Global stability and Hopf bifurcation of a diffusive predator–prey model with hyperbolic mortality and prey harvesting

Jingfu Zhao,Sanyun Li,Yan Li

doi:10.15388/na.2017.5.5

Abstract

This paper is concerned with a predator&ndashprey model with hyperbolic mortality and prey harvesting. The parameter regions for the stability and instability of the unique positive constant solution of ODE and PDE are derived, respectively, especially the global asymptotical stability of positive constant equilibrium of the diffusive model is obtained by iterative technique. The stability and direction of periodic solutions of ODE and PDE are investigated by center manifold theorem and normal form theory, respectively. Numerical simulations are carried out to depict our theoretical analysis.

Highlights

Predator–prey models are basic differential equation models for describing the interactions between two species and are of great interest to researchers in mathematics and ecology
There are many different kinds of functional response for different kinds of species to model the phenomena of predation such as Holling I–III type, Ivlev type, Beddington–DeAngelis type, the Crowley–Martin type, and the recent well-known ratio dependence type, which was first proposed by Arditi and Ginzburg
Global stability and Hopf bifurcation of a diffusive predator–prey model harvesting, proportional harvesting, and nonlinear harvesting are currently investigated by many authors, see [5, 8, 13]

Summary

Introduction

Predator–prey models are basic differential equation models for describing the interactions between two species and are of great interest to researchers in mathematics and ecology Both the functional response and harvesting can affect dynamical properties of biological and mathematical models. In paper [11], the authors considered a predator–prey model with hyperbolic mortality as follows: suv ut. We treat λ (or equivalently h) as Hopf bifurcation parameter and do analysis of stability and Hopf bifurcation to demonstrate the important role of prey harvesting in the model. We focus on the important role of the prey harvesting, while paper [10] is concerned with the role of the time delay, and we obtain the global asymptotical stability of the unique positive constant equilibrium of the diffusive model in term of the iteration technique.

Hopf bifurcation of ODE

Hopf bifurcation of PDE

Numerical simulations