Hopf bifurcation analysis in a predator\u2013prey model with two time delays and stage structure for the prey

Miao Peng,Zhengdi Zhang

doi:10.1186/s13662-018-1705-9

Abstract

In this paper, a stage-structured predator–prey model with Holling type III functional response and two time delays is investigated. By analyzing the associated characteristic equation, its local stability and the existence of Hopf bifurcation with respect to both delays are studied. Based on the normal form method and center manifold theorem, the explicit formulas are derived to determine the direction of Hopf bifurcation and the stability of bifurcating period solutions. Finally, the effectiveness of theoretical analysis is verified via numerical simulations. This study may be helpful in understanding the behavior of ecological environment.

Highlights

The dynamics of predator–prey models is one of important subjects in ecology and mathematical ecology, and many factors, such as time delay, disease, harvesting, functional response, etc., can affect it in the natural world.In recent years, some researchers have discussed predator–prey models with these factors
5 Conclusions In this paper, we have studied the problem of Hopf bifurcation analysis in a delayed predator–prey model with stage structure for the prey
By setting the same group of parameter values, according to the existing two time delays and discussing four different cases, we know that the interior equilibrium will lose its original stability and a Hopf bifurcation occurs, and a family of periodic solutions bifurcate E∗ when the time delay passes though some critical values

Summary

Introduction

The dynamics of predator–prey models is one of important subjects in ecology and mathematical ecology, and many factors, such as time delay, disease, harvesting, functional response, etc., can affect it in the natural world.In recent years, some researchers have discussed predator–prey models with these factors. 2, the local stability of the positive equilibrium and the existence of Hopf bifurcation for system (1.2) are studied. We shall discuss the local stability of a linearized system at the positive equilibrium and the existence of Hopf bifurcations for system (1.2).

Results

Conclusion