Prey-predator Dynamics under Herd Behavior of Prey

In prey-predator system prey populations take various defensive mechanism to save themselves from predator and/or to get advantage in inter-species competition. In this work, we analyzed the dynamics of preypredator system with two prey species and single predator population in which two prey populations are fighting for the same resources. Here, it is assumed that one prey population exhibits herd behavior as their own defense mechanism and second prey population releases toxin elements which gives them advantage of inter-species competition. From the analysis of our model we observed that the strategy of herd behavior as self defense mechanism is stronger than the toxin producing strategy. Also due to herd behavior of prey the model shows ecologically meaningful dynamics near origin.


I. INTRODUCTION
Predator-prey interactions is an important area in ecology and mathematical ecology for which many problems still remain open [1].Lotka [2]-Volterra [3] model was the first in this context to describe the interaction of species.After that many complex models are developed to study prey-predator systems.In the last few decades a number of prey-predator models have been studied extensively, where several possible dynamics have been considered, starting from the famous work of Lotka and Volterra to the recent works [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] which depart from classical assumptions.Several researcher has shown that interaction between prey-predator and corresponding response function plays an important role in the dynamics of prey-predator model.Till now several classical response function namely mass action Holling, type-II, type-III, and type-IV, Tanner ratio dependence [19], Beddington-DeAngelis [20] has been introduced.Cosner at el [21] have nicely shown how different types of response function can be introduced when prey-predator forms different kind of spatial pattern.In a recent work of Ajraldi et al [22] took a different view of interaction among the species.They argued the interactions not just of individuals of two populations that intermingle on a common ground, but consider a more elaborated social model, in which the individuals of one population gather together in herds, to wander about in search of food sources and for defensive purposes.The concept of group defense has already been considered, via suitable assumptions on the form and type of functional responses of the prey modeled in very general terms.Specifically, there is a threshold on the size of herd of the prey beyond which the predators' hunting capabilities begin to fall.In other words, the larger the prey population is, the smaller the success of hunting and the corresponding return rate are for predators.A similar reasoning was given by Chottapadhyay at el [23] in to the formulation of a plankton model in which toxic phytoplankton releases poison through the surface of a threedimensional patch.More recently, Braza [24] considered a predator-prey model in which the prey exhibits herd behavior, so that the predator interacts with the prey along the outer corridor of the herd of prey.As a mathematical consequence of the herd behavior, they considered competition models and predator-prey systems in which interaction terms use the square root of the prey population rather than simply the prey population.
Braza shown that the origin to be either locally stable or unstable, depending on the location of the values of the predator and prey populations in the phase plane which is ecologically more sensible.On the other hand, several researcher have formalized different mathematical models taking account of spread of disease in prey population [25][26][27][28], but not much intensive attention has been given to the disease in predator which itself can give huge advantage to the prey population.In this paper we modified the model as proposed by Ajraldi et al. [19] and Braza [21] and incorporate the idea of spread of disease in predator as suggested by Haque and Venturino [29].Here we analyze the dynamical behavior of prey predator model where predator papulation is suffering from disease and prey exhibits herd behavior (as assumed by Ajraldi et al. [19] and Braza [21]).We have analyzed the behaviors of the model system near the origin, and in other equilibrium points.The stability near the origin has been studied from a new angle with the help of non-linear analysis (using appropriate powers of the variable) which allows more realistic results.Finally, we find the impact of herd behavior mechanism of prey population to the model system both analytically and numerically.

II. Mathematical Model:
We consider a prey-predator model where the predator population suffers from disease and prey is gathered in a group so it exhibits a herd behavior.Let is prey population density, is susceptible predator population density and is infected predator population density.We consider that the prey population grows logistically in the absence of predator with intrinsic growth rate and be the carrying capacity of the prey population.
The susceptible predator has another food source rather than so we introduce logistic growth term in susceptible predator population and let is the intrinsic growth rate of the susceptible predator.Also let be the carrying capacity of predator population which is shared by both susceptible predators and infected predators.The infections are spread out according to the law of mass action and the rate of infection is denoted by .The infected predator dies out at a rate .Here we also assume that only susceptible predators consumes the prey and in the corresponding response function the square root of the prey population was taken due to herd structure of prey as suggested by Braza by [24].With these assumptions we can write the model system as follows:

Boundedness of the system:
Let us define W= .The time derivatives = .Therefore we obtained as (1 ) .
Applying the result of differential inequality [31] we obtain which implies that 0 as t .
Hence all the solution of (2.1) are bounded.

Equilibria and Existence
The system of equations (2.2) has six equilibrium points, namely

Stability Analysis
In this section we perform stability analysis our model system.We first calculate the Jacobian matrix of the system (2.2) which is as follows: (3.1) First we note that the prey only equilibrium point (0, 1, 0) is always unstable, which is clear if we evaluate the Jacobian matrix (3.1) at (0, 1, 0).This is ecologically coherent since at this stage just introduction of small amount of predator population will bust the dynamics and therefore prey only equilibrium point (0, 1, 0) will not be stable at all.Also through a routine calculation we can show that the predator only equilibrium point (0, 1, 0) is always stable.This is in fact happening due to the alternative food source of the predator, which essentially gives lots of support to the predator population when prey supply is in crisis.
Next we notice that zero-population equilibrium point and its steady state is very interesting.The stability of the zero-population equilibrium point cannot be determined by simply evaluating the Jacobian matrix (3.1) at , since the term is indeterminate.Basically, system (2.2) is not linearizable at (0, 0, 0), because square root is involved at the response function.Thus a natural course of action is to rescale the variable , say , so that the square root singularity can be removed.The authors in [22] did this with their models and capture the expected dynamics near the origin.However, for the predator-prey model the dynamics near the origin are subtle, and such a re-scaling can serve to mask the true dynamics near the origin, rather than readily illuminating them (see Braza [24] and Matia at el [19]).It is best to highlight the effect of the square root term by a local nonlinear analysis of the equations (2.2) to uncover the singular dynamics near the origin.For the stability analysis near origin it is reasonable to neglect the infected predators population as its mother resource namely suspectable predator is very small.Furthermore, for the true dynamics near origin the assumption of alternative food source must be dropped otherwise the suspectable predator population will get undue advantage due to this trust of alternative food source.Thus under these two realistic assumptions our model system (2.2) takes the following simple form: To determine the behavior near the origin we must consider equation (3.2) for which and case it is clear that and .So near origin the model system (3.3) can be reduced to Here, it may be tempting to use just the linear term so that which would imply that the origin is a saddle, but that is correct when .It may be the case that is bigger or the same size as .To do the analysis correctly, we first start with so that with , and consider .There are three distinguishing cases: , and .The first case in which gives saddle behavior near the origin, as already noted.This can be explained in the following way: If the prey population is considerable smaller than the predator population, the prey first goes extinct, causing the predator to follow the same suit.This makes perfect ecological sense, yet it is a shortcoming of other models in which, as a consequence of the origin being a saddle and the prey population recovers no matter how small it is relative to the predator population.In contrast, the net effect of the square root model is that the (square root) interaction term effectively imposes the equivalent of a minimum sustaining level on the populations to sustain themselves, and that is entirely reasonable.The case q=2 is the intermediary case.In this case two scenarios namely extinction and saddle can happen depending on other local conditions.These can be established through the routine calculation as shown in [24].The stability condition of other equilibrium points of our model system has been stated in the following lemma:

IV. NUMERICAL SIMULATION AND DISCUSSIONS
It is to be noted that in the aforesaid Lemmas analytically we established the conditions for stabilities of disease free equilibrium point ( and interior equilibrium point , , which are not only complicated to understand but also very complex to interpret the system dymamics from it.Thus, we

2 . 1
Positive invariance of the system: Let us put equation (2.2) in a vector form by setting .easy to cheek in the above equation that whenever choosing Y( such that =0, then 0, Y(t) ,(i=1, 2, 3).Due to lemma (Yang et al.[30]) any solution of the above equation with Y( , say Y(t)=Y(t, Y( )), such that Y( ) for all t>0.
0, 0), (0, 1, 0) exist for all parametric values.The equilibrium point 0, , ) will exist if m > n; that is if .The equilibrium point ( is exists if Where can be obtained from the equation =0.It clear that this equation must has a positive root if .can be found from the equation =0, whose one positive root is grunted whenever .

5 )
The curve represented by the equation (3.5) is part of a parabola that starts at and terminates on the axis at , the inequality holds true since .The implication of this is that a trajectory in the phase plane with terminates at at some positive value of , after which the predator declines to zero because of .

Lemma 3 . 1 :
The We proof this by using Routh-Hertz criterion.At that point the variational matrix is

Figure 1 :
Figure 1: Depict the stability of the interior equilibrium point , .The model system (2.1) has been solved using following set of values of the parameters: ,, , , , , and .

Figure 2 :
Figure 2: Depict the stability of the planner equilibrium point ( .The model system (2.1) has been solved using following set of values of the parameters: , , , , , , and .