Delay-induced Hopf bifurcation in a diffusive Holling\u2013Tanner predator\u2013prey model with ratio-dependent response and Smith growth

Huiping Fang,Yongfeng Wu,Ling Hu

doi:10.1186/s13662-018-1726-4

Abstract

In this paper, we study a diffusion Holling–Tanner predator–prey model with ratio-dependent functional response and Simth growth subject to a homogeneous Neumann boundary condition. Firstly, we use iteration technique and eigenvalue analysis to get the local stability and a Hopf bifurcation at the positive equilibrium. Secondly, by choosing the constant related to delay as bifurcation parameter we obtain periodic solutions near the positive equilibrium. Besides, by using center manifold theory and normal form theory we reflect the stability with Hopf bifurcating periodic solution and bifurcating direction.

Highlights

Mathematical ecology is a fast and active branch of biomathematics, and the dynamics of biological models are very rich
We investigate the stability of the positive equilibrium, delay-induced Hopf bifurcation, and the properties of Hopf bifurcation such as the direction of the bifurcation and stability of the bifurcating periodic solutions
We only investigate the effect of the delay on the stability of the positive equilibrium E∗(u∗, v∗) of system (1.2), where u∗ = α + η – β, α + η + βc v∗ = u∗, η under condition (H): β < α + η

Summary

Introduction

Mathematical ecology is a fast and active branch of biomathematics, and the dynamics of biological models are very rich. The population dynamics growth restriction is based on the unused available resources [12,13,14,15,16] This model, known as the Holling–Tanner model, has been studied for both its mathematical properties and its efficacy for describing real ecological systems such as mite/spider mite, lynx/hare, sparrow/sparrow hawk, and so one by Holling [2], Tanner [3] and Wollkind, Collings, and Logan [4]. Yue and Wang [16] studied the stability and Hopf bifurcation of a diffusive Holling–Tanner predator–prey model with smith growth subject to Neumann boundary condition:. The Hopf bifurcation of the diffusive predator–prey system with delay subject to homogeneous Neumann boundary conditions was studied in [20,21,22,23]. We investigate the stability of the positive equilibrium, delay-induced Hopf bifurcation, and the properties of Hopf bifurcation such as the direction of the bifurcation and stability of the bifurcating periodic solutions

The stability of equilibrium and the existence of Hopf bifurcation

Direction of Hopf bifurcation and stability of bifurcating periodic solution

Conclusions