Stability Analysis and Hopf Bifurcation for ODE System of Predator-Prey Model with Mutual Interference

In this paper, the temporal and spatial patterns of a diffusive predator-prey model with mutual interference under homogeneous Neumann boundary conditions were studied. Specifically, first, taking the intrinsic growth rate of the predator as the parameter, we give a computational and theoretical analysis of Hopf bifurcation on the positive equilibrium for the ODE system. As well, we have discussed the conditions for determining the bifurcation direction and the stability of the bifurcating periodic solutions.


Introduction
The interaction with predator-prey populations and their possible outcomes are probably the most studied topics in ecology, because of their existence and universal relevance, will continue to be one of the dominant topics in both ecology and mathematical ecology [1]. In spite of the predator-prey theory has undergone significant developments in the past several years, many long-standing mathematical and ecological problems still deserve researchers' attention [1] [2] [3] [4]. The functional response of predators to their prey density refers to the change in the density of attached prey per unit time per predator with the change in prey density. It is the average number of prey killed per individual predator per unit of time. For the functional response functions, there are many types. The most commonly used functional response function, which was suggested by Holling (1959) and called Michaelis-Menten type or Holling type II functional response [5]. And it is takes the form: ( ) mu p u a bu = + . where u represents the density of the prey population, and the positive constants m (units: 1/times) and a (units: 1/prey) describe the effects of capture rate and handling time, respectively, on the feeding rate.
However, Prey-dependent functional responses cannot describe the interaction between predators, and have been challenged by the biology and physiology communities [6] [7]. Some biologists have argued that in many situations, especially when predators have to search for food (and therefore, have to share or compete for food), the functional response in a prey-predator model should be predator-dependent. There is significant evidence that predator dependence in functional response occurs frequently in laboratories and natural systems [8] [9].
Given that many experiments and observations show predators do indeed interfere with each other's activities to trigger competition effects and that prey changes its behavior under increased predator threat, models with a functional predatordependent response are reasonable alternatives to models with prey-dependent functional response.
Starting with this argument and the traditional prey-only model, to describe the mutual interference of a predator, Beddington (1975) and DeAngelis (1975) [10] [11] [12] [13] proposed that an individual from a population of over two predators not only allocate time in searching for and processing their prey but also takes time in encountering with other predator. This result in the so-called It is assumed that the predator feeding rate decreases by higher predator density even when prey density is high, and therefore the effects of predator interfe-rence in the feeding rate remain important all the time whether an individual predator is handling or searching for a prey at a time. And in [17]. The authors consider a Leslie-Gower predator-prey model with a functional Crowley-Martin response describing predator mutual interference, which its takes the form: Initially, their analysis focuses on local asymptotic stability and Hopf bifurcation of the spatially homogeneous model based on ODE system for the reaction-diffusion model.
Based on the above discussion, and therefore. In this paper, we consider the following predator-prey model with mutual interference with Beddington-De Angelis functional response: where ( ) u t and ( ) v t represent population densities of prey and predator at time t, respectively. r, K, s, m, a, b, and h are positive constants, K is the carrying capacity of the prey, and r and s denote to their intrinsic growth rate, respectively, u h is a function on the prey population size (h is a measure of the food quality of the prey for conversion into predator growth depending on the density of the population). For the model (1.2), we mainly discuss about Hopf bifurcation on the positive equilibrium for the ODE system.
The aim of this article is to show that the diffusive predator-prey model (1.2) shows different spatial, temporal and spatiotemporal patterns across the mechanisms described above.
For simplicity, applying the following scaling: is a bounded smooth domain and ν is the outward unit normal vector on ∂Ω . The constants 1 0 d > and 2 0 d > are the diffusion coefficients of ( ) , u x t and ( ) , v x t , which represent the natural dispersive force of movement of the prey and predator density, respectively. The condition of the homogeneous Neumann boundary means that the two species have zero-flux across the boundary ∂Ω . The initial conditions In section two of this paper, we investigate the asymptotical behaviour of the interior equilibrium and occurrence of Hopf bifurcation of the local ODE system of (1.3).

Conclusion
In this article, we showed that the predator-prey model with mutual interference exhibits a rich and interesting dynamic behavior. We first studied the local stability and Hopf bifurcation in the corresponding ODE system. Then we studied the existence and direction of Hopf bifurcation and the stability of the bifurcating periodic solution in the reaction-diffusion system.