Ecological Complexity 18 (2014) 67–73

Original Research Article

On dynamical behavior of the sugarcane borer – Parasitoidagroecosystem

Marat Rafikov *, Jean Carlos Silveira

Centro de Engenharia, Modelagem e Ciencias Sociais Aplicadas, Universidade Federal do ABC, Santo Andre, SP, Brazil

A R T I C L E I N F O

Article history:

Received 2 December 2012

Received in revised form 23 November 2013

Accepted 2 December 2013

Available online 4 January 2014

Keywords:

Mathematical model

Biological control

Sugarcane borer

Larvae parasitoid

Optimal control strategy

A B S T R A C T

The interaction between the sugarcane borer (Diatraea saccharalis) and its most important larvae

parasitoid (Cotesia flavipes) was investigated by aid of a mathematical model. The steady states of the

system are determined, and the dynamical behavior of the larvae, parasitized larvae and parasitoid

populations is examined. The traditional biological control strategy by liberation of the high-density

parasitoid populations is investigated. The numerical simulations show that in some cases this strategy

does not control the sugarcane borer efficiently. The linear feedback control strategy, based on the

optimal control theory, is proposed to indicate how the natural enemies should be introduced in the

environment.

� 2013 Elsevier B.V. All rights reserved.

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Ecological Complexity

jo ur n al ho mep ag e: www .e lsev ier . c om / lo cate /ec o co m

1. Introduction

Brazil is the world’s second largest producer of ethanol fuel. Theeconomic interest in sugarcane production in recent years hasincreased significantly. Sugarcane, being a long duration crop with10–15 months as a plant crop, cultivated on a commercial scale inextensive areas, offers almost a monoculture condition and nearperfect habitat for different pests. To improve this ecologicalsituation it is necessary to develop the efficient ways to controlthese pests.

The sugarcane borer Diatraea sacharalis is reported to be themost important sugarcane pest in south-east region of Brazil (Parraet al., 2002). The sugarcane borer builds internal galleries in thesugarcane plants causing direct damages that result in apical buddeath, weight loss and atrophy. Indirect damages occur when thereis contamination by yeasts that cause red rot in the stalks, eithercausing contamination or inverting the sugar, increasing yield lossin both sugar and alcohol (Macedo and Botelho, 1988).

To increase the crop productivity, the biological pest control isof great significance. Biological control is the reduction of pestpopulations by their natural enemies; namely, predators, para-sitoids, and pathogens (DeBach, 1974). Parasitoids are species that

* Corresponding author at: UFABC – Universidade Federal do ABC, CECS, Av. dos

Estados, 5001, Bairro Bangu, Santo Andre CEP 09.210-580, SP, Brazil.

Tel.: +55 11 987689744.

E-mail addresses: marat.rafikov@ufabc.edu.br, marat9119@yahoo.com.br

(M. Rafikov), jean.silveira@yahoo.com.br (J.C. Silveira).

1476-945X/$ – see front matter � 2013 Elsevier B.V. All rights reserved.

http://dx.doi.org/10.1016/j.ecocom.2013.12.003

develop within or on the host and ultimately kill it. Thus,parasitoids are popular biological control agents of crop pests.Commonly, they are reared in laboratories and periodicallyreleased into the field at high-density (Barclay et al., 1970). Amain goal of pest control is to maintain the density of the pestpopulation at an equilibrium level below the economic injury level.

There is an important larvae parasitoid of the sugarcane borer, awasp named Cotesia flavipes, which is widely used in biologicalcontrol in Brazil (Parra et al., 2002). In spite of this control beingconsidered successful in Brazil, there are some areas where C.

flavipes does not control the sugarcane borer efficiently. This canhappen if the steady state of the agroecosystem is above theeconomic injury level or if the pest population size oscillates abovethis level. According to Tang and Cheke (2008), the economic injurylevel is the lowest population density that will cause economicdamage. When implementing a biological pest control strategy,action is required once the density of pests reaches a critical levelin the field. Therefore, if economic decisions regarding pests areappropriately understood and adopted, profits can be increasedand environmental quality maintained (Tang and Cheke, 2008).

The mathematical modeling of these agroecosystems can helpto study their dynamics and choose the biological control strategy.Thus, a good strategy of biological pest control, based onmathematical modeling, can increase the ethanol production.The applications of prey–predator and host–parasitoid models forbiological control were reviewed in (Chatterjee et al., 2009; Gamezet al., 2009, 2010; Venturino et al., 2006, 2008).

In this paper we want to model the interactions between thesugarcane borer (Diatraea saccharalis) and its larvae parasitoid

M. Rafikov, J.C. Silveira / Ecological Complexity 18 (2014) 67–7368

C. flavipes in order to determine the biological control strategy. Theproposed model is essentially very closely related to ecoepidemicmodels with infected prey (Venturino, 2002, 2007; Haque andVenturino, 2006, 2007).

Based on results of Rafikov et al. (2008), the linear feedbackcontrol strategy is proposed to indicate how the natural enemiesshould be introduced in the environment. It should be mentionedthat the linear feedback control methodology proposed in (Rafikovet al., 2008), has already been applied in (Gamez et al., 2009, 2010;Rafikov et al., 2009) to different biological systems.

2. Mathematical model of interactions between the sugarcaneborer and its parasitoid C. flavipes

The agroecosystem we consider consists of the larvae sugarcaneborer population, the larvae population parasitized by C. flavipes

and parasitoid C. flavipes population. The host–parasitoid modelwith parasitized host in which all the parameters are assumed tobe nonnegative, has the following form

dH

dt¼ r 1 � H

K

� �H � n1H � bHP

dI

dt¼ bHP � m2I � n2I

dP

dt¼ gn2I � m3P

(1)

were H is the larvae density of the sugarcane borer (hostpopulation), I is the density of the larvae population parasitizedby C. flavipes (infected host population) and P is the density of thelarvae parasitoid C. flavipes; r is the intrinsic growth rate; K is thecarrying capacity of the environment; m2 and m3 are mortalityrates of the parasitized larvae populations and parasitoid; n1 is thefraction of the larvae population which molts into pupae stage attime t; n2 is the fraction of the parasitized larvae from which theadult parasitoids emerge at time t; b is the intrinsic rate ofparasitism and g is a number of the adult parasitoids which emergefrom one parasitized larvae at time t.

The first and second equations describe the evolution of thesugarcane borer and the variation of the larvae populationparasitized by C. flavipes, respectively.

The equation for the sugarcane borer accounts for logisticgrowth, first term, while the second term represents thetransformation the larvae population into pupae population andthe last term represents the parasitism of the sugarcane borerpopulation by C. flavipes, modeled via a mass action law.

The first term of the equation for the parasitized populationmodels the population growth caused by parasitism, the secondand third terms represent the mortality and the transformation ofthe parasitized larvae population into parasitoids, respectively.

The dynamics of parasitoids is given in third equation, in whicha negative Malthus growth describes their mortality. The positiveterm accounts for the growth due to parasitism, where theconstant g models the conversion factor of the parasitized larvaespecie into new parasitoids.

3. Equilibrium points and their stability

3.1. Equilibrium points

The equilibrium points can be obtained by setting to zero theright hand sides of (1). We obtain therefore the following points:

E1 ¼ ð0; 0; 0ÞE2 ¼

K

rðr � n1Þ; 0; 0

� �E3 ¼ ðH3; I3; P3Þ;

(2)

where

H3 ¼ðm2 þ n2Þm3

bgn2; P3 ¼

1

br 1 � ðm2 þ n2Þm3

bgn2K

� �� n1

� �; I3

¼ m3

gn2P3:

3.2. Stability analysis of the equilibrium points

Next, we consider the local stability of each of the equilibriumpoints. The Jacobian of system (1) is

J ¼r � 2r

KH � n1 � b P 0 �b H

b P �m2 � n2 b H0 g n2 �m3

264

375 (3)

At the equilibrium E1 = (0, 0, 0), the eigenvalues the Jacobian (3) are

l1 ¼ r � n1; l2 ¼ �m2 � n2; l3 ¼ �m3:

It follows then that equilibrium point E1 = (0, 0, 0) is stable if

r < n1: (4)

The ecological interpretation of the inequality (4) is following. Allspecies go to extinction if the intrinsic growth rate is smaller thanthe fraction of the parasitized larvae n2.

The equilibrium point E2 = ((K/r)(r � n1), 0, 0) exists if r > n1.The characteristic equation of the Jacobian at E2 is given by

ðn1 � r � lÞ ðl2 þ a1l þ a2Þ ¼ 0

where

a1 ¼ m2 þ n2 þ m3 > 0

a2 ¼ m3ðm2 þ n2Þ � b g n2K

rðr � n1Þ:

The eigenvalue l1 = n1 � r is negative if

r > n1: (5)

According to the Routh-Hurwitz criterion the eigenvalues of theequation l2 + a1l + a2 = 0 have negative real parts if a1 > 0 anda2 > 0. The coefficient a2 > 0 if

b <m3ðm2 þ n2Þrg n2Kðr � n1Þ

: (6)

It follows then that equilibrium point E2 is stable if and only if theinequalities (5) and (6) are satisfied. In this case, the naturalenemies go to extinction, and the pest population goes to itsmaximum level. It is possible that the biological control will beneeded.

The equilibrium point E3 = (H3, I3, P3) exists if

b >m3ðm2 þ n2Þrg n2Kðr � n1Þ

: (7)

The characteristic equation of the Jacobian at E3 is given by

l3 þ a1l2 þ a2l þ a3 ¼ 0 (8)

where

a1 ¼ m2 þ n2 þ m3 þrx�1bK

> 0

a2 ¼ ðm2 þ n2 þ m3Þrx�1bK

> 0

a3 ¼ bx�3ðm2 þ n2Þm3 > 0:

(9)

M. Rafikov, J.C. Silveira / Ecological Complexity 18 (2014) 67–73 69

According to the Routh-Hurwitz criterion the eigenvalues of theEq. (8) have negative real parts if a1 > 0, a2 > 0, a3 > 0 andD = a1a2 � a3 > 0. From last inequality we have

b <rm3ðm2 þ n2Þ

gn2BK

where

B ¼ � c1

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðc1Þ2

4þc2

s

c1 ¼ m2 þ n2 þ m3 þðm2 þ n2Þm3

m2 þ n2 þ m3

c2 ¼ðm2 þ n2Þm3ðr � n1Þ

m2 þ n2 þ m3

(10)

It follows then the equilibrium point E3 is locally stable if andonly if

m3ðm2 þ n2Þrg n2Kðr � n1Þ

< b <m3ðm2 þ n2Þr

g n2BK(11)

In this case, all species coexist in the monoculture habitat and thepest density can exceeds the economic injury level.

In addition, based on above mentioned inequalities we canobtain more results.

In fact comparing (4), (5) and (6), one can say that the systempersists if r > n1.

Also, comparing (6) and (7) we can conclude that there is atranscritical bifurcation for which E2 emanates from E1.

3.3. Global stability analysis of the equilibrium point E3

We define a Lyapunov function

V ¼Z H

H3

y � H3

ydy þ

Z I

I3

y � I3

ydy þ ðP � P3Þ2

2(12)

It can be easily verified that the function V is zero at theequilibrium point E3 and is positive for all other positive values ofH, I and P.

The time derivative of V along the trajectories of (1) is

V ¼ d11ðH � H3Þ2 þ d12ðH � H3ÞðI � I3Þ þ d13ðH � H3ÞðP � P3Þ

þ d22ðI � I3Þ2 þ d23ðI � I3ÞðP � P3Þ þ d33ðP � P3Þ2

or

V ¼ eT De (13)

where the elements of the matrix D are

d11 ¼ � r

K; d12 ¼

bP

I; d013 ¼ �b;

d22 ¼ �m2 þ n2

I; d23 ¼

bH3

Iþgn2; d033 ¼ �m3;

and the elements of the vector e are

e1 ¼ H � H3; e2 ¼ I � I3; e3 ¼ P � P3:

Matrix D in (13) is negative definite. Then the time derivative of V

along the trajectories of (1) is negative definite, and theequilibrium point E3 is globally stable.

3.4. Hopf bifurcation analysis

From (10) we have

b <m3ðm2 þ n2Þr

gn2BK¼ bcr

and we can conclude that the intrinsic rate of parasitism b plays animportant role on the global stability of the system (1). When bcrosses bcr, the positive equilibrium becomes unstable and Hopfbifurcation may occur.

Now we analyze the bifurcation of the model (1) assuming b asthe bifurcation parameter. The traditional Hopf bifurcationcriterion is stated in terms of the properties of eigenvalues. Liu(1994) presented a criterion of Hopf bifurcation without using theeigenvalues of the characteristic equation. The Liu’s result,specified for the current purposes, is presented in the Appendix 1.

Applying the Liu’s result to the characteristic Eq. (8), from (9)we observe that a1(b) > 0 and a3(b) > 0 for all positive values of b.Solving the equation

DðbÞ ¼ a1ðbÞa2ðbÞ � a3ðbÞ ¼ 0;

we obtain

b� ¼ rm3ðm2 þ n2Þgn2BK

;

where B is given by (10).Considering the condition (2) of the Liu’s criterion, we have

dD

db

� �b¼b�

¼ � B2

b�2� 2B3

b�3< 0

where

B2 ¼ � rðm2 þ n2 þ m3Þ2ðm2 þ n2Þm3

gn2K;

B3 ¼ � r2ðm2 þ n2 þ m3Þðm2 þ n2Þ2m23

g2n22K2

:

Hence, according to the Liu’s criterion a simple Hopf bifurcationoccurs at b = b*.

4. Numerical simulations

For numerical simulations of interactions between the sugar-cane borer and its parasitoid we use the following values of modelcoefficients: n1 = 1/50, n2 = 1/16, g = 40, m2 = 0.036, m3 = 0.5,K = 25, 000.

These values we obtain based on data from different availableliterature about the use of the larvae parasitoid C. flavipes againstthe sugarcane borer D. saccharalis these (Macedo and Botelho,1988; Parra et al., 2002, etc.).

The main objective of the biological pest control is to maintainthe pest population in an equilibrium level below the economicinjury level. According to Tang and Cheke (2008), the pest densitiesequal or exceeding the economic injury level cause economicdamage. Economic threshold is population density at whichcontrol measures should be determined to prevent an increasingpest population from reaching the economic injury level. Thus,parasitoids and predators are commonly reared in laboratories andperiodically liberated in high-density populations when the pestpopulation reaches the economic threshold level.

When the condition r < n1 is satisfies the equilibrium E1 isstable and other points are unstable. All populations go toextinction in this case.

The value of the parameter b is important for determination thestability of the equilibrium points E2 and E3. When r > n1 and bsatisfies the condition b < (m3(m2 + n2)r)/(gn2K(r � n1)), the equi-librium E2 is stable and other points are unstable. In this case, theparasitized larvae and parasitoid populations go to extinction, andthe larvae populations go to positive equilibrium levels.

M. Rafikov, J.C. Silveira / Ecological Complexity 18 (2014) 67–7370

When b satisfies the condition (11)

m3ðm2 þ n2Þrgn2Kðr � n1Þ

< b <m3ðm2 þ n2Þr

gn2BK;

the positive equilibrium point E3 is stable and the populationscoexist in a common environment.

When the equilibrium points E2 and E3 are stable, the sugarcaneborer larvae density H can take on values larger than the economicinjury level pest density HEIL = 2500 pests/ha (Parra et al., 2002).Densities above this level cause economic damages to thesugarcane crops. In this case, it is necessary to apply the biologicalcontrol.

When b satisfies the conditionb >

m3ðm2þn2Þrgn2BK ¼ bcr;the positive equilibrium point E3 loses

stability and a stable limit cycle occurs (Fig. 1).

Fig. 1. Evolution of the larvae, parasitized larvae and parasitoid populations for

b = 0.000008. In this case, the positive equilibrium point E3 loses stability and a

stable limit cycle occurs. The sugarcane borer larvae density H takes on values larger

than the economic injury level for this pest HEIL = 2500 pests/ha.

Fig. 2. Biological control application by introduction 40,000 parasitoids/ha at first

day is not sufficient to maintain the pest population below the economic injury

level value when the stable limit cycle occurred.

From Fig. 1 one can see that the sugarcane borer larvae densityH takes on values larger than the economic injury level for this pestHEIL = 2500 pests/ha. In this case, it is necessary to apply thebiological control.

Mathematically, the liberation of the high-density parasitoidpopulations can be interpreted by impulsive control function U

that produces the discontinuous augmentation of the naturalenemy population.

From Fig. 2 one can see that the liberation of 40,000 parasitoids/ha, applied in the initial day, is not sufficient to maintain the pestpopulation below the economic injury level value when the stablelimit cycle occurred.

5. Optimization of the biological control

As previously mentioned, when the equilibrium points E2 andE3 are stable or the stable limit cycle occurred, the sugarcane borerlarvae density H can take on values larger than the economic injurylevel pest density. Fig. 2 presents the example when the traditionalbiological control application by liberation of the high-densityparasitoid populations does not control the sugarcane borerefficiently.

In this section, we determine the sugarcane borer controlstrategy based on the optimal control theory. This control mustmove the controlled system to the steady state where the larvaedensity is stabilized without causing economic damages, andwhere the parasitoid population is stabilized at the level enough tocontrol the pests.

The dynamic system (1) with control has the following form:

dH

dt¼ r 1 � H

K

� �H � n1H � bHP

dI

dt¼ bHP � m2I � n2I

dP

dt¼ gn2I � m3P þ U

(14)

The control function U in (14) consists of two parts

U ¼ u� þ u; (15)

where u* is constant control which maintains the larvaepopulation at the desired pest population density level H* beloweconomic injury level, that is H* < HEIL and u is feedback controlwhich stabilizes the ecosystem at desired steady state. The desiredpositive steady state with control satisfies the followingequations

r 1 � H�

K

� �H� � n1H� � bH�P� ¼ 0

bH�P� � m2I� � n2I� ¼ 0gn2I� � m3P� þ u� ¼ 0

(16)

From the first equation of the system (16) we obtain the parasitoiddensity value which is necessary to maintain the larvae populationat level H*

P� ¼ r

br 1 � x�1

K

� �� n1

b(17)

From the second equation of the system (16) we obtain theparasitized larvae density value which is necessary to maintain thelarvae population at level H*

I� ¼ bH�P�

m2 þ n2(18)

From the last equation of the system (16) we obtain the value of theconstant control u*

u� ¼ m3P� � gn2I� (19)

Fig. 3. Evolution of the dynamic system (13) with optimal control. The optimal

control (26) is designed to drive the trajectory of the system (14) to desired steady

state (H*, I*, P*).

Fig. 4. Dynamics of the optimal control strategy. The proposed optimal control

strategy determines that a great amount of parasitoid have to be introduced in the

initial days.

M. Rafikov, J.C. Silveira / Ecological Complexity 18 (2014) 67–73 71

In the general case, the desired steady-state (H*, I*, P*) of thesystem (14) controlled by u* can be unstable. In this case thefeedback control u can be made so that the desired state becomesasymptotically stable.

Defining the following new variables

y ¼H � H�

I � I�

P � P�

24

35; u ¼ U � u� (20)

and substituting (20) into (14) and admitting (16), we get thefollowing perturbed system:

y ¼ A y þ hðyÞ þ Bu (21)

where the matrices A and B are

A ¼r � 2rH�

K�n1 � bP� 0 �bH�

bP� �m2 � n2 bH�

0 gn2 �m3

264

375; B ¼

001

24

35 (22)

and the vector h(y) has a form:

hðyÞ ¼� r

Ky2

1 � b y1y3

b y1y3

0

264

375 (23)

The feedback control u can be determined applying two theorems,presented in the Appendix 2. According to these theorems thelinear feedback control is

u ¼ �R�1BT P y; (24)

where P the symmetric, positive definite matrix, is the solution ofthe matrix algebraic Riccati equation

PA þ AT P � PBR�1BT P þ Q ¼ 0 (25)

From the theorems one can conclude that the perturbeddynamical system (21) controlled by linear feedback control u isasymptotically stable, and hence, the system (14), controlled by(15) tends to the desired steady state (H*, I*, P*).

We illustrate the application of the optimal pest controlstrategy (24) on the agroecosystem which consists of sugarcaneborer and its parasitoid. We will stabilize the ecosystem (14) atthe desired steady state H* = 2000 larvae/ha, I* = 3158 parasit-ized larvae/ha, P* = 19,441 parasitoids/ha. The values of I* and P*

were calculated from (18) and (17), respectively. In this case,u* = 1826 parasitoids/day, and the matrices A and B have thefollowing form

A ¼�0:0153 0 �0:0160:1555 �0:0985 0:016

0 2:5 �0:5

24

35; B ¼

001

24

35

Choosing

Q ¼1 0 00 1 00 0 1

24

35; R ¼ ½1�

we obtain

P ¼171:47 24:21 �1:8224:21 30:61 1:4�1:81 1:4 0:66

24

35

from the solution of the Riccati equation (25) using the MATLABcommand LQR.

Finally, we can conclude that the optimal strategy has thefollowing form:

U ¼ 1826 þ 1:82 y1 � 1:4 y2 � 0:66y3 (26)

The optimal control (26) is designed to drive the trajectory of thesystem (14) to desired steady state (H*, I*, P*), as shown in Fig. 3.Dynamics of the optimal control function (26) is presented inFig. 4.

Numerical simulations showed that the function l(y),defined by

lðyÞ ¼ yT Qy � hTðyÞPy � yT PhðyÞ;

was positive for all considered initial condition values, but it isnecessary to perform more investigations to prove if this functionis positive definite in a positive space.

M. Rafikov, J.C. Silveira / Ecological Complexity 18 (2014) 67–7372

6. Concluding remarks

We have presented mathematical model to describethe possible interactions among the sugarcane borer (D.

saccharalis) and its most important larvae parasitoid (C. flavipes).This model is closely related to ecoepidemic models withinfected prey (Venturino, 2002, 2007; Haque and Venturino,2006, 2007). Our investigation mainly focuses on the facts thatthere are some areas in Brazil where C. flavipes does not controlthe sugarcane borer efficiently. In the model description weincorporated the biological control function that represents theintroduction of parasitoid population in agroecosystem. Thetraditional biological control strategy by liberation of the high-density parasitoid populations is investigated. We also proposedthe biological control strategy based on the optimal controltheory.

We determined the system’s equilibria and performed thestability analysis of them. The analytical and numerical investiga-tions show when the equilibrium points E2 and E3 are stable or thestable limit cycle occurred, the sugarcane borer larvae density H

can take on values larger than the economic injury level pestdensity. Moreover, from the numerical experiments we carriedout, we may conclude that the traditional biological controlapplication by liberation of the high-density parasitoid popula-tions does not control the sugarcane borer efficiently in thesecases. The numerical simulations reported in Fig. 2 shows that theliberation of 40,000 parasitoids/ha, applied in the initial day is notsufficient to maintain the pest population below the economicinjury level value.

The linear feedback control strategy, based on the optimalcontrol theory is proposed to indicate how the natural enemiesshould be introduced in the environment in order to stabilizes thelarvae population at level H* < HEIL. This strategy permits to avoiddramatic explosions of the pest population.

The proposed optimal control strategy determines that a greatamount of parasitoid have to be introduced in the initial days(Fig. 4). This fact suggests that the proposed feedback controlstrategy can be integrated into existing biological controltechnologies, combining the feedback control with the traditionalbiological control application by liberation of the high-densityparasitoid populations.

Existing technologies for implementation of the biologicalcontrol strategies in large agroecosystems are based on thehypothesis that the environment is homogeneous. The proposedmodel, using ordinary differential equations, satisfies thishypothesis. For modeling spatial structure and heterogeneitycan be used traditional models of partial differential equations orrecent approaches such as the evolutionary dynamics of groupinteractions on structured populations (Perc and Szolnoki, 2010,2013).

Acknowledgments

The authors would like to thank the referees for their carefulreading of the original paper and their valuable comments andsuggestions that improved the presentation of this paper.

The first author thanks Fundacao de Amparo a Pesquisa doEstado de Sao Paulo (FAPESP) and Conselho Nacional de Pesquisas(CNPq) for the financial supports on this research.

Appendix 1

Liu’s criterion. If the characteristic of the positive equilibriumpoint is given by l3 + a1(b)l2 + a2(b)l + a3(b) = 0, where a1(b),a2(b) and a3(b) are smooth functions of b in an open interval about

b* 2 R such that:

a1ðb�Þ > 0; Dðb�Þ ¼ a1ðb�Þa2ðb�Þ � a3ðb�Þ

¼ 0; a3ðb�Þ > 0 (A1)

dD

db

� �b¼b�

6¼ 0; (A2)

then a simple Hopf bifurcation occurs at b = b*.

Appendix 2

Theorem 1. If there exist constant matrices Q, and R, positive definite,

being Q symmetric, such as that the function (Rafikov et al., 2008)

lðyÞ ¼ yT Qy � hTðyÞPy � yT PhðyÞ;

is positive definite then the linear feedback control

u ¼ �R�1BT P y

is optimal, in order to transfer the nonlinear system (21) from an

initial to final state

yð1Þ ¼ 0

minimizing the functional

J ¼Z 1

0½lðyÞ þ uT Ru�dt

where P the symmetric, positive definite matrix, is the solution of the

matrix algebraic Riccati equation

PA þ AT P � PBR�1BT P þ Q ¼ 0

In addition, with the feedback control u, there exists a neighborhood

G0 � G, G � Rn, of the origin such that if y0 2 G0, the solution y(t) = 0,

t I 0, of the controlled system (19) is locally asymptotically stable, and

Jmin ¼ yT0Pð0Þ y0: Finally, if G = R

n then the solution y(t) = 0, t I 0, of

the controlled system (21) is globally asymptotically stable.

The next theorem determines the positive definiteness of the

function l(y) in the neighborhood G0 of the origin for the system (21).

Theorem 2. For any matrix P and

Q ¼q11 0 00 q22 00 0 q33

24

35; hðyÞ ¼

� r

Ky2

1 � b y1y3

b y1y3

0

264

375

function l(y) defined in (22) is positive definite at the neighborhood G0

of the origin (0, 0, 0).

Proof. In the considered case, for all y we have

lðyÞ ¼ q11y21 þ q22y2

2 þ q33y23 � 2ðy1G1 þ y2G2 þ y3G3Þ

where

G1 ¼ � p11r

Ky2

1 þ by1y3

� �þ p12by1y2

G2 ¼ � p12r

Ky2

1 þ by1y3

� �þ p22by1y2

M. Rafikov, J.C. Silveira / Ecological Complexity 18 (2014) 67–73 73

G3 ¼ � p13r

Ky2

1 þ by1y3

� �þ p23by1y2:

Its first order partial derivatives are

@l

@y1

¼ 2q11y1 � 2 G1 þ y1@G1

@y1

þ y2@G2

@y1

þ y3@G3

@y1

� �

@l

@y2

¼ 2q22y2 � 2G2

@l

@y3

¼ 2q33y3 � 2 y1@G1

@y3

þ y2@G2

@y3

þ y3@G3

@y3

þ G3

� �

It is obvious that

@l

@y1

ð0Þ ¼ @l

@y2

ð0Þ ¼ @l

@y3

ð0Þ ¼ 0

and for the Hessian of l(y) at the origin we have

Hð0Þ ¼2q11 0 0

0 2q22 00 0 2q33

24

35

which is positive definite, implying that the origin is a strict local

minimum point of function l(y) and that this function is positive

definite at the neighborhood G0 of the origin. &

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