Abstract
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.
Highlights
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]
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]
The Beddington-DeAngelis functional response and the analogous response with diffusion in a constant environment have received much attention in the literatures [14] [15]. Such as Crowley and Martin [16] assumed that interference among predators occurs regardless of whether a particular predator is looking for prey or handling prey and proposed a functional response, which is called the Crowley-Martin-type functional response, and its takes the form:
Summary
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]. Given that many experiments and observations show predators do 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. The Beddington-DeAngelis functional response and the analogous response with diffusion in a constant environment have received much attention in the literatures [14] [15] Such as Crowley and Martin [16] assumed that interference among predators occurs regardless of whether a particular predator is looking for prey or handling prey and proposed a functional response, which is called the Crowley-Martin-type functional response, and its takes the form:. We consider the following predator-prey model with mutual interference with Beddington-De Angelis functional response: du dt au 2 muv + buv cv. 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)
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