International Journal of Mathematical, Engineering and Management Sciences
Vol. 5, No. 4, 769-786, 2020
https://doi.org/10.33889/IJMEMS.2020.5.4.061
769
Controlling Pest by Integrated Pest Management: A Dynamical
Approach
Vandana Kumari
Department of Mathematics,
Amity Institute of Applied Science,
Amity University, Sector-125, Noida, U.P., India.
E-mail: [email protected]
Sudipa Chauhan Department of Mathematics,
Amity Institute of Applied Science,
Amity University,Sector-125, Noida, U.P., India.
Corresponding author: [email protected]
Joydip Dhar Mathematical Modelling and Simulation Laboratory,
Atal Bihari Vajpayee Indian Institute of Information Technology and Management,
Gwalior, M.P., India.
E-mail: [email protected]
(Received August 28, 2019; Accepted January 28, 2020)
Abstract
Integrated Pest Management technique is used to formulate a mathematical model by using biological and chemical
control impulsively. The uniform boundedness and the existence of pest extinction and nontrivial equilibrium points is
discussed. Further, local stability of pest extinction equilibrium point is studied and it has been derived that if ๐ โค ๐๐๐๐ฅ,
the pest extinction equilibrium point is locally stable and for ๐ > ๐๐๐๐ฅ , the system is permanent. It has also been
obtained that how delay helps in eradicating pest population more quickly. Finally, analytic results have been validated
numerically.
Keywords- Plant-pest-natural enemy, Boundedness, Local stability, Permanence.
1. Introduction Plants as we all know conflict between and pests has been a root cause of concern in our ecology
from almost two decades. Rescuing crops from predator pests such as insects has become a tedious
task for farmers. With the advent in science and technology, effective measures have been
discovered to deal with predator pest effectively like introducing natural enemies and chemical
pesticides in relevent environment. It is a well known fact that excessive use of chemical pesticide
such as organochlorine (DDT and toxaphene) is hazardous both for animals and human being as
studied by authors (James, 1997). Therefore, Integrated pest management came into scenario in
which selective pesticides control pests as natural predators when regulation through biological
means fails. Many biological food web models to control pests have been discussed by many
scholars (Changguo et al., 2009; Liu et al., 2013; Jatav et al., 2014; Song et al., 2014) where they
took assumptions of either impulsive release of natural enemies or chemical pesticides. Authors
(Jatav and Dhar, 2014) studied a model in which they formulated a mathematical model and
obtained a threshold value below which pests gets eradicated. Later, many more IPM approach
International Journal of Mathematical, Engineering and Management Sciences
Vol. 5, No. 4, 769-786, 2020
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inclined models were proposed where impulsive control strategies for pest eradication were
introduced and to name a few are (Tang et al., 2005; Akman et al., 2015; El-Shafie, 2018; Paez
Chavez et al., 2018). They studied various prospect of IPM method and its application. Scholars
(Zhang et al., 2004) did comparison between IPM method and classical method for pest control and
obtained that IPM strategy is better than any classical method to control pests. Recently, Yu et al.
(2019) introduced IPM method for predatorโprey model with Allee effect and stochastic effect
respectively where they obtained thresholds based on biological and chemical control. However,
in all the papers discussed above no-one discussed significanlty about delays, in particularly
gestation delay which in a real situation always exist.
Hence, keeping in mind the above alma matter, we have formulated our model in reference to the
previous models and studied the dynamics of the new system with delay. The highlight of the paper
is that how delay parameter helps in reducing the pest population more quickly in comparison to
the system without delay. The results would be extremely beneficial for those crops where pest
population are growing exponentially due to favourable habitable condition. A relevent biological
example to our model is as follows:
Australian herb is always at the verge of being attacked by green Lacewing Larvae, which is a well
known pest. Encapsulating biological controls like mealy bugs followed by chemical control such
as chlorothalonil has shown remarkable results which advocates our approach of hybrid technique.
The organisation of the paper is as follows: In Section 2, 3 model formulation and preliminary
lemmas are discussed. In Section 4, local stability of pest extinction is achieved followed by
permanence in Section 5. Finally, in the last two sections numerical simulation is done for
validation of analtical results with conclusion.
2. Mathematical Model We have proposed our mathematical model by the following set of differential equations:
๐๐
๐๐ก= ๐(๐ โ ๐) โ ๐1๐๐
๐๐
๐๐ก= ๐1๐1๐๐ โ ๐2๐(๐ก โ ๐)๐2(๐ก โ ๐)๐
โ๐1๐ โ ๐ท๐
๐๐1
๐๐ก= ๐2๐2๐(๐ก โ ๐)๐2(๐ก โ ๐)๐
โ๐1๐ โ (๐ท3 + ๐0)๐1
๐๐2
๐๐ก= ๐0๐1 โ ๐ท3๐2
}
๐ก โ ๐๐ (1)
๐(๐ก+) = ๐
๐(๐ก+) = (1 โ ๐ฟ)๐
๐1(๐ก+) = ๐1 + ๐1
๐2(๐ก+) = ๐2 + ๐2
}
= ๐๐ (2)
The model completes with the following initial conditions:
๐(๐) = ๐1(๐), ๐(๐) = ๐2(๐), ๐1 = ๐1(๐), ๐2 = ๐2(๐) , ๐๐(0) > 0 , ๐๐(0) > 0 , ๐ โ [โ๐, 0] ,
(๐ = 1,2) , where (๐1, ๐2, ๐1, ๐2) โ ๐ถ([โ๐, 0], โ+4 , the Banach space of continuous
functions mapping on the interval [โ๐, 0]into โ+4 . The graphical representation of the model is
International Journal of Mathematical, Engineering and Management Sciences
Vol. 5, No. 4, 769-786, 2020
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as follows in Figure 1. Negative and positive sign represents outgoing and incoming rates.
Figure 1. Graphical representation of model
The parameters/variables used in the model are explained in detail in Table 1 mentioned below and
for convenience ๐ก is removed from the variables throughout the paper.
Table. 1 Meaning of parameters /variables
Parameters/Variables Meaning
๐1(๐ก) Immature natural enemy
๐ growth rate of plant population
๐2(t) Mature natural enemy
๐ Time delay
๐(๐ก) Plant population
๐1 Rate at which plant population is decreasing to pest population
๐1 Growth rate of pest population
D Mortality Rate
๐2 Rate at which pest population is decreasing
๐2 Rate at immature natural enemy population
๐0 Mortality rate of immature natural enemy
๐ท3 Mortality rate of mature natural enemy
๐ Period of impulse
๐1 Amount of pulse release of immature natural enemy
๐2 Amount of pulse release of mature natural enemy
0 โค ๐ฟ < 1 harvesting rate of pest through chemical pesticide
๐(๐ก) Pest population
3. Preliminary Lemmas In this section, we have given a few Lemmas, which will be useful for our main result.
Lemma 3.1 Let us consider the system
๐คโฒ(๐ก) = ๐ โ ๐๐ค(๐ก), ๐ก โ ๐๐,
(3)
๐ค(๐ก+) = ๐ค(๐ก) + ๐, ๐ก = ๐๐, ๐ = 1,2,3โฆ. (4)
Then the system has a positive periodic solution ๏ฟฝฬ๏ฟฝ(๐ก)and for any solution ๐ค(๐ก) of the system
(3),we have,
|๐ค(๐ก) โ ๏ฟฝฬ๏ฟฝ(๐ก)| โ 0,
for ๐ก โ โ, where, for
๐ก โ (๐๐ก, (๐ + 1)๐], ๏ฟฝฬ๏ฟฝ(๐ก) =๐
๐+๐๐๐ฅ๐(โ๐(๐กโ๐๐))
1โ๐๐ฅ๐(โ๐๐) with ๏ฟฝฬ๏ฟฝ(0+) =
๐
๐+
๐
1โ๐๐ฅ๐(โ๐๐).
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The boundedness is given lemma 3.2.
Lemma 3.2 There exists a constant ๐ > 0 s.t ๐(๐ก) โค ๐, ๐(๐ก) โค ๐, ๐1(๐ก) โค ๐, ๐2(๐ก) โค ๐, for (1 โ 2) with t being sufficiently large where
๐ =๐0
๏ฟฝฬ๏ฟฝ+(๐1 + ๐2)๐๐ฅ๐(๏ฟฝฬ ๏ฟฝ๐ก)
๐๐ฅ๐(๏ฟฝฬ ๏ฟฝ๐ก) โ 1> 0.
Now, we will discuss the pest extinction case and our impulsive system (1 โ 2) reduces to:
๐๐1(๐ก)
๐๐ก= โ(๐ท3 + ๐0)๐1(๐ก)
๐๐2(๐ก)
๐๐ก= ๐0๐1(๐ก) โ ๐ท3๐2(๐ก)
}
๐ก โ ๐๐, (5)
๐1(๐ก+) = ๐1 + ๐1
๐2(๐ก+) = ๐2 + ๐2 } ๐ก = ๐๐, (6)
For the system (5 โ 6), we integrate it over the interval (๐๐, (๐ + 1)๐] , and by means of
stroboscopic mapping we get, ๐1((๐ + 1)๐+) = ๐๐ฅ๐( โ (๐ท3 + ยต0)๐) ๐1(๐๐
+) + ๐1
Thus the corresponding periodic solution of (5 โ 6) in ๐ก โ (๐๐, (๐ + 1)๐] is,
๏ฟฝฬ๏ฟฝ1(๐ก) =๐1๐๐ฅ๐(โ(๐ท3 + ๐0)(๐ก โ ๐๐))
1 โ ๐๐ฅ๐(โ(๐ท3 + ๐0)๐)
with
๏ฟฝฬ๏ฟฝ1(0+) =
๐11 โ ๐๐ฅ๐(โ(๐ท3 + ๐0)๐)
and is stable globally. Substituting ๏ฟฝฬ๏ฟฝ1(๐ก) into (5 โ 6), we obtain the following subsystem:
๐๐2(๐ก)
๐๐ก= ๐0๏ฟฝฬ๏ฟฝ1(๐ก) โ ๐ท3๐2(๐ก), ๐ก โ ๐๐
๐2(๐ก+) = ๐2 + ๐2, ๐ก = ๐๐
} (7)
Further, integrating (7) in the interval (๐๐, (๐ + 1)๐], we get,
๏ฟฝฬ๏ฟฝ2(๐ก) =โ๐1๐๐ฅ๐(โ(๐ท3 + ๐0)(๐ก โ ๐๐))
1 โ ๐๐ฅ๐(โ(๐ท3 + ๐0)๐)+(๐1 + ๐2)๐๐ฅ๐(โ๐ท3(๐ก โ ๐๐))
1 โ ๐๐ฅ๐(โ๐ท3๐),
with initial value
๏ฟฝฬ๏ฟฝ2(0+) =
โ๐11 โ ๐๐ฅ๐(โ(๐ท3 + ๐0)๐)
+(๐1 + ๐2)
1 โ ๐๐ฅ๐(โ๐ท3๐),
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which is stable globally.
Moreover, due to the absence of pest, the subsystem of (1 โ 2) can also be considered as follows:
๐๐(๐ก)
๐๐ก= ๐(๐ โ ๐) (8)
With ๐ = 0 as unstable equilibrium and ๐ = ๐ as globally stable. Therefore, the two periodic
solutions of (1 โ 2) are (0,0, ๏ฟฝฬ๏ฟฝ1, ๏ฟฝฬ๏ฟฝ2) and (๐, 0, ๏ฟฝฬ๏ฟฝ1, ๏ฟฝฬ๏ฟฝ2).
4. Local Stability of Pest Extinction Case
This section will discuss the local stability analysis of the equilibrium point with pest population.
Theorem 4.1 Let (๐, ๐, ๐1, ๐2) be a solution of (1 โ 2), Then
(i) (0,0, ๏ฟฝฬ๏ฟฝ1, ๏ฟฝฬ๏ฟฝ2) is unstable.
(ii) (๐, 0, ๏ฟฝฬ๏ฟฝ1, ๏ฟฝฬ๏ฟฝ2) is locally asymptotically stable iff ๐ โค ๐๐๐๐ฅ, where
๐๐๐๐ฅ =1
(๐1๐1 โ ๐){๐๐๐
1
(1 โ ๐ฟ)+ ๐โ๐1๐๐2(
๐ท3๐2 + ๐0(๐1 + ๐2)
๐ท3(๐ท3 + ๐0))}, ๐1๐1 > ๐ (9)
Proof: (i) Here, we define,
๐ = ๐1 , ๐ = ๐2 , ๐1 = ๏ฟฝฬ๏ฟฝ1 + ๐3, ๐2 = ๏ฟฝฬ๏ฟฝ2 + ๐4
where, ๐1(๐ก), ๐2(๐ก), ๐3(๐ก), ๐4(๐ก)are perturbation in ๐, ๐, ๐1, ๐2 then the systemโs linearized
form becomes:
๐๐1(๐ก)
๐๐ก= โ๐๐1(๐ก)
๐๐2(๐ก)
๐๐ก= โ(๐ท + ๐2๏ฟฝฬ๏ฟฝ2(๐ก)๐
โ๐1๐)๐2(๐ก)
๐๐3(๐ก)
๐๐ก= ๐2๐2๐2(๐ก)๏ฟฝฬ๏ฟฝ2(๐ก)๐
โ๐1๐ โ (๐ท3 + ๐0)๐3(๐ก)
๐๐4(๐ก)
๐๐ก= ๐0๐3(๐ก) โ ๐ท3๐4(๐ก)
}
๐ก โ ๐๐ (10)
๐1(๐ก+) = ๐1(๐ก)
๐2(๐ก+) = (1 โ ๐ฟ)๐2(๐ก)
๐3(๐ก+) = ๐3(๐ก) + ๐1
๐4(๐ก+) = ๐4(๐ก) + ๐2
}
= ๐๐ (11)
Let ๐(๐ก) be the fundamental matrix of (10 โ 11), then ๐(๐ก) must satisfy,
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๐๐(๐ก)
๐๐ก=
[ ๐ 0 0 00 โ(๐ท + ๐2๏ฟฝฬ๏ฟฝ2(๐ก)๐
โ๐1๐) 0 0
0 ๐2๐2๏ฟฝฬ๏ฟฝ2(๐ก)๐โ๐1๐ โ(๐ท3 + ๐0) 0
0 0 ๐0 โ๐ท3]
๐(๐ก) = ๐ด๐(๐ก) (12)
Thus, the monodromy matrix of (10 โ 11) is
๐ =
[ 1 0 0 00 1 โ ๐ฟ 0 00 0 1 00 0 0 1
]
๐(๐ก)
From (12), we get ๐(๐ก) = ๐(0)๐๐ฅ๐ (โซ๐
0๐ด๐๐ก), where ๐(0) is an identity matrix and hence
the eigen values corresponding to matrix ๐ are as follows:
๐3 = ๐๐ฅ๐ (โ(๐ท3 + ๐0))๐ < 1, ๐4 = ๐๐ฅ๐(โ๐ท3๐) < 1, ๐1 = ๐๐ฅ๐(๐๐) > 1,
๐2 = (1 โ ๐ฟ)๐๐ฅ๐โซ๐
0
(โ(๐ท + ๐2๏ฟฝฬ๏ฟฝ2(๐ก)๐โ๐1๐)) ๐๐ก < 1.
Therefore, according to the Floquet theory (Bainov and Sineonov, 1993) the pest eradication
periodic solution is unstable as |๐1| > 1.
Remark 1: The effect of delay can be easily seen in the value of ๐๐๐๐ฅ which helps in reducing its
value.
(ii) The local stability of (๐, 0, ๏ฟฝฬ๏ฟฝ1(๐ก), ๏ฟฝฬ๏ฟฝ2(๐ก)) is proved in the similar fashion. We define ๐ = ๐ +๐1(๐ก), ๐ = ๐2(๐ก), ๐1 = ๏ฟฝฬ๏ฟฝ1(๐ก) + ๐3(๐ก), ๐2 = ๏ฟฝฬ๏ฟฝ2(๐ก) + ๐4(๐ก) and the system (1 โ 2)โฒ๐ linearized
form is as follows:
๐๐1(๐ก)
๐๐ก= โ๐๐1(๐ก) โ ๐1๐2
๐๐2(๐ก)
๐๐ก= (๐1๐1 โ ๐ท โ ๐2๏ฟฝฬ๏ฟฝ2(๐ก)๐
โ๐1๐)๐2(๐ก)
๐๐3(๐ก)
๐๐ก= ๐2๐2๐2(๐ก)๏ฟฝฬ๏ฟฝ2(๐ก)๐
โ๐1๐ โ (๐ท3 + ๐0)๐3(๐ก)
๐๐4(๐ก)
๐๐ก= ๐0๐3(๐ก) โ ๐ท3๐4(๐ก)
}
๐ก โ ๐๐ (13)
๐1(๐ก+) = ๐1(๐ก),
๐2(๐ก+) = (1 โ ๐ฟ)๐2(๐ก)
๐3(๐ก+) = ๐3(๐ก) + ๐1
๐4(๐ก+) = ๐4(๐ก) + ๐2
}
๐ก = ๐๐ (14)
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Let ๐(๐ก) be the fundamental matrix of (13 โ 14), then ๐(๐ก) must satisfy
๐๐(๐ก)
๐๐ก=
[ โ๐ โ๐1 0 0
0 ๐1๐1 โ ๐ท โ ๐2๏ฟฝฬ๏ฟฝ2(๐ก)๐โ๐1๐ 0 0
0 ๐2๐2๏ฟฝฬ๏ฟฝ2(๐ก)๐โ๐1๐ โ(๐ท3 + ๐0) 0
0 0 ๐0 โ๐ท3]
๐(๐ก)
๐๐(๐ก)
๐๐ก= ๐ด๐(๐ก) (15)
Thus, the monodromy matrix of (13 โ 14) is
๐ =
[ 1 0 0 00 1 โ ๐ฟ 0 00 0 1 00 0 0 1
]
๐(๐ก).
From (15) , we get ๐(๐ก) = ๐(0)๐๐ฅ๐(โซ๐
0๐ด๐๐ก), where๐(0) is an identity matrix. Then the
characteristic values obtained for ๐ are as follows:
๐1 = ๐๐ฅ๐(โ๐๐) < 1, ๐2 = (1 โ ๐ฟ)๐๐ฅ๐โซ๐
0
(๐1๐1 โ ๐ท โ ๐2๏ฟฝฬ๏ฟฝ2(๐ก)๐โ๐1๐) < 1,
๐3 = ๐๐ฅ๐((โ(๐ท3 + ๐0) โ ๐)๐) < 1, ๐4 = ๐๐ฅ๐(โ๐ท3๐) < 1.
Therefore, pest eradication periodic solution of (1 โ 2) is locally asymptotically stable as per
Floquet theory (Bainov and Sineonov, 1993) if and only if |๐2| โค 1 which implies ๐ โค ๐๐๐๐ฅ.
Hence, the theorem is proved.
5. Permanence In this section, we will discuss permanence of system (1 โ 2).
Theorem 5.1 The system (1-2) is permanent if ๐ > ๐๐๐๐ฅ.
Proof. Suppose (๐, ๐, ๐1, ๐2) is the solution of the system (1 โ 2), ๐ก being removed for
convenience, We have already proved that ๐(๐ก) โค ๐, ๐(๐ก) โค ๐, ๐1(๐ก) โค ๐ and ๐2(๐ก) โค ๐ โ
๐ก. From, (1 โ 2) we have ๐๐
๐๐กโฅ ๐(๐ โ ๐1๐โ ๐) which implies that ๐(๐ก) > ๐ โ ๐1๐ โ ๐1
for all large t. For small ๐4 > 0, we choose ๐1 = 1 โ ๐ > 0 and also define,
๐2 =โ๐1๐๐ฅ๐(โ(๐ท3 + ๐0)(๐ก โ ๐๐))
1 โ ๐๐ฅ๐(โ(๐ท3 + ๐0)๐)โ ๐4 > 0,
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๐3 =โ๐1๐๐ฅ๐(โ(๐ท3 + ๐0)(๐ก โ ๐๐))
1 โ ๐๐ฅ๐(โ(๐ท3 + ๐0)๐)+(๐1 + ๐2)๐๐ฅ๐(โ๐ท3(๐ก โ ๐๐))
1 โ ๐๐ฅ๐(โ๐ท3๐)โ๐4๐0๐ท3
โ ๐4 > 0.
Now, the system (1 โ 2) can be rewritten as:
๐๐1(๐ก)
๐๐ก= โ(๐ท3 + ๐0)๐1(๐ก)
๐๐2(๐ก)
๐๐ก= ๐0๐1(๐ก) โ ๐ท3๐2(๐ก)
}
๐ก โ ๐๐, (16)
๐1(๐ก+) = ๐1 + ๐1
๐2(๐ก+) = ๐2 + ๐2 } ๐ก = ๐๐. (17)
The system (16 โ 17) is same as (5 โ 6), using same technique, we can easily find that ๐1(๐ก) >๐2 and ๐2(๐ก) > ๐3 โ t. Hence, for proving the permanence we have only have to prove ๐4 >0, such that ๐(๐ก) โฅ ๐4โ t which will be done in two steps.
Step 1: Let ๐(๐ก) โฅ ๐4 is false โ a ๐ก1 โ (0,โ) s.t ๐(๐ก) < ๐4 โ ๐ก > ๐ก1. Using this
supposition, we get subsystem of (1 โ 2):
๐๐1(๐ก)
๐๐กโค ๐2๐2๐๐4๐
โ๐1๐ โ (๐ท3 + ๐0)๐1, ๐ก โ ๐๐
๐1(๐ก+) = ๐1(๐ก) + ๐1, ๐ก = ๐๐, ๐ = 1,2,3โฆโฆ.
Let us assume the comparison system:
๐๏ฟฝฬ ๏ฟฝ1(๐ก)
๐๐กโค ๐2๐2๐๐4๐
โ๐1๐ โ (๐ท3 + ๐0)๏ฟฝฬ ๏ฟฝ1(๐ก), ๐ก โ ๐๐
๏ฟฝฬ ๏ฟฝ1(๐ก+) = ๏ฟฝฬ ๏ฟฝ1(๐ก) + ๐1, ๐ก = ๐๐, ๐ = 1,2,3. . . .
} (18)
Using lemma 3.1, equation (18) has periodic solution
(๐ก)๏ฟฝฬฬ ๏ฟฝ1 =๐2๐2๐4๐๐๐ฅ๐(โ๐1๐)
๐ท3 + ๐0+๐1๐๐ฅ๐(โ(๐ท3 + ๐0)(๐ก โ ๐๐))
1 โ ๐๐ฅ๐(โ(๐ท3 + ๐0)๐)
which is globally asymptotically stable. Then, โ an ๐5 > 0 s.t
๐1(๐ก) โค ๏ฟฝฬฬ ๏ฟฝ1(๐ก) <๐2๐2๐4๐๐๐ฅ๐(โ๐1๐)
๐ท3 + ๐0+๐1 exp(โ(๐ท3 + ๐0)(๐ก โ ๐๐))
1 โ exp(โ(๐ท3 + ๐0)๐)+ ๐5 > 0.
For sufficiently large ๐ก. Thus we find the following subsystem of (1 โ 2):
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๐๐2(๐ก)
๐๐ก= ๐0 (
๐2๐2๐4๐๐๐ฅ๐(โ๐1๐)
๐ท3 + ๐0+๐1 exp(โ(๐ท3 + ๐0)(๐ก โ ๐๐))
1 โ exp(โ(๐ท3 + ๐0)๐)+ ๐5) โ ๐ท3๐2, ๐ก โ ๐๐
๐2(๐ก+) = ๐2 + ๐2, ๐ก = ๐๐, ๐ = 1,2,3. . . . . . .
}
(19)
Consider the comparison system (19) as follows:
๐๏ฟฝฬ ๏ฟฝ 2(๐ก)
๐๐ก= ๐0(
๐2๐2๐4๐๐๐ฅ๐(โ๐1๐)
๐ท3 + ๐0+๐1๐๐ฅ๐(โ(๐ท3 + ๐0)(๐ก โ ๐๐))
1 โ ๐๐ฅ๐(โ(๐ท3 + ๐0)๐)+ ๐5)(๐ก) โ ๐ท3๏ฟฝฬ ๏ฟฝ2(๐ก), ๐ก โ ๐๐
๏ฟฝฬ ๏ฟฝ2(๐ก+) = ๏ฟฝฬ ๏ฟฝ2(๐ก) + ๐2, ๐ก = ๐๐, ๐ = 1,2,3. . . . . . .
}
(20)
Similarly, system (20) also has a periodic solution
๐2(๐ก) < ๏ฟฝฬฬ ๏ฟฝ2(๐ก) <โ๐1๐๐ฅ๐(โ(๐ท3 + ๐0)(๐ก โ ๐๐))
1 โ ๐๐ฅ๐(โ(๐ท3 + ๐0)๐)+(๐1 + ๐2)๐๐ฅ๐(โ๐ท3(๐ก โ ๐๐))
1 โ ๐๐ฅ๐(โ๐ท3๐)
+๐0๐ท3(๐2๐2๐4๐๐๐ฅ๐(โ๐1๐)
(๐ท3 + ๐0)+ ๐5)
๏ฟฝฬฬ ๏ฟฝ2(๐ก) <โ๐1 exp(โ(๐ท3 + ๐0)(๐ก โ ๐๐))
1 โ exp(โ(๐ท3 + ๐0)๐)+(๐1 + ๐2) exp(โ๐ท3(๐ก โ ๐๐))
1 โ exp(โ๐ท3๐)
+๐0
๐ท3(๐2๐2๐4๐๐๐ฅ๐(โ๐1๐)
(๐ท3 + ๐0)๐5) (21)
which is globally asymptotically stable and โ an ๐6 > 0 s.t
๐2(๐ก) < ๏ฟฝฬฬ ๏ฟฝ2(๐ก) <โ๐1๐๐ฅ๐(โ(๐ท3 + ๐0)(๐ก โ ๐๐))
1 โ ๐๐ฅ๐(โ(๐ท3 + ๐0)๐)+(๐1 + ๐2)๐๐ฅ๐(โ๐ท3(๐ก โ ๐๐))
1 โ ๐๐ฅ๐(โ๐ท3๐)
+๐0๐ท3(๐2๐2๐4๐๐๐ฅ๐(โ๐1๐)
(๐ท3 + ๐0)+ ๐5) + ๐6.
It shows that โ a ๐1 > 0 s.t for ๐๐ < ๐ก โค (๐ + 1)๐, we are having the following subsystem of
(1 โ 2): ๐๐(๐ก)
๐๐กโฅ [๐1๐1๐1 โ ๐2(๏ฟฝฬฬ ๏ฟฝ2(๐ก) + ๐6)๐
โ๐1๐ โ ๐ท]๐, ๐ก โ ๐๐
๐(๐ก+) = (1 โ ๐ฟ)๐(๐ก), ๐ก = ๐๐, ๐๐๐, ๐ก > ๐1} (22)
Integrating the system, (22) on (๐๐, (๐ + 1)๐], ๐ โฅ ๐1 (here,๐1 is the nonnegative integer and
๐1๐ โฅ ๐1), then we obtain that,
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๐((๐ + 1)๐) โฅ (1 โ ๐ฟ)๐(๐๐+)๐๐ฅ๐(โซ(๐+1)๐
๐๐
(๐1๐1๐1 โ ๐2(๏ฟฝฬฬ ๏ฟฝ2(๐ก) โ ๐6)๐โ๐1๐ โ๐ท)๐๐ก)
= ๐(๐๐+)๏ฟฝฬ ๏ฟฝ
where, ๏ฟฝฬ ๏ฟฝ(1 โ ๐ฟ)๐(๐๐+)๐๐ฅ๐(โซ(๐+1)๐
๐๐(๐1๐1๐1 โ ๐2(๏ฟฝฬฬ ๏ฟฝ2(๐ก) โ ๐6)๐
โ๐1๐ โ ๐ท)๐๐ก) > 1, as, ๐ >
๐๐๐๐ฅ, therefore, for๐5 > 0, we obtain that,
(๐1๐1๐1 โ ๐2๐6๐๐ฅ๐(โ๐1๐) โ ๐ท)๐ โ๐2๐0๐๐ฅ๐(โ๐1๐)
๐ท3(๐2๐4๐๐๐ฅ๐(โ๐1๐)
(๐ท3 + ๐0)โ ๐5) โ
๐2(๐1
(๐ท3 + ๐0)
(๐1 + ๐2) exp(โ๐1๐)
๐ท3) โ ๐๐๐(
1
1 โ ๐ฟ) > 1.
Thus, ๐((๐1 + ๐)๐) โฅ ๐(๐1๐+)๏ฟฝฬ ๏ฟฝ๐ โ โ as ๐ โ โ, which violoates our assumption ๐(๐ก) <
๐4, for every ๐ก > ๐ก2. Hence there exists a ๐ก2 > ๐ก1 s.t ๐(๐ก2) โฅ ๐4.
Step 2: If ๐(๐ก) โฅ ๐4 โ ๐ก โฅ ๐ก2, then our aim will be fulfilled. On the contrary let us assume
that ๐(๐ก) < ๐4 for some ๐ก > ๐ก2. Let ๐กโ = inf{๐ก|๐(๐ก) < ๐4, ๐ก > ๐ก2}, then there will be two
cases:
Case 1: Let๐กโ = ๐1๐, ๐1 โ ๐+ . In this case ๐(๐ก) โฅ ๐4 for ๐ก โ [๐ก2, ๐ก
โ) and (1 โ ๐ฟ)๐4 โค๐(๐กโ+ = (1 โ ๐ฟ)๐(๐กโ) < ๐4) . Let ๐2 = ๐2๐ + ๐3๐, where ๐2 = ๐2
โฒ + ๐2โฒโฒ, ๐2
โฒ , ๐2 โฒโฒ and ๐3
satisfy these inequalities:
๐2โฒ๐ > โ1
๐ท3 + ๐0๐๐
๐5๐ + ๐1
,
๐2โฒโฒ๐ > โ1
๐ท3 + ๐0๐๐
๐6๐+ ๐2
,
(1 โ ๐ฟ)๐2+๐3exp (๐๐2๐)exp (๐3๐) > 1,
๐ = ๐1๐1๐1 โ ๐2๐6๐๐ฅ๐(โ๐1๐) โ ๐ท < 0. Now, we claim that โ a time ๐ก2โฒ โ (๐กโ, ๐กโ + ๐2) such
that ๐(๐ก2โฒ ) โฅ ๐4, if it is not true, then ๐(๐ก2
โฒ ) < ๐4, ๐ก2โฒ โ (๐กโ, ๐กโ + ๐2). If the system (18) is taken
with initial value ๏ฟฝฬ ๏ฟฝ1(๐กโ+) = ๐1(๐ก
โ+), then from lemma (3.1) for ๐ก โ (๐๐, (๐ + 1)๐],
we have
๏ฟฝฬ ๏ฟฝ1(๐ก) = (๏ฟฝฬ ๏ฟฝ1(๐กโ+) โ
๐2๐2๐4๐๐๐ฅ๐(โ๐1๐)
๐ท3+๐0+
๐1
1โ๐๐ฅ๐(โ(๐ท3+๐0)๐))๐๐ฅ๐(โ(๐ท3 + ๐0)(๐ก โ ๐ก
โ)) + ๏ฟฝฬฬ ๏ฟฝ1(๐ก),
for ๐1 โค ๐ โค ๐1 + ๐2 + ๐3 which shows that |๏ฟฝฬ ๏ฟฝ1(๐ก) โ ๏ฟฝฬฬ ๏ฟฝ1(๐ก)| โค (๐ + ๐1)๐๐ฅ๐(โ(๐ท3 +
๐0)(๐ก โ ๐1๐)) < ๐5, and ๐1(๐ก) โค ๏ฟฝฬ ๏ฟฝ1(๐ก) < ๏ฟฝฬฬ ๏ฟฝ1(๐ก) + ๐5 for ๐กโ + ๐2
โฒ ๐ โค ๐ก โค ๐กโ + ๐2.
Now, from the system (18) with initial values ๏ฟฝฬ ๏ฟฝ2(๐กโ + ๐2
โฒ ๐) = ๐2(๐กโ + ๐2
โฒ ๐) โฅ 0 and again from
lemma (3.1), we have |๏ฟฝฬ ๏ฟฝ1(๐ก) โ ๏ฟฝฬฬ ๏ฟฝ1(๐ก)| < (๐ + ๐2)๐๐ฅ๐(๐ท3(๐ก โ (๐1 +๐2โฒ)๐)) <
๐6, and ๐2(๐ก) โค ๏ฟฝฬ ๏ฟฝ2(๐ก) < ๏ฟฝฬฬ ๏ฟฝ2(๐ก) + ๐6 for ๐กโ + ๐2
โฒ ๐ + ๐2โฒโฒ๐ โค ๐ก โค ๐กโ + ๐2, which shows that
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system (22) holds for [๐กโ + ๐2๐, ๐กโ + ๐2].
Integrating equation (22) on [๐กโ + ๐2๐, ๐กโ + ๐2], we have
๐((๐1 + ๐2 + ๐3)๐) โฅ ๐((๐1 + ๐2)๐)(1 โ ๐ฟ)๐3๐๐ฅ๐(๐3๐) (23)
In addition from the system (1 โ 2), we have
๐๐(๐ก)
๐๐ก= (๐1๐1๐1 โ ๐2๐๐
โ๐1๐ โ ๐ท) ๐(๐ก) = ๐๐(๐ก), ๐ก โ ๐๐
๐(๐ก+) = (1 โ ๐ฟ)๐, ๐ก = ๐๐, ๐ = 1,2,3. . . .} (24)
On integrating (24) in the interval [๐โ, (๐1 + ๐2)๐], it is obtained that
๐((๐1 + ๐2)๐) โฅ ๐4(1 โ ๐ฟ)๐2๐๐ฅ๐(๐๐2๐) (25)
Now substitute (25) into (24), we get that
๐((๐1 + ๐2 + ๐3)๐) โฅ ๐4(1 โ ๐ฟ)๐2+๐3๐๐ฅ๐(๐3๐)๐๐ฅ๐(๐๐2๐) > ๐4 (26)
which contradicts to our supposition, so there exists a time ๐ก2โฒ โ [๐กโ, ๐กโ + ๐2] such that๐2
โฒ โฅ
๐4. Let ๏ฟฝฬ๏ฟฝ = inf{๐ก|๐ก โฅ ๐กโ, ๐(๐ก) โฅ ๐4} ,since 0 < ๐ฟ < 1, ๐(๐๐+) = (1 โ ๐ฟ)๐(๐๐) <
๐(๐๐)and๐(๐ก) < ๐4, ๐ก โ (๐กโ, ๏ฟฝฬ๏ฟฝ). Thus,๐(๏ฟฝฬ๏ฟฝ) = ๐4.
Suppose๐ก โ (๐กโ + (๐ โ 1)๐, ๐โ + ๐๐] (๐is a positive integer) and ๐ โค ๐2 + ๐3, from the system
(24), we have
๐(๐ก) โฅ ๐(๐กโ + (๐ โ 1)๐)๐๐ฅ๐(๐(๐ก โ ๐กโ โ (๐ โ 1))๐)
๐(๐ก) โฅ ๐(๐๐+)๐๐ฅ๐(๐๐(๐ โ 1))(1 โ ๐ฟ)๐โ1๐๐ฅ๐(๐๐)
๐(๐ก) โฅ ๐4(1 โ ๐ฟ)๐๐๐ฅ๐(๐๐๐)
๐(๐ก) โฅ ๐4(1 โ ๐ฟ)(๐2 + ๐3)๐๐ฅ๐((๐2 + ๐3)๐๐) โ ๏ฟฝฬ ๏ฟฝ4
for ๐ก > ๏ฟฝฬ๏ฟฝ. The same argument can be continued since ๐(๏ฟฝฬ๏ฟฝ) โฅ ๐4. Hence ๐(๐ก) โฅ ๏ฟฝฬ ๏ฟฝ4โ๐ก > ๐ก2.
Case 2: If ๐กโ โ ๐๐, then ๐(๐กโ) = ๐4 and ๐(๐ก) โฅ ๐4, ๐ก โ [๐ก2, ๐กโ]. Suppose ๐กโ โ (๐1
โฒ๐, (๐1โฒ +
1)๐], we are having two subcases for ๐ก โ [๐กโ, (๐1โฒ + 1)๐] as given below:
Case a: ๐(๐ก) โค ๐4, ๐ก๐[๐กโ, (๐1
โฒ + 1)๐], we claim that there exists a ๐ก3ํ[(๐1โฒ + 1)๐, (๐1
โฒ +
1)๐ + ๐2] s.t ๐(๐ก3) > ๐4. Otherwise, integrating system (24) on the interval [(๐1โฒ + 1 +
๐2)๐, (๐1โฒ + 1 + ๐2 + ๐3)๐] , we have, ๐((๐1
โฒ + 1 + ๐2 + ๐3)๐) โฅ ๐((๐1โฒ + 1 + ๐2)๐)(1 โ
๐ฟ)๐3๐๐ฅ๐(๐3๐)
Since ๐(๐ก) โค ๐4, ๐ก โ [๐กโ, (๐1
โฒ + 1)๐], therefore, (13) holds on [๐กโ, (๐1โฒ + ๐2 + ๐3)๐].
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Thus,
๐((๐1โฒ + 1 + ๐2)๐) = ๐(๐ก
โ)(1 โ ๐ฟ)๐2๐๐ฅ๐(๐(๐1โฒ + 1 + ๐2)๐ โ ๐ก
โ)
๐((๐1โฒ + 1 + ๐2)๐) โฅ ๐4(1 โ ๐ฟ)
๐2๐๐ฅ๐(๐๐2๐)
and
๐((๐1โฒ + 1 + ๐2 + ๐3)๐) โฅ ๐4(1 โ ๐ฟ)
๐2+๐3๐๐ฅ๐(๐๐2๐)๐๐ฅ๐(๐3๐) > ๐4
which negates the assumption. Let ๏ฟฝฬ๏ฟฝ = inf {๐ก|๐ โฅ ๐4, ๐ก > ๐กโ}, then ๐(๏ฟฝฬ๏ฟฝ) = ๐4 and ๐ < ๐4, ๐ก โ
(๐กโ, ๏ฟฝฬ๏ฟฝ). Choose ๐ก โ (๐1โฒ๐ + (๐โฒ โ 1)๐, ๐1
โฒ๐ + ๐โฒ๐] โ (๐กโ, ๏ฟฝฬ๏ฟฝ), ๐โฒ is a positive integer and ๐โฒ < 1 +
๐2 + ๐3, we have
๐(๐ก) โฅ ๐((๐1โฒ + ๐โฒ โ 1)๐+)๐๐ฅ๐(๐(๐ก โ (๐1
โฒ + ๐โฒ โ 1)๐))
๐(๐ก) โฅ (1 โ ๐ฟ)๐โฒโ1๐(๐กโ)๐๐ฅ๐(๐(๐ก โ ๐กโ))
๐(๐ก) โฅ ๐4(1 โ ๐ฟ)๐2+๐3๐๐ฅ๐(๐(๐2 + ๐3 + 1)๐).
Hence, ๐ โฅ ๏ฟฝฬ ๏ฟฝ4 for ๐ก โ (๐กโ, ๏ฟฝฬ๏ฟฝ). For ๐ก > ๏ฟฝฬ๏ฟฝ, we can proceed in the same manner since ๐(๏ฟฝฬ๏ฟฝ) โฅ ๐4.
Case b: If โ a ๐ก โ (๐กโ, (๐1โฒ + 1)๐) s.t ๐(๐ก) โฅ ๐4. Let ๏ฟฝฬ๏ฟฝ = inf (๐ก|๐(๐ก) โฅ ๐4, ๐ก > ๐ก
โ), then
๐(๐ก) < ๐4 for ๐ก โ [๐กโ, ๐กฬ ) and ๐(๐กฬ ) = ๐4. For ๐ก โ [๐กโ, ๐กฬ ) (24) holds. On integrating (24) on
๐กโ, ๏ฟฝฬ๏ฟฝ, we obtain
๐ โฅ ๐(๐กโ) โฅ ๐๐ฅ๐(๐(๐ก โ ๐กโ) โฅ ๐4๐๐ฅ๐(๐๐) > ๏ฟฝฬ ๏ฟฝ4
Since, ๐(๏ฟฝฬ๏ฟฝ) โฅ ๐4 for ๐ก > ๏ฟฝฬ๏ฟฝ, we can proceed in the same manner. Hence, we have ๐(๐ก) โฅ ๏ฟฝฬ ๏ฟฝ4 for
all ๐ก > ๐ก2. Therefore we can conclude that ๐(๐ก) โฅ ๏ฟฝฬ ๏ฟฝ4 for all ๐ก โฅ ๐ก2 in both cases.
6. Numerical Section For the intended process, we have taken data per week in view of the short term life cycle of the
insect population under investigation. Our aim is to validate the analytical results numerically. We
have considered numerical values for the following set of parameters in reference to (Jatav and
Dhar, 2014) as mentioned in Table 2.
Table 2. Parametric values
Parameters ๐0 r a1 b1 d1 ๐ a2 b2 D D3
Values 50 1 1 0.1 0.3 0.2 0.3 0.5 0.03 25
Using the above parametric values, we obtained the threshold value ๐๐๐๐ฅ for the parameters per
week as 0.8 . It is proved that (๐, 0, ๏ฟฝฬ๏ฟฝ1(๐ก), ๏ฟฝฬ๏ฟฝ1(๐ก))is locally asymptotically stable if ๐ = 0.5 <๐๐๐๐ฅ as stated above in the theorem 4.1 (Figure 2-5). Further, it is also verified that the system
(๐ด โ ๐ต) is permanent if ๐ = 4 > ๐๐๐๐ฅ(Figure 6-9) which is inline with theorem 5.1. It is also
shown that if there is no biological control, that is, ๐1 = 0 and ๐2 = 0, ๐1 = 0 and ๐2 >0 or ๐1 > 0 and ๐2 = 0, then both plants and pest population survives.This concludes, that
solely using chemical pesticide cannot eradicate pest population (Figure 10-14).
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Figure 2. Plant population (๐(๐ก)) existing
Figure 3. Pest population (๐(๐ก)) vanishes
Figure 4. Periodic behaviour of ๐1(๐ก)
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Figure 5. Periodic behaviour of (๐2(๐ก))
Figure 6. Plant population (๐(๐ก)) exists
Figure 7. Pest population (๐(๐ก)) survives
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Figure 8. Immature natural enemies (๐1(๐ก)) exists
Figure 9. Behaviour of mature natural enemies (๐2(๐ก))
Figure 10. Existence of the pest population(๐(๐ก)) for ๐1, ๐2 = 0
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Figure 11. Immature natural enemy (๐1(๐ก)) vanishes for๐1, ๐2 = 0
Figure 12. Mature natural enemy (๐2(๐ก)) for ๐1, ๐2 = 0
Figure 13. Plant population (๐(๐ก)) is stable for๐1 = 100, ๐2 = 50
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Figure 14. Pest population (๐(๐ก)) declines for๐1 = 100, ๐2 = 50
7. Conclusion In this paper, we have examine the effects of hybrid approach to control the pests by release of
natural enemies and pesticides impulsively. It is evident that pest population can become extinct
when large amount of the natural enemies are released impulsively. Thus, integrated pest
management reduces pest quickly rather than using any one of the methods. Hence, in this paper,
we have shown that by incorporating delay in the pests, we are able to control the pest population
but to a lower threshold value which in a way is helpful as it is leading to early reduction in the pest
which is not only economic but it also prevents pest resistance to crops. Incorporating delay
lowered the threshold level from to ๐๐๐๐ฅ = 7 to ๐๐๐๐ฅ = 0.8 for the same set of parameters as in
(Jatav & Dhar, 2014). Thus, we can conclude that various control measures should be applied
collectively for the eradication of pest. Such a practice improves economy as it is cost effective and
synonymous with sustainable development.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgements
The first author would like to express her sincere thanks to her guide, co-guide for their constant guidance and support
and special thanks to all the reviewers and editor.
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