The stability and bifurcation analysis of a discrete Holling-Tanner model

Hui Cao,Zongmin Yue,Yicang Zhou

doi:10.1186/1687-1847-2013-330

Abstract

A discrete predator-prey model with Holling-Tanner functional response is formulated and studied. The existence of the positive equilibrium and its stability are investigated. More attention is paid to the existence of a flip bifurcation and a Neimark-Sacker bifurcation. Sufficient conditions for those bifurcations have been obtained. Numerical simulations are conducted to demonstrate our theoretical results and the complexity of the model.

Highlights

Differential equations and difference equations are two typical mathematical approaches to modeling population dynamical systems
We study the dynamical behaviors of model ( )
We have studied the dynamical behaviors of a discrete prey-predator model with HollingTanner functional response

Summary

Introduction

Differential equations and difference equations are two typical mathematical approaches to modeling population dynamical systems. There are less results on dynamical behaviors of discrete predator-prey models. The flip bifurcation and the Neimark-Sacker bifurcation are two important phenomena of discrete population model dynamics. The unique positive equilibrium point E(u∗, u∗) of model ( ) is asymptotically stable if and only if condition ( ) or condition ( ) holds. Remark Inequality ( ) or ( ) gives stability conditions for the equilibrium E(u∗, u∗) of model ( ). The numerical simulations demonstrate that the positive equilibrium E(u∗, u∗) of model ( ) may be globally asymptotically stable if the conditions in Theorem . I are the complex eigenvalues of the linearized matrix J of model ( ) at the positive equilibrium. The solution series and the phase portrait in Figure show that the positive equilibrium E( . The solution series and the phase portrait in Figure show that the positive equilibrium E( . , . ) of model ( ) may be globally asymptotically stable

Flip bifurcation

Conclusion and discussion