Filippov system (3) can be also rewritten asfollows
= 119865 (119885119880119867) (A6)
where the control 119880119867is defined as
119880119867=
0 119867 (119885) lt 0
1198801(119885 119905) 119867 (119885) gt 0
(A7)
where 1198801is a continuous function In the controlled system
(ie system 1198782) the control 119880
119867= 1198801is applied and in the
free system (ie system 1198781) 119880119867= 0 is applied Assume that
a sliding mode exists on manifold Σ that is Σ2is nonempty
Solving the following algebraic equation
=120597119867
120597119885119865 (119885U
119867) = 0 (A8)
with respect to 119880119867
on this manifold gives the solutiondenoted by 119880
lowast(119885 119905) which is referred to as equivalent
control Substituting for 119880119867in system (A6) yields
which determines the sliding mode dynamics of the Filippovsystem (3)
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China (NSFC 11171199) and by the FundamentalResearch Funds for the Central Universities (GK201003001)
References
[1] R M May ldquoLimit cycles in predator-prey communitiesrdquo Sci-ence vol 177 no 4052 pp 900ndash902 1972
[2] Y Kuang and E Beretta ldquoGlobal qualitative analysis of a ratio-dependent predator-prey systemrdquo Journal ofMathematical Biol-ogy vol 36 no 4 pp 389ndash406 1998
[3] W Murdoch C Briggs and R Nisbet Consumer-Resource Dy-namics Princeton University Press New York NY USA 2003
[4] H I Freedman Deterministic Mathematical Models in Popula-tion Ecology vol 57 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1980
[5] G T Skalski and J F Gilliam ldquoFunctional responseswith preda-tor interference viable alternatives to theHolling type IImodelrdquoEcology vol 82 no 11 pp 3083ndash3092 2001
[6] R Arditi and L R Ginzburg ldquoCoupling in predator-preydynamics ratio-dependencerdquo Journal ofTheoretical Biology vol139 no 3 pp 311ndash326 1989
[7] S-B Hsu T-W Hwang and Y Kuang ldquoGlobal analysis ofthe Michaelis-Menten-type ratio-dependent predator-prey sys-temrdquo Journal of Mathematical Biology vol 42 no 6 pp 489ndash506 2001
[8] C Jost O Arino and R Arditi ldquoAbout deterministic extinctionin ratio-dependent predator-prey modelsrdquo Bulletin of Mathe-matical Biology vol 61 no 1 pp 19ndash32 1999
[9] D Xiao and S Ruan ldquoGlobal dynamics of a ratio-dependentpredator-prey systemrdquo Journal of Mathematical Biology vol 43no 3 pp 268ndash290 2001
[10] Z Lu X Chi and L Chen ldquoImpulsive control strategies inbiological control of pesticiderdquo Theoretical Population Biologyvol 64 no 1 pp 39ndash47 2003
[11] S Tang and R A Cheke ldquoModels for integrated pest controland their biological implicationsrdquo Mathematical Biosciencesvol 215 no 1 pp 115ndash125 2008
[12] J C Van Lenteren and J Woets ldquoBiological and integrated pestcontrol in greenhousesrdquo Annual Review of Entomology vol 33pp 239ndash250 1988
[13] B Dai H Su and D Hu ldquoPeriodic solution of a delayed ratio-dependent predator-prey model with monotonic functionalresponse and impulserdquo Nonlinear Analysis Theory Methods ampApplications vol 70 no 1 pp 126ndash134 2009
[14] X Liu G Li and G Luo ldquoPositive periodic solution for a two-species ratio-dependent predator-prey system with time delayand impulserdquo Journal of Mathematical Analysis and Applica-tions vol 325 no 1 pp 715ndash723 2007
[15] G Jiang Q Lu and L Qian ldquoComplex dynamics of a Hollingtype II prey-predator system with state feedback controlrdquoChaos Solitons amp Fractals vol 31 no 2 pp 448ndash461 2007
[16] H Baek and Y Lim ldquoDynamics of an impulsively controlledMichaelis-Menten type predator-prey systemrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 16 no4 pp 2041ndash2053 2011
[17] T Zhao Y Xiao and R J Smith ldquoNon-smooth plant diseasemodels with economic thresholdsrdquo Mathematical Biosciencesvol 241 no 1 pp 34ndash48 2013
[18] V I Utkin Sliding Modes in Control and Optimization Com-munications and Control Engineering Series Springer BerlinGermany 1992
[19] M I S Costa E Kaszkurewicz A Bhaya and L Hsu ldquoAchiev-ing global convergence to an equilibrium population in pred-ator-prey systems by the use of a discontinuous harvestingpolicyrdquo Ecological Modelling vol 128 no 2-3 pp 89ndash99 2000
[20] B L Van De Vrande D H Van Campen and A De KrakerldquoApproximate analysis of dry-friction-induced stick-slip vibra-tions by a smoothing procedurerdquo Nonlinear Dynamics vol 19no 2 pp 157ndash169 1999
[21] S H Doole and S J Hogan ldquoA piecewise linear suspensionbridge model nonlinear dynamics and orbit continuationrdquoDynamics and Stability of Systems vol 11 no 1 pp 19ndash47 1996
[22] Yu A Kuznetsov S Rinaldi and A Gragnani ldquoOne-parameterbifurcations in planar Filippov systemsrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol13 no 8 pp 2157ndash2188 2003
[23] M di Bernardo C J Budd A R Champneys et al ldquoBifurca-tions in nonsmooth dynamical systemsrdquo SIAM Review vol 50no 4 pp 629ndash701 2008
[24] A F Filippov Differential Equations with Discontinuous Right-hand Sides vol 18 of Mathematics and Its Applications (SovietSeries) Kluwer Academic Dordrecht The Netherlands 1988
[25] S Tang J Liang Y Xiao and R A Cheke ldquoSliding bifurcationsof Filippov two stage pest control models with economicthresholdsrdquo SIAM Journal on Applied Mathematics vol 72 no4 pp 1061ndash1080 2012
Abstract and Applied Analysis 11
[26] M I da Silveira Costa and M E M Meza ldquoApplication of athreshold policy in the management of multispecies fisheriesand predator cullingrdquoMathematical Medicine and Biology vol23 no 1 pp 63ndash75 2006
[27] M E Mendoza Meza A Bhaya E Kaszkurewicz and M IDa Silveira Costa ldquoThreshold policies control for predator-preysystems using a control Liapunov function approachrdquoTheoreti-cal Population Biology vol 67 no 4 pp 273ndash284 2005
[28] F Dercole A Gragnani and S Rinaldi ldquoBifurcation analysisof piecewise smooth ecological modelsrdquoTheoretical PopulationBiology vol 72 no 2 pp 197ndash213 2007
[29] W Wang ldquoBackward bifurcation of an epidemic model withtreatmentrdquo Mathematical Biosciences vol 201 no 1-2 pp 58ndash71 2006
[30] I Noy-Meir ldquoStability of grazing systems an application ofpredator-prey graphsrdquo The Journal of Animal Ecology vol 63no 2 pp 459ndash481 1975
[31] R M Colombo and V Krivan ldquoSelective strategies in foodwebsrdquo Mathematical Medicine and Biology vol 10 no 4 pp281ndash291 1993
[32] V Krivan ldquoOptimal foraging and predator-prey dynamicsrdquoTheoretical Population Biology vol 49 no 3 pp 265ndash290 1996
Figure 3 Boundary node bifurcation for Filippov system (3) The parameter values are fixed as follows 119886 = 07 119887 = 06 119903 = 15 119896 = 12120573 = 05 120575 = 025 119902
1= 025 and 119902
2= 01
0 5 10
5
10
15
20
25
30
35
40
45
E1V
E2R
x
y
(a) ET = 4
0
5
10
15
20
25
30
35
0 5 10x
yE2B(EP)
E1V
(b) ET = 46667
0
5
10
15
20
25
0 5 10x
yE2V
E1V
EP
(c) ET = 58
Figure 4 Boundary node bifurcation for Filippov system (3) The parameter values are fixed as follows 119886 = 07 119887 = 06 119903 = 15 119896 = 12120573 = 05 120575 = 025 119902
1= 025 and 119902
2= 01
such as in science and engineering [19ndash21 23] In this paperwe employ the sliding analysis of Filippov system to describeand investigate the long term dynamical behavior of theratio-dependent-type prey-predator model with Michaelis-Menten-type functional response so that we can use theFilippov system to model intervention of pest control policyFirstly we investigate the sliding mode domain sliding modedynamics Secondly the null-isoclines and the existence of
four types of equilibria for Filippov system including regularvirtual boundary and pseudoequilibrium and the tangentpoints are discussed in detail Moreover the stability ofpseudoequilibrium is also studied Thirdly we have investi-gated the local sliding bifurcations including regularvirtualequilibrium bifurcation and boundary node bifurcationsFurther global touching bifurcation is also studied by numer-ical techniques
Abstract and Applied Analysis 9
0 1 2 30
5
10
15
x
y
E1R
(a) ET = 15
0 2 40
5
10
15
x
y
E1R
(b) ET = 259
0 2 4 60
5
10
15
x
y
E1R
(c) ET = 35
Figure 5 Globally touching bifurcation for Filippov system (3) The parameter values are fixed as follows 119886 = 051 119887 = 04 119903 = 12 119896 = 6120573 = 06863 120575 = 005 119902
1= 04 and 119902
2= 01
In the process of pest control we should apply all kindsof control strategies so as to prevent multiple pest outbreaksor make sure that the density of pest stabilizes at a desiredlevel of ET In order to realize these goals on the one handwe can determine a set of parameters such that not only all theequilibria of subsystems 119878
1and 1198782are virtual equilibria which
ensures that the pseudoequilibria exist but also the pseudoe-quilibrium is globally stable From Theorem 6 we providedthe conditions of the existence of pseudoequilibrium 119864
1198752
forthe system (3) and showed that it is locally stable if it existsFurther we showed that the local stability of pseudoequilib-rium 119864
1198752
implies global stability by numerical simulation Onthe other hand globally touching bifurcation indicates thatthe density of pest can be successfully maintained below theET by designing suitable threshold policy strategies (shownin Figure 5) Therefore our control objective can be achievedfully in the above two cases which can be used to pest controlin crop livestock sectors and forestry
Although impulsive prey-predator models with the ratio-dependent- or Michaelis-Menten-type response functionhave been studied in [13ndash16] the Filippov systems have manyadvantages in describing interventions including sprayingpesticides and releasing natural enemies compared withimpulsive models In this work the number of natural enemyto be released is proportional to its number It is interestingthat releasing number of natural enemy can be described byconstant independently of the existing numbers of pest andnatural enemy Moreover in practice considering environ-mental energy resource finiteness we should choose the totalnumber of both populations as a guide to switch the systemwhich is called the weighted escapement policy (WEP) [1930ndash32] Therefore in the future work we will focus on theabove two cases which could result in richer dynamics
Appendix
Methods for Analysis of the Sliding Solution
Filippov Convex Method The Filippov method associatesthe following convex combination 119865
119878(119885) of the two vectors
1198651198781
(119885) and 1198651198782
(119885) with each nonsingular sliding point 119885 isin
Σ2 that is
119865119878(119885) = (1 minus 120582 (119885)) 119865
1198781(119885) + 120582 (119885) 119865
1198782(119885) (A1)
where
120582 (119885) =⟨119867119885 1198651198781
⟩
⟨119867119885 1198651198781
minus F1198782
⟩ (A2)
119865119878(119885) is tangent to Σ
2and 0 le 120582(119885) le 1
Thus the sliding mode dynamics can be determined by
= 119865119878(119885) 119885 isin Σ
2 (A3)
which is smooth on a one-dimensional sliding interval of Σ2
The solution of the above equation is the sliding solutionThe equation 120582(119885) = 0 (or 120582(119885) = 1) indicates that the
flow is determined by 1198651198781
(or 1198651198782
) aloneTherefore the slidingmode domain can be defined as
Σ119878= 119885 = (119909 119910) isin Σ | 0 le 120582 (119885) le 1 (A4)
which is equivalent to Σ119878= 119885 isin Σ | 120590(119885) le 0 Denote
Σ+
119878= 119885 isin Σ | 120582 (119885) = 1 Σ
minus
119878= 119885 isin Σ | 120582 (119885) = 0
(A5)
10 Abstract and Applied Analysis
by the boundary of the sliding mode domain so the vectorfields are tangent to boundary Σ+
119878or Σminus119878
Utkin Equivalent Control Method It follows from Utkinrsquosworks [18] on sliding mode dynamics along the manifoldΣ we note that Filippov system (3) can be also rewritten asfollows
= 119865 (119885119880119867) (A6)
where the control 119880119867is defined as
119880119867=
0 119867 (119885) lt 0
1198801(119885 119905) 119867 (119885) gt 0
(A7)
where 1198801is a continuous function In the controlled system
(ie system 1198782) the control 119880
119867= 1198801is applied and in the
free system (ie system 1198781) 119880119867= 0 is applied Assume that
a sliding mode exists on manifold Σ that is Σ2is nonempty
Solving the following algebraic equation
=120597119867
120597119885119865 (119885U
119867) = 0 (A8)
with respect to 119880119867
on this manifold gives the solutiondenoted by 119880
lowast(119885 119905) which is referred to as equivalent
control Substituting for 119880119867in system (A6) yields
which determines the sliding mode dynamics of the Filippovsystem (3)
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China (NSFC 11171199) and by the FundamentalResearch Funds for the Central Universities (GK201003001)
References
[1] R M May ldquoLimit cycles in predator-prey communitiesrdquo Sci-ence vol 177 no 4052 pp 900ndash902 1972
[2] Y Kuang and E Beretta ldquoGlobal qualitative analysis of a ratio-dependent predator-prey systemrdquo Journal ofMathematical Biol-ogy vol 36 no 4 pp 389ndash406 1998
[3] W Murdoch C Briggs and R Nisbet Consumer-Resource Dy-namics Princeton University Press New York NY USA 2003
[4] H I Freedman Deterministic Mathematical Models in Popula-tion Ecology vol 57 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1980
[5] G T Skalski and J F Gilliam ldquoFunctional responseswith preda-tor interference viable alternatives to theHolling type IImodelrdquoEcology vol 82 no 11 pp 3083ndash3092 2001
[6] R Arditi and L R Ginzburg ldquoCoupling in predator-preydynamics ratio-dependencerdquo Journal ofTheoretical Biology vol139 no 3 pp 311ndash326 1989
[7] S-B Hsu T-W Hwang and Y Kuang ldquoGlobal analysis ofthe Michaelis-Menten-type ratio-dependent predator-prey sys-temrdquo Journal of Mathematical Biology vol 42 no 6 pp 489ndash506 2001
[8] C Jost O Arino and R Arditi ldquoAbout deterministic extinctionin ratio-dependent predator-prey modelsrdquo Bulletin of Mathe-matical Biology vol 61 no 1 pp 19ndash32 1999
[9] D Xiao and S Ruan ldquoGlobal dynamics of a ratio-dependentpredator-prey systemrdquo Journal of Mathematical Biology vol 43no 3 pp 268ndash290 2001
[10] Z Lu X Chi and L Chen ldquoImpulsive control strategies inbiological control of pesticiderdquo Theoretical Population Biologyvol 64 no 1 pp 39ndash47 2003
[11] S Tang and R A Cheke ldquoModels for integrated pest controland their biological implicationsrdquo Mathematical Biosciencesvol 215 no 1 pp 115ndash125 2008
[12] J C Van Lenteren and J Woets ldquoBiological and integrated pestcontrol in greenhousesrdquo Annual Review of Entomology vol 33pp 239ndash250 1988
[13] B Dai H Su and D Hu ldquoPeriodic solution of a delayed ratio-dependent predator-prey model with monotonic functionalresponse and impulserdquo Nonlinear Analysis Theory Methods ampApplications vol 70 no 1 pp 126ndash134 2009
[14] X Liu G Li and G Luo ldquoPositive periodic solution for a two-species ratio-dependent predator-prey system with time delayand impulserdquo Journal of Mathematical Analysis and Applica-tions vol 325 no 1 pp 715ndash723 2007
[15] G Jiang Q Lu and L Qian ldquoComplex dynamics of a Hollingtype II prey-predator system with state feedback controlrdquoChaos Solitons amp Fractals vol 31 no 2 pp 448ndash461 2007
[16] H Baek and Y Lim ldquoDynamics of an impulsively controlledMichaelis-Menten type predator-prey systemrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 16 no4 pp 2041ndash2053 2011
[17] T Zhao Y Xiao and R J Smith ldquoNon-smooth plant diseasemodels with economic thresholdsrdquo Mathematical Biosciencesvol 241 no 1 pp 34ndash48 2013
[18] V I Utkin Sliding Modes in Control and Optimization Com-munications and Control Engineering Series Springer BerlinGermany 1992
[19] M I S Costa E Kaszkurewicz A Bhaya and L Hsu ldquoAchiev-ing global convergence to an equilibrium population in pred-ator-prey systems by the use of a discontinuous harvestingpolicyrdquo Ecological Modelling vol 128 no 2-3 pp 89ndash99 2000
[20] B L Van De Vrande D H Van Campen and A De KrakerldquoApproximate analysis of dry-friction-induced stick-slip vibra-tions by a smoothing procedurerdquo Nonlinear Dynamics vol 19no 2 pp 157ndash169 1999
[21] S H Doole and S J Hogan ldquoA piecewise linear suspensionbridge model nonlinear dynamics and orbit continuationrdquoDynamics and Stability of Systems vol 11 no 1 pp 19ndash47 1996
[22] Yu A Kuznetsov S Rinaldi and A Gragnani ldquoOne-parameterbifurcations in planar Filippov systemsrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol13 no 8 pp 2157ndash2188 2003
[23] M di Bernardo C J Budd A R Champneys et al ldquoBifurca-tions in nonsmooth dynamical systemsrdquo SIAM Review vol 50no 4 pp 629ndash701 2008
[24] A F Filippov Differential Equations with Discontinuous Right-hand Sides vol 18 of Mathematics and Its Applications (SovietSeries) Kluwer Academic Dordrecht The Netherlands 1988
[25] S Tang J Liang Y Xiao and R A Cheke ldquoSliding bifurcationsof Filippov two stage pest control models with economicthresholdsrdquo SIAM Journal on Applied Mathematics vol 72 no4 pp 1061ndash1080 2012
Abstract and Applied Analysis 11
[26] M I da Silveira Costa and M E M Meza ldquoApplication of athreshold policy in the management of multispecies fisheriesand predator cullingrdquoMathematical Medicine and Biology vol23 no 1 pp 63ndash75 2006
[27] M E Mendoza Meza A Bhaya E Kaszkurewicz and M IDa Silveira Costa ldquoThreshold policies control for predator-preysystems using a control Liapunov function approachrdquoTheoreti-cal Population Biology vol 67 no 4 pp 273ndash284 2005
[28] F Dercole A Gragnani and S Rinaldi ldquoBifurcation analysisof piecewise smooth ecological modelsrdquoTheoretical PopulationBiology vol 72 no 2 pp 197ndash213 2007
[29] W Wang ldquoBackward bifurcation of an epidemic model withtreatmentrdquo Mathematical Biosciences vol 201 no 1-2 pp 58ndash71 2006
[30] I Noy-Meir ldquoStability of grazing systems an application ofpredator-prey graphsrdquo The Journal of Animal Ecology vol 63no 2 pp 459ndash481 1975
[31] R M Colombo and V Krivan ldquoSelective strategies in foodwebsrdquo Mathematical Medicine and Biology vol 10 no 4 pp281ndash291 1993
[32] V Krivan ldquoOptimal foraging and predator-prey dynamicsrdquoTheoretical Population Biology vol 49 no 3 pp 265ndash290 1996
Figure 5 Globally touching bifurcation for Filippov system (3) The parameter values are fixed as follows 119886 = 051 119887 = 04 119903 = 12 119896 = 6120573 = 06863 120575 = 005 119902
1= 04 and 119902
2= 01
In the process of pest control we should apply all kindsof control strategies so as to prevent multiple pest outbreaksor make sure that the density of pest stabilizes at a desiredlevel of ET In order to realize these goals on the one handwe can determine a set of parameters such that not only all theequilibria of subsystems 119878
1and 1198782are virtual equilibria which
ensures that the pseudoequilibria exist but also the pseudoe-quilibrium is globally stable From Theorem 6 we providedthe conditions of the existence of pseudoequilibrium 119864
1198752
forthe system (3) and showed that it is locally stable if it existsFurther we showed that the local stability of pseudoequilib-rium 119864
1198752
implies global stability by numerical simulation Onthe other hand globally touching bifurcation indicates thatthe density of pest can be successfully maintained below theET by designing suitable threshold policy strategies (shownin Figure 5) Therefore our control objective can be achievedfully in the above two cases which can be used to pest controlin crop livestock sectors and forestry
Although impulsive prey-predator models with the ratio-dependent- or Michaelis-Menten-type response functionhave been studied in [13ndash16] the Filippov systems have manyadvantages in describing interventions including sprayingpesticides and releasing natural enemies compared withimpulsive models In this work the number of natural enemyto be released is proportional to its number It is interestingthat releasing number of natural enemy can be described byconstant independently of the existing numbers of pest andnatural enemy Moreover in practice considering environ-mental energy resource finiteness we should choose the totalnumber of both populations as a guide to switch the systemwhich is called the weighted escapement policy (WEP) [1930ndash32] Therefore in the future work we will focus on theabove two cases which could result in richer dynamics
Appendix
Methods for Analysis of the Sliding Solution
Filippov Convex Method The Filippov method associatesthe following convex combination 119865
119878(119885) of the two vectors
1198651198781
(119885) and 1198651198782
(119885) with each nonsingular sliding point 119885 isin
Σ2 that is
119865119878(119885) = (1 minus 120582 (119885)) 119865
1198781(119885) + 120582 (119885) 119865
1198782(119885) (A1)
where
120582 (119885) =⟨119867119885 1198651198781
⟩
⟨119867119885 1198651198781
minus F1198782
⟩ (A2)
119865119878(119885) is tangent to Σ
2and 0 le 120582(119885) le 1
Thus the sliding mode dynamics can be determined by
= 119865119878(119885) 119885 isin Σ
2 (A3)
which is smooth on a one-dimensional sliding interval of Σ2
The solution of the above equation is the sliding solutionThe equation 120582(119885) = 0 (or 120582(119885) = 1) indicates that the
flow is determined by 1198651198781
(or 1198651198782
) aloneTherefore the slidingmode domain can be defined as
Σ119878= 119885 = (119909 119910) isin Σ | 0 le 120582 (119885) le 1 (A4)
which is equivalent to Σ119878= 119885 isin Σ | 120590(119885) le 0 Denote
Σ+
119878= 119885 isin Σ | 120582 (119885) = 1 Σ
minus
119878= 119885 isin Σ | 120582 (119885) = 0
(A5)
10 Abstract and Applied Analysis
by the boundary of the sliding mode domain so the vectorfields are tangent to boundary Σ+
119878or Σminus119878
Utkin Equivalent Control Method It follows from Utkinrsquosworks [18] on sliding mode dynamics along the manifoldΣ we note that Filippov system (3) can be also rewritten asfollows
= 119865 (119885119880119867) (A6)
where the control 119880119867is defined as
119880119867=
0 119867 (119885) lt 0
1198801(119885 119905) 119867 (119885) gt 0
(A7)
where 1198801is a continuous function In the controlled system
(ie system 1198782) the control 119880
119867= 1198801is applied and in the
free system (ie system 1198781) 119880119867= 0 is applied Assume that
a sliding mode exists on manifold Σ that is Σ2is nonempty
Solving the following algebraic equation
=120597119867
120597119885119865 (119885U
119867) = 0 (A8)
with respect to 119880119867
on this manifold gives the solutiondenoted by 119880
lowast(119885 119905) which is referred to as equivalent
control Substituting for 119880119867in system (A6) yields
which determines the sliding mode dynamics of the Filippovsystem (3)
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China (NSFC 11171199) and by the FundamentalResearch Funds for the Central Universities (GK201003001)
References
[1] R M May ldquoLimit cycles in predator-prey communitiesrdquo Sci-ence vol 177 no 4052 pp 900ndash902 1972
[2] Y Kuang and E Beretta ldquoGlobal qualitative analysis of a ratio-dependent predator-prey systemrdquo Journal ofMathematical Biol-ogy vol 36 no 4 pp 389ndash406 1998
[3] W Murdoch C Briggs and R Nisbet Consumer-Resource Dy-namics Princeton University Press New York NY USA 2003
[4] H I Freedman Deterministic Mathematical Models in Popula-tion Ecology vol 57 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1980
[5] G T Skalski and J F Gilliam ldquoFunctional responseswith preda-tor interference viable alternatives to theHolling type IImodelrdquoEcology vol 82 no 11 pp 3083ndash3092 2001
[6] R Arditi and L R Ginzburg ldquoCoupling in predator-preydynamics ratio-dependencerdquo Journal ofTheoretical Biology vol139 no 3 pp 311ndash326 1989
[7] S-B Hsu T-W Hwang and Y Kuang ldquoGlobal analysis ofthe Michaelis-Menten-type ratio-dependent predator-prey sys-temrdquo Journal of Mathematical Biology vol 42 no 6 pp 489ndash506 2001
[8] C Jost O Arino and R Arditi ldquoAbout deterministic extinctionin ratio-dependent predator-prey modelsrdquo Bulletin of Mathe-matical Biology vol 61 no 1 pp 19ndash32 1999
[9] D Xiao and S Ruan ldquoGlobal dynamics of a ratio-dependentpredator-prey systemrdquo Journal of Mathematical Biology vol 43no 3 pp 268ndash290 2001
[10] Z Lu X Chi and L Chen ldquoImpulsive control strategies inbiological control of pesticiderdquo Theoretical Population Biologyvol 64 no 1 pp 39ndash47 2003
[11] S Tang and R A Cheke ldquoModels for integrated pest controland their biological implicationsrdquo Mathematical Biosciencesvol 215 no 1 pp 115ndash125 2008
[12] J C Van Lenteren and J Woets ldquoBiological and integrated pestcontrol in greenhousesrdquo Annual Review of Entomology vol 33pp 239ndash250 1988
[13] B Dai H Su and D Hu ldquoPeriodic solution of a delayed ratio-dependent predator-prey model with monotonic functionalresponse and impulserdquo Nonlinear Analysis Theory Methods ampApplications vol 70 no 1 pp 126ndash134 2009
[14] X Liu G Li and G Luo ldquoPositive periodic solution for a two-species ratio-dependent predator-prey system with time delayand impulserdquo Journal of Mathematical Analysis and Applica-tions vol 325 no 1 pp 715ndash723 2007
[15] G Jiang Q Lu and L Qian ldquoComplex dynamics of a Hollingtype II prey-predator system with state feedback controlrdquoChaos Solitons amp Fractals vol 31 no 2 pp 448ndash461 2007
[16] H Baek and Y Lim ldquoDynamics of an impulsively controlledMichaelis-Menten type predator-prey systemrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 16 no4 pp 2041ndash2053 2011
[17] T Zhao Y Xiao and R J Smith ldquoNon-smooth plant diseasemodels with economic thresholdsrdquo Mathematical Biosciencesvol 241 no 1 pp 34ndash48 2013
[18] V I Utkin Sliding Modes in Control and Optimization Com-munications and Control Engineering Series Springer BerlinGermany 1992
[19] M I S Costa E Kaszkurewicz A Bhaya and L Hsu ldquoAchiev-ing global convergence to an equilibrium population in pred-ator-prey systems by the use of a discontinuous harvestingpolicyrdquo Ecological Modelling vol 128 no 2-3 pp 89ndash99 2000
[20] B L Van De Vrande D H Van Campen and A De KrakerldquoApproximate analysis of dry-friction-induced stick-slip vibra-tions by a smoothing procedurerdquo Nonlinear Dynamics vol 19no 2 pp 157ndash169 1999
[21] S H Doole and S J Hogan ldquoA piecewise linear suspensionbridge model nonlinear dynamics and orbit continuationrdquoDynamics and Stability of Systems vol 11 no 1 pp 19ndash47 1996
[22] Yu A Kuznetsov S Rinaldi and A Gragnani ldquoOne-parameterbifurcations in planar Filippov systemsrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol13 no 8 pp 2157ndash2188 2003
[23] M di Bernardo C J Budd A R Champneys et al ldquoBifurca-tions in nonsmooth dynamical systemsrdquo SIAM Review vol 50no 4 pp 629ndash701 2008
[24] A F Filippov Differential Equations with Discontinuous Right-hand Sides vol 18 of Mathematics and Its Applications (SovietSeries) Kluwer Academic Dordrecht The Netherlands 1988
[25] S Tang J Liang Y Xiao and R A Cheke ldquoSliding bifurcationsof Filippov two stage pest control models with economicthresholdsrdquo SIAM Journal on Applied Mathematics vol 72 no4 pp 1061ndash1080 2012
Abstract and Applied Analysis 11
[26] M I da Silveira Costa and M E M Meza ldquoApplication of athreshold policy in the management of multispecies fisheriesand predator cullingrdquoMathematical Medicine and Biology vol23 no 1 pp 63ndash75 2006
[27] M E Mendoza Meza A Bhaya E Kaszkurewicz and M IDa Silveira Costa ldquoThreshold policies control for predator-preysystems using a control Liapunov function approachrdquoTheoreti-cal Population Biology vol 67 no 4 pp 273ndash284 2005
[28] F Dercole A Gragnani and S Rinaldi ldquoBifurcation analysisof piecewise smooth ecological modelsrdquoTheoretical PopulationBiology vol 72 no 2 pp 197ndash213 2007
[29] W Wang ldquoBackward bifurcation of an epidemic model withtreatmentrdquo Mathematical Biosciences vol 201 no 1-2 pp 58ndash71 2006
[30] I Noy-Meir ldquoStability of grazing systems an application ofpredator-prey graphsrdquo The Journal of Animal Ecology vol 63no 2 pp 459ndash481 1975
[31] R M Colombo and V Krivan ldquoSelective strategies in foodwebsrdquo Mathematical Medicine and Biology vol 10 no 4 pp281ndash291 1993
[32] V Krivan ldquoOptimal foraging and predator-prey dynamicsrdquoTheoretical Population Biology vol 49 no 3 pp 265ndash290 1996
by the boundary of the sliding mode domain so the vectorfields are tangent to boundary Σ+
119878or Σminus119878
Utkin Equivalent Control Method It follows from Utkinrsquosworks [18] on sliding mode dynamics along the manifoldΣ we note that Filippov system (3) can be also rewritten asfollows
= 119865 (119885119880119867) (A6)
where the control 119880119867is defined as
119880119867=
0 119867 (119885) lt 0
1198801(119885 119905) 119867 (119885) gt 0
(A7)
where 1198801is a continuous function In the controlled system
(ie system 1198782) the control 119880
119867= 1198801is applied and in the
free system (ie system 1198781) 119880119867= 0 is applied Assume that
a sliding mode exists on manifold Σ that is Σ2is nonempty
Solving the following algebraic equation
=120597119867
120597119885119865 (119885U
119867) = 0 (A8)
with respect to 119880119867
on this manifold gives the solutiondenoted by 119880
lowast(119885 119905) which is referred to as equivalent
control Substituting for 119880119867in system (A6) yields
which determines the sliding mode dynamics of the Filippovsystem (3)
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China (NSFC 11171199) and by the FundamentalResearch Funds for the Central Universities (GK201003001)
References
[1] R M May ldquoLimit cycles in predator-prey communitiesrdquo Sci-ence vol 177 no 4052 pp 900ndash902 1972
[2] Y Kuang and E Beretta ldquoGlobal qualitative analysis of a ratio-dependent predator-prey systemrdquo Journal ofMathematical Biol-ogy vol 36 no 4 pp 389ndash406 1998
[3] W Murdoch C Briggs and R Nisbet Consumer-Resource Dy-namics Princeton University Press New York NY USA 2003
[4] H I Freedman Deterministic Mathematical Models in Popula-tion Ecology vol 57 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1980
[5] G T Skalski and J F Gilliam ldquoFunctional responseswith preda-tor interference viable alternatives to theHolling type IImodelrdquoEcology vol 82 no 11 pp 3083ndash3092 2001
[6] R Arditi and L R Ginzburg ldquoCoupling in predator-preydynamics ratio-dependencerdquo Journal ofTheoretical Biology vol139 no 3 pp 311ndash326 1989
[7] S-B Hsu T-W Hwang and Y Kuang ldquoGlobal analysis ofthe Michaelis-Menten-type ratio-dependent predator-prey sys-temrdquo Journal of Mathematical Biology vol 42 no 6 pp 489ndash506 2001
[8] C Jost O Arino and R Arditi ldquoAbout deterministic extinctionin ratio-dependent predator-prey modelsrdquo Bulletin of Mathe-matical Biology vol 61 no 1 pp 19ndash32 1999
[9] D Xiao and S Ruan ldquoGlobal dynamics of a ratio-dependentpredator-prey systemrdquo Journal of Mathematical Biology vol 43no 3 pp 268ndash290 2001
[10] Z Lu X Chi and L Chen ldquoImpulsive control strategies inbiological control of pesticiderdquo Theoretical Population Biologyvol 64 no 1 pp 39ndash47 2003
[11] S Tang and R A Cheke ldquoModels for integrated pest controland their biological implicationsrdquo Mathematical Biosciencesvol 215 no 1 pp 115ndash125 2008
[12] J C Van Lenteren and J Woets ldquoBiological and integrated pestcontrol in greenhousesrdquo Annual Review of Entomology vol 33pp 239ndash250 1988
[13] B Dai H Su and D Hu ldquoPeriodic solution of a delayed ratio-dependent predator-prey model with monotonic functionalresponse and impulserdquo Nonlinear Analysis Theory Methods ampApplications vol 70 no 1 pp 126ndash134 2009
[14] X Liu G Li and G Luo ldquoPositive periodic solution for a two-species ratio-dependent predator-prey system with time delayand impulserdquo Journal of Mathematical Analysis and Applica-tions vol 325 no 1 pp 715ndash723 2007
[15] G Jiang Q Lu and L Qian ldquoComplex dynamics of a Hollingtype II prey-predator system with state feedback controlrdquoChaos Solitons amp Fractals vol 31 no 2 pp 448ndash461 2007
[16] H Baek and Y Lim ldquoDynamics of an impulsively controlledMichaelis-Menten type predator-prey systemrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 16 no4 pp 2041ndash2053 2011
[17] T Zhao Y Xiao and R J Smith ldquoNon-smooth plant diseasemodels with economic thresholdsrdquo Mathematical Biosciencesvol 241 no 1 pp 34ndash48 2013
[18] V I Utkin Sliding Modes in Control and Optimization Com-munications and Control Engineering Series Springer BerlinGermany 1992
[19] M I S Costa E Kaszkurewicz A Bhaya and L Hsu ldquoAchiev-ing global convergence to an equilibrium population in pred-ator-prey systems by the use of a discontinuous harvestingpolicyrdquo Ecological Modelling vol 128 no 2-3 pp 89ndash99 2000
[20] B L Van De Vrande D H Van Campen and A De KrakerldquoApproximate analysis of dry-friction-induced stick-slip vibra-tions by a smoothing procedurerdquo Nonlinear Dynamics vol 19no 2 pp 157ndash169 1999
[21] S H Doole and S J Hogan ldquoA piecewise linear suspensionbridge model nonlinear dynamics and orbit continuationrdquoDynamics and Stability of Systems vol 11 no 1 pp 19ndash47 1996
[22] Yu A Kuznetsov S Rinaldi and A Gragnani ldquoOne-parameterbifurcations in planar Filippov systemsrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol13 no 8 pp 2157ndash2188 2003
[23] M di Bernardo C J Budd A R Champneys et al ldquoBifurca-tions in nonsmooth dynamical systemsrdquo SIAM Review vol 50no 4 pp 629ndash701 2008
[24] A F Filippov Differential Equations with Discontinuous Right-hand Sides vol 18 of Mathematics and Its Applications (SovietSeries) Kluwer Academic Dordrecht The Netherlands 1988
[25] S Tang J Liang Y Xiao and R A Cheke ldquoSliding bifurcationsof Filippov two stage pest control models with economicthresholdsrdquo SIAM Journal on Applied Mathematics vol 72 no4 pp 1061ndash1080 2012
Abstract and Applied Analysis 11
[26] M I da Silveira Costa and M E M Meza ldquoApplication of athreshold policy in the management of multispecies fisheriesand predator cullingrdquoMathematical Medicine and Biology vol23 no 1 pp 63ndash75 2006
[27] M E Mendoza Meza A Bhaya E Kaszkurewicz and M IDa Silveira Costa ldquoThreshold policies control for predator-preysystems using a control Liapunov function approachrdquoTheoreti-cal Population Biology vol 67 no 4 pp 273ndash284 2005
[28] F Dercole A Gragnani and S Rinaldi ldquoBifurcation analysisof piecewise smooth ecological modelsrdquoTheoretical PopulationBiology vol 72 no 2 pp 197ndash213 2007
[29] W Wang ldquoBackward bifurcation of an epidemic model withtreatmentrdquo Mathematical Biosciences vol 201 no 1-2 pp 58ndash71 2006
[30] I Noy-Meir ldquoStability of grazing systems an application ofpredator-prey graphsrdquo The Journal of Animal Ecology vol 63no 2 pp 459ndash481 1975
[31] R M Colombo and V Krivan ldquoSelective strategies in foodwebsrdquo Mathematical Medicine and Biology vol 10 no 4 pp281ndash291 1993
[32] V Krivan ldquoOptimal foraging and predator-prey dynamicsrdquoTheoretical Population Biology vol 49 no 3 pp 265ndash290 1996
[26] M I da Silveira Costa and M E M Meza ldquoApplication of athreshold policy in the management of multispecies fisheriesand predator cullingrdquoMathematical Medicine and Biology vol23 no 1 pp 63ndash75 2006
[27] M E Mendoza Meza A Bhaya E Kaszkurewicz and M IDa Silveira Costa ldquoThreshold policies control for predator-preysystems using a control Liapunov function approachrdquoTheoreti-cal Population Biology vol 67 no 4 pp 273ndash284 2005
[28] F Dercole A Gragnani and S Rinaldi ldquoBifurcation analysisof piecewise smooth ecological modelsrdquoTheoretical PopulationBiology vol 72 no 2 pp 197ndash213 2007
[29] W Wang ldquoBackward bifurcation of an epidemic model withtreatmentrdquo Mathematical Biosciences vol 201 no 1-2 pp 58ndash71 2006
[30] I Noy-Meir ldquoStability of grazing systems an application ofpredator-prey graphsrdquo The Journal of Animal Ecology vol 63no 2 pp 459ndash481 1975
[31] R M Colombo and V Krivan ldquoSelective strategies in foodwebsrdquo Mathematical Medicine and Biology vol 10 no 4 pp281ndash291 1993
[32] V Krivan ldquoOptimal foraging and predator-prey dynamicsrdquoTheoretical Population Biology vol 49 no 3 pp 265ndash290 1996
