Global Stability of Predator-Prey System with Alternative Prey

A predator-prey model in presence of alternative prey is proposed. Existence and local stability conditions for interior equilibrium points are derived. Global stability conditions for interior equilibrium points are also found. Bifurcation analysis is done with respect to predator's searching rate and handling time. Bifurcation analysis confirms the existence of global stability in presence of alternative prey.


Introduction
e classical predator-prey model based on the logistic growth principle and Hollings predation theory is as follows: where and represent the density of prey and predator species with carrying capacity . e constant denotes the food intake rate of predator, denotes the food conversion rate to predator, and is the predator's death rate. e constant , being predator's searching rate, and handling time on , respectively. In this model, there is no protection for prey from predator and predator's survival depends on prey alone. Here the predator species totally depends on the prey species and so there is high predation pressure on the prey species. As a result, the prey species has high extinction risk for different searching rate and handling time which is shown in Figure 1. In nature, when the prey population falls below a certain level, the predator searches alternative prey and returns only when the prey population rises to required level. ere are large numbers of three or more species food chain system [1,2] instead of two species system for the survival of prey species. Van Baalen et al. [3] showed the switching fashion from prey species to alternative prey for persistence of predator-prey system. Plants bene�t from providing food to predators even when it is also edible to herbivores which is discussed by Van Rijn et al. [4]. Harwood and Obrycki [5] investigated the role of alternative prey in sustaining predator populations. e role of alternative food for biological pest control in predator-prey system is investigated by many scientists [6,7]. Sahoo [8] studied a food chain model with different functional responses and different growth rates in presence of additional food for construction of real food chain model. Recently, Sahoo [9] showed that additional food is very important for survival of consumer species in an ecosystem. e consequences of providing a predator with additional food and the corresponding effects on the predator-prey dynamics with monotonic and nonmonotonic functional response and its utility in biological control is comparatively studied by Sahoo [10]. But, all of them assumed that the additional food is not dynamic but maintained at a speci�c constant level either by the nature or by an external agency. In this context, I have proposed a predator-prey model with alternative prey (a dynamic additional food for predator). is model is similar to two prey one predator model. e following assumptions are done to formulate the model.
(a) Let be the prey density, let be the density of alternative prey and the density of the predator is .
(b) Both preys are distributed uniformly in the habitat.
(c) e prey and alternative prey grow as per logistic equation in the absence of predators. (d) e predator-prey and predator-alternative prey capture rates are of Holling type II.
(e) e constant is predator's handling time on and , ℎ are predator's searching rate and handling time on , respectively.
With the above assumptions, we formulate the following model as where 1 and are the carrying capacity of prey ( ) and alternative prey ( ), respectively; the constants and are predator's ( ) food intake rate on prey and alternative prey respectively. e constants and are conversion rates of prey and alternative food to predator, respectively; is constant death rate for predator.
Here we assume that predator's food intake rate on prey ( ) is much more greater than that of alternative prey (i.e, ). e parameters and ℎ characterize the alternative prey. is formulation implies that the density of prey ( ) and alternative prey ( ) are scaled with respect to search rate of the predators, this can be done without loss of generality. e system has to be analyzed with the following initial conditions: , , . e main objective of this paper is to investigate the dynamic properties and behaviors of the system. Here I shall analyze the dynamics of the system with respect to predator's ( ) searching rate and handling time (ℎ) on alternative prey . is paper is organized as follows. In Section 2, we show the dissipativeness of the system. e local stability and global stability of the interior equilibrium points of the system are examined in Section 3. Moreover, we discuss the numerical experiment of our system in Section 4. Finally, conclusion is written in Section 5.

Positive Invariance. Let
, , ∈ and where + → and ∈ ∞ + . en system (2) becomeṡ (   4) with ∈ + . It is easy to verify that whenever choosing ∈ such that then [ (for 1, , ). Now any solution of (4) with ∈ + , say , , is such that ∈ + for all (Nagumo, [11]). erefore, erefore, where 1 1 : Applying the theory of differential inequality we obtain For , we have 0 < < 2 . Hence all the solutions of the system (2) that initiate in 3 are con�ned in the region ( ) 3 ∶ 2 , for any 0 , which means that all species are uniformly bounded for any initial value in 3 . is proves the theorem.

Results and Discussion
From Figures 2, 3, and 4 we observe that the system (2)  high handling time as well as searching rate. erefore our system is globally stable for any handling time and searching rate. Now, I have done the bifurcation analysis of the system with respect to predator's searching rate and handling time ℎ taking ecological parameters values 1 3 , 2 2 5, 5, 5 , 5, , and 25 which is �xed throughout the bifurcation analysis. Figure 5 is the bifurcation diagram of the system with respect to searching rate with �xed handling time ℎ 5. From Figure 5, I observe that prey population has extinction risk for lower values of searching rate and for higher searching rate it reaches steady state. On the other hand, alternative prey and predator population show periodic behaviour for lower searching rate and for higher searching rate they go to steady states. erefore, alternative prey and predator population have no extinction risk; they survive in the system always.
Bifurcation analysis with respect to handling time ℎ for �xed searching rate 5, is shown in Figure 6. From Figure 6, I observe that prey population extinct for low values of handling time ℎ but for high values of handling time ℎ the system (2) settles down to steady state. Alternative prey and predator population have no extinction risk, they survive in the system always. For higher searching rate at , the Figure 7 shows that the prey population extinction risk increases for higher values of handling time ℎ. erefore, the increase of searching rate shows prey's extinction from the system for higher values of handling time compare to low searching rate.

Conclusions
I have proposed a predator-prey model in presence of alternative prey. I have derived the condition of local asymptotic stability and global stability of the interior equilibrium points. eoretically, I have shown the global stability under certain condition. Numerically, I have done bifurcation analysis of the system with respect to predator's searching rate and handling time ℎ. From bifurcation diagrams we observe that the system's dynamics are either periodic or stable. e periodic behaviour of the system indicates the existence of stability of the system. From bifurcation analysis, I can conclude that when searching rate is very low, the prey populations are easily captured by predators, and, therefore, prey population has high extinction risk while the alternative prey has no extinction risk. Also it is observed that the prey populations will survive in the system if the searching rate of predator is very high. Similar dynamics are shown with respect to handling time-taking �xed-searching rate. For higher predator's handling time, prey population, and alternative prey go to steady state. erefore, I can conclude that the predator population never extinct for presence of alternative prey.