Prey-Predator Model with Two-Stage Infection in Prey: Concerning Pest Control

1Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, West Bengal 711103, India 2Shibpur Sri Ramkrishna Vidyalaya, 92 Kshetra Banerjee Lane, Shibpur, Howrah 711102, India 3Department of Mathematics, Abhedananda Mahavidyalaya, Sainthia, West Bengal 731234, India 4Department of Mathematics, Sundarban Hazi Desarat College, Pathankhali, West Bengal 743611, India


Introduction
It is of great interest to consider the effects of pesticide on ecological communities from both an environmental and economical point of view with the rapid development of modern technology, industry, and agriculture.The main foods of mankind are different types of crops or vegetables.Due to the limitations of land for planting, our main interest is not to destroy even a single mole of food.But in nature many times the leaves or branches of those crop trees are consumed by so many small insects, animals, weeds, and so forth for their survival indicating the agriculture resource would severely be damaged from the environment.These insects or animals or weeds are known as pests.In a word an organism which harms agriculture by means of feeding on crops or parasitizing livestock is known to us as a pest.The term pest not only is used to refer to harmful animals but also relates to all other harmful organisms, namely, fungi, bacteria, virus parasites, and so forth.It is also possible that an animal may be a pest in one case but beneficial or domesticated in another situation.But in this paper by the word pest we mean those livelihoods which are involved in the cause of the damage to the human populations directly or indirectly.
The control of pest is a worldwide problem.The most common technique of the pest control is use of chemical insecticides to control the pest population.The use of chemical insecticides is an attempt to control pest directly at low cost (see Uboh et al. [1]).But in many research works it is shown that these chemicals have many environmental effects that include chemical residing in crop and in the agricultural ecosystems.In nature, other than pest, there are some other types of population species known as natural enemy, which predates those pests.For an example, there are some birds, frog, and so forth which feed on agricultural pests and thus those species can be treated as predator species to the pests.So these populations can be used as one of the biological controls for the pest.A prey-predator model can be used to study pest control phenomenon (see the works of Tang et al. [2], Ghosh et al. [3], Ghosh and Bhattacharya [4], Pei et al. [5], Liang et al. [6], Apreutesei [7], Kar et al. [8], Jana and Kar [9,10], and Mondal et al. [11] and references therein).To control the pest species without using any pesticide control, use of natural enemy is one of the important ways.Again the pest population may be affected due to the infection of some bacterial or viral diseases.So another important natural way to control pest population is the consideration of the effect of some infectious diseases among the targeted pest population.Sun and Chen [12] and Tan and Chen [13] used infection to control the pest population.Some viral or bacterial infection caused the reduction of pest species significantly.Wang and Song [14] used mathematical model by infected pest to control the pest population.Hadeler and Freedman [15], Venturino [16,17], Kar and Mondal [18], Liu and Wang [19], Guo and Chen [20], Y. Zhang and Q. Zhang [21], Wang et al. [22], and Wang and Chen [23] also worked on prey-predator model with the disease in the prey populations.In our present paper we consider two stages of infected pest, namely, infected pest in the first stage and infected pest in the last stage.The two stages of the infection are considered on the assumption that the predator population would consume infected pest of the first stage but the predator would not consume the pest population at the second stage.
Again a prey-predator system plays an important role when an alternative source of food to the predator population is supplied (see Pahari and Kar [24], Srinivasu and Prasad [25], Srinivasu [26], Sabelis and van Rijn [27], Harwood and Obryeki [28], etc.).To control the pest, the effect of alternative food takes some attention compared to ordinary predatorprey dynamics in the context of biological conservation.For both the conservation of predator and reduction of the pest, an alternative source of food must be given to the natural enemy of the pest (see Jana and Kar [10]).
The paper is organized in the following schemes: In Section 2, we formulate the pest control model on the basis of some assumptions and in Section 3, we analyze the complex dynamics of that model.In Section 4, we analyze the model if predator populations would be provided with some supplementary food.In Section 5, we present some numerical simulations for validating our model and in the final section we give some outcomes from our whole paper.

Model Formulation and Description
The following assumptions are made to build the mathematical model for pest control.

P1.
In this paper, we consider mainly two types of species: one is prey, that is, pest, denoted by , and the other is predator, denoted by .But in the presence of disease the pest population is divided into two classes, namely, susceptible () and infected ().Again the infected pest population is classified into two stages: one is the early stage or first stage of infected pest ( 1 ) and the other is the later stage or last stage of infected pest ( 2 ).Therefore, at any time  the total pest population is () = () +  1 () +  2 ().
P2.We consider that the susceptible pest population grows logistically in the absence of the disease (see Das et al. [29]).So if  (> 0) is the intrinsic growth rate and  (> 0) is the carrying capacity, then the growth equation of susceptible pest is P3. Infected pest populations can be divided into two classes, namely, infected pest in the first stage and infected pest in the last stage.The susceptible pest becomes infected through direct contact with the early stage and later stage of infected pest at rates  1 and  2 , respectively, following the mass action law.After primary infection the pest population becomes more infected by the same disease because the infection takes some time to spread among the population.In this scenario,  is assumed as active rate of the disease.The infected pest dies linearly in two stages at rates  and , respectively.Therefore the  −  model for the pest population can be written as ( P4.The predator population consumes only the susceptible pest and the early stage of infected pest and consciously avoids the latter stage of infected pest for its better survival.We choose the predation function as Holling type II functional response /( + ), where  is the capturing rate and  is the half saturation constant, to predate the susceptible pest as some searching time is needed for capturing and handling the prey.But as the primary infected pest population is somewhat weaker than the susceptible pest, no more searching time is needed by the predators for catching the infected pest  1 and thereby the predation function is assumed as Holling type I functional response in the form  1 , where  is the maximum capturing rate.
P5. Conversion factors for the predator are assumed as ℎ 1 and ℎ 2 from the susceptible and primary infected pest, respectively.Also, to improve this model, we incorporate the density dependent factor  for predators, where  is the density dependent mortality rate.Let  be the death rate of the predator.

Analysis of the System
In this section we will study the different dynamical behavior like uniform boundedness, existence of different equilibria and their stability, and so forth.

Boundedness of the System
Proof.See Appendix A.
For (5), we can deduce the following cases.
Case 2. If  1 <  <  0 , then  2 < 0 and  3 > 0, where Therefore, (5) will have two positive roots provided  2 2 > 4 1  3 .In this case, the largest value of  * will be considered and thus the unique interior equilibrium will be obtained.
Thus, the positive value of  * is given by Furthermore, positivity of  * 1 implies that  > ℎ 1 .

Stability Analysis
Theorem 2. The trivial equilibrium  0 is always unstable.
Proof.All the eigenvalues at the trivial equilibrium are , −( + ), −, and −.Therefore, the trivial equilibrium is unstable.
Remark 3. It is concluded from the analysis of Theorem 2 that the trivial equilibrium is a saddle point with one-dimensional unstable and three-dimensional stable manifold.Biologically, it means that the predator population may become extinct from the environment in absence of pest population.Proof.The characteristic equation at  1 is where  = ( − (ℎ 1  − ))/( + ),  1 =  +  +  −  1 , and If  <   , then two eigenvalues are always real and negative, and another two eigenvalues will be negative or having negative real parts, hence the theorem.
Remark 5.The above stability analysis biologically indicates that the pest population increases rapidly towards its carrying capacity in absence of disease and predators while the environmental carrying capacity for susceptible pest population is below some threshold value   .Moreover, the stability of  1 indicates that the predator-free equilibrium  2 does not exist.Now, we discuss the stability criteria of predator-free equilibrium.The characteristic equation at  2 is where If  < 0, then one root of (8) will be real and negative.The other three roots are given by the second part of (8).Clearly, all   ( = 1, 2, 3) are positive and function of .So, we assume  =  as bifurcation parameter.Note that  ∈ (0, ).
We shall discuss the Hopf bifurcation using the approach of Bhattacharyya and Bhattacharya [30].
Theorem 6.If  < 0, then the predator-free equilibrium  2 is locally asymptotically stable for  >  * and unstable for  >  * .Furthermore, the system enters into a single Hopf bifurcation around  2 when  =  * .
Remark 7. Theorem 6 biologically indicates that the infected pest populations coexist with the susceptible pest population in absence of predator and exhibits oscillatory balance behavior, and for large value of the carrying capacity the susceptible pest population may persist for a long time without using pesticide.Also, if  is the period of the bifurcating periodic orbits close to  =  * , then we get Theorem 8.The disease-free equilibrium  3 = ( 3 , 0, 0,  3 ) is locally asymptotically stable if   > 0, for  = 1, 2, 3 and , where the symbols are defined in the proof.
Proof.See Appendix B. Now, the global stability analysis of the system around the disease-free equilibrium will be examined in the next theorem using the approach of reference name.Remark 9. From Theorem 8, it is evident that the disease may become extinct from pest population and in this situation predator population can not consume pest easily.Therefore, the interaction between susceptible pest and predator will be ensured during long time for their survival.
Proof.The details of the proof are given in Appendix C. Remark 11.Theorem 10 provides us with a threshold parameter  * such that, for  <  * , the interior equilibrium is stable and the system bifurcates towards a periodic solution of small amplitude when  is increased.In this scenario, we can observe an oscillatory nature of the system.The period  of the bifurcating periodic orbits close to  =  * is given by

Modified Model
Here we study the system when some additional food is to be supplied from some alternating source to the predator populations and assume that the maximum growth rate due to this food is .So here predator populations get their food from two sources, namely, from the prey population and from the alternative source of food.Therefore only the final differential equation of the predator population will be changed and other equations will remain in the same form.So combining these with the previous assumptions we construct our final revised model as follows: The initial conditions are (0) > 0,  1 (0) > 0,  2 (0) > 0, and (0) > 0.
Among all these equilibria the coordinates of the first four equilibria are the same as system (3).The main difference between system (3) and system ( 13) is the fact that in the later system there is one more equilibrium point which does not exist in previous system (3).The equilibrium point   (0, 0,  2 ) is the new equilibrium point and around this equilibrium the system will be totally pest-free.Further this equilibrium point exists if the maximum growth rate of the predator population due to the application of additional food () is of higher value than the death rate of the predator population.In the next theorem, we discuss the local asymptotic stability of the pest-free equilibrium   .
Proof.The characteristic equation of the system around   is as follows: Therefore it is clear that if  >  and  < ( − ), that is, if  > / +  holds, then all the eigenvalues are negative, hence the theorem.

Numerical Simulation
In this section we give some numerical simulations regarding the stability and bifurcation of system (3) around the interior equilibrium  * .For simulation let us take  = 1.5,  = 100,  1 = 0.24,  2 = 0.26,  = 0.5,  = 0.62,  = 0.6,  = 0.25,  = 0.1,  = 0.12, ℎ 1 = 0.8, ℎ 2 = 0.5,  = 0.3, and  = 0.05.We see that all the conditions for existence of interior equilibrium hold and the interior equilibrium point is  * = (6.45366,1.08759, 2.26581, 7.82435).Now, all the eigenvalues at  * are −2.75187,−0.538162, −0.154566 ± 0.353358.Therefore, according to Routh-Hurwitz criterion the interior equilibrium is locally asymptotically stable, depicted in Figure 1.This figure implies that all the trajectories oscillate initially and after some times they all converge their equilibrium points.Now if the numerical value of the parameter  2 is increased then the oscillation in the solution curve increases.In Figure 2, we see that for  2 = 0.281 oscillation increases but system (3) remains asymptotically stable around  * .Again for  2 = 0.282 and the other parameter value the same as previous we have an unstable interior equilibrium  * (see Figure 3).Further increment of  2 , say,  2 = 2, and the other parameter value the same as that before make the system extinction of predator population and system is unstable around  2 (see Figure 4).
Next to compare systems (3) and ( 13), taking all the other parameters fixed if we take the value of  2 = 0.276 and  = 0.01 make the increment in oscillation but system (14) remains asymptotically stable at the interior equilibrium  * (see Figure 5) but further increment of  2 , say,  2 = 0.281, and all other parameters having the same values make the system unstable around  * (see Figure 6).For further increment of , say, for  = 0.375, the new pest-free equilibrium is created and it becomes asymptotically stable and this phenomenon is depicted in Figure 7. Therefore the major important observation of the modified model is that if the additional food is supplied as a sufficient amount then the system may be pest-free.

Conclusions
In the present paper, we have formulated and discussed a prey-predator model by dividing the prey compartment into three classes, where the disease spreads and developed in two stages.Five equilibria are obtained, namely, trivial, axial, predator-free, infection-free, and interior equilibrium, and the dynamical behavior of the system is investigated.The important critical value   of the carrying capacity is obtained and it has been found that when the environmental carrying capacity for susceptible pest population is below   , the pest population sharply increases in absence of infection and predator.Furthermore, we have seen that the system enters into a single Hopf bifurcation around the predator-free equilibrium and for large value of the carrying capacity the susceptible pest population may persist for a long time without using pesticide.
Also, our global stability analysis of the system around the disease-free equilibrium indicates that the interaction between susceptible pest and predator will be ensured during long time for their survival.We proved that the interior equilibrium is locally asymptotically stable under some parametric conditions and the system enters into the Hopf bifurcation around the interior equilibrium while the density dependent mortality rate passes through the threshold value  * .
Our main objective is to provide a pest-free system.In this regard, stability of the system at the pest-free equilibrium point is the most targeted job.But in the first model (i.e., system (3)), we observe that there is no pestfree equilibrium point whereas the modified system has a pest-free equilibrium point.In this regard, we may conclude that the additional food on the natural enemy of the pest population has a strong impact to remove the pest population.Now, we assume ℎ 2 ≥ ℎ 1 and in this case, we consider (A.5) Then for  < min(, , ), we have The maximum value of (1 − /) +  is ( + ) 2 /4 = Ω (say).Then (A.6) can be written as Applying the theorem of differential inequality [33], we obtain

B. Local Stability of the Disease-Free Equilibrium
The characteristic equation of system (3) around the diseasefree equilibrium is where Hence for local stability of the disease-free equilibrium  3 , the following Routh-Hurwitz criterion must be satisfied: (i)   > 0 ( = 1, 2, 3, 4),

C. Local Stability and Hopf Bifurcation of the Interior Equilibrium
The characteristic equation of system (3) around the interior equilibrium is where  By Routh-Hurwitz criterion, the interior equilibrium  * is locally asymptotically stable if   > 0 ( = 1, 2, 3, 4) and . Now, we shall find out the conditions for which the system enters into Hopf bifurcation around the interior equilibrium  * .
Let us define a function Γ : (0, ∞) →  of  as follows: The necessary and sufficient conditions for Hopf bifurcation to occur are that there exists  =  * such that If  3 and  4 are complex conjugate, then from above we have Re( 3 ) = − 1 < 0, but if they are real roots, then we get  3 < 0 and  4 < 0. Further, since Γ() is a continuous function of all its roots, then there exists an open interval ( * − ,  * + ) containing , where  1 and  2 are complex conjugate for .Suppose

Figure 5 :
Figure 5: Stable solution curve when we supplied an alternating source of food (i.e., revised model) with  = 0.01 and  2 = 0.276.

Figure 7 :
Figure 7: Pest-free solution curves of the revised model for  2 = 2 and  = 0.376.