Bogdanov–Takens bifurcation in a predator–prey model

Zhihua Liu,Pierre Magal,Dongmei Xiao

doi:10.1007/s00033-016-0724-1

Abstract

In this paper, we investigate a class of predator–prey model with age structure and discuss whether the model can undergo Bogdanov–Takens bifurcation. The analysis is based on the normal form theory and the center manifold theory for semilinear equations with non-dense domain combined with integrated semigroup theory. Qualitative analysis indicates that there exist some parameter values such that this predator–prey model has an unique positive equilibrium which is Bogdanov–Takens singularity. Moreover, it is shown that under suitable small perturbation, the system undergoes the Bogdanov–Takens bifurcation in a small neighborhood of this positive equilibrium.

Highlights

In this article, we will analyze the following predator–prey system with an age structure∂u(t, a) + ∂u(t, a) = −μu(t, a), for a ≥ 0, ∂t u(t, 0) = η V ∂a (t)+∞ 0 β(a)u(t, h + V 2(t) a)da dV (t) = rV (t) dt 1 − V (t) K − (t)+∞ 0 u(t, a)da h + V 2(t)
To the best of our knowledge, there is no result on Bogdanov–Takens bifurcation for an age-structured model which is an hyperbolic partial differential equations with nonlinear and non-local boundary conditions
We first look for some conditions of parameters which guarantee system (1.1) has a positive equilibrium who is Bogdanov–Takens singularity by applying the normal form theory developed by Liu, Magal, and Ruan [29] for the non-densely defined abstract Cauchy problems, we choose suitable small perturbation of parameters such that system (1.1) can undergo the Bogdanov–Takens bifurcation

Summary

Introduction

We will analyze the following predator–prey system with an age structure. To the best of our knowledge, there is no result on Bogdanov–Takens bifurcation for an age-structured model which is an hyperbolic partial differential equations with nonlinear and non-local boundary conditions. We first look for some conditions of parameters which guarantee system (1.1) has a positive equilibrium who is Bogdanov–Takens singularity by applying the normal form theory developed by Liu, Magal, and Ruan [29] for the non-densely defined abstract Cauchy problems, we choose suitable small perturbation of parameters such that system (1.1) can undergo the Bogdanov–Takens bifurcation. 4, the eigenvalue problem for the linearized system of (1.1) around the unique positive equilibrium is investigated and the normal form and center manifold theory for semilinear equations with non-dense domain is used to carry out the analysis of the Bogdanov–Takens bifurcation.

Preliminary

Equilibria and linearized equation

Bogdanov–Takens singularity

Bogdanov–Takens bifurcation

Conclusion