Dynamics of a diffusive predator–prey model with herd behavior

Yan Li,Fengrong Zhang,Sanyun Li

doi:10.15388/namc.2020.25.15723

Abstract

This paper is devoted to considering a diffusive predator–prey model with Leslie–Gower term and herd behavior subject to the homogeneous Neumann boundary conditions. Concretely, by choosing the proper bifurcation parameter, the local stability of constant equilibria of this model without diffusion and the existence of Hopf bifurcation are investigated by analyzing the distribution of the eigenvalues. Furthermore, the explicit formula for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are also derived by applying the normal form theory. Next, we show the stability of positive constant equilibrium, the existence and stability of periodic solutions near positive constant equilibrium for the diffusive model. Finally, some numerical simulations are carried out to support the analytical results.

Highlights

A fundamental goal of theoretical ecology is to understand the interactions between different species, and between species and natural environment
The dynamics of a diffusive predator–prey model with Leslie–Gower term and herd behavior are studied under the homogeneous Neumann boundary conditions
By choosing the appropriate bifurcation parameter, the stability of the constant solutions of ordinary differential equations (ODEs) and the existence of Hopf bifurcation are discussed by analyzing the corresponding characteristic equation

Summary

Introduction

A fundamental goal of theoretical ecology is to understand the interactions between different species, and between species and natural environment. In paper [23], taking into account the inhomogeneous distribution of the prey and predators in different spatial locations within a fixed bounded domain Ω ⊂ R at any given time, and the natural tendency of each species to diffuse to areas of smaller population concentration, the authors considered the following model:. In paper [27], the authors considered the following spatial predator–prey model with herd behavior:.

Hopf bifurcation of ODE

Hopf bifurcation of PDE

Numerical simulations and discussion

Conclusions