Stability and Hopf bifurcation of a diffusive Gompertz population model with nonlocal delay effect

Xiuli Sun,Baochuan Tian,Luan Wang

doi:10.14232/ejqtde.2018.1.22

Abstract

Stability and Hopf bifurcation of a diffusive Gompertz population model with nonlocal delay effect

Highlights

The Gompertz equation is one of the models that are often used to describe the dynamics of the populations, including cellular populations of tumour growth, see [18, 26, 28,29,30, 37]
Since Laird et al [30] showed that the Gompertz model could describe the normal growth of an organism such as the guinea pig over an incredible 10000-fold range of the growth in [26], the Gompertz equation is often used in the formulation of equations describing the population dynamics and to describe the inner growth of tumour
Through analyzing the distribution of eigenvalues of the infinitesimal generator associated with the linearized system, we show the stability of spatially nonhomogeneous steady-state solutions and the existence of Hopf bifurcation with the changes of the time delay

Summary

Introduction

The Gompertz equation is one of the models that are often used to describe the dynamics of the populations, including cellular populations of tumour growth, see [18, 26, 28,29,30, 37]. In [10], Chen and Yu analyzed the following reaction–diffusion equation with spatiotemporal delay and homogeneous Dirichlet boundary condition: d. The stability of the bifurcated positive equilibrium was investigated They proved that, for the given spatiotemporal delay, the bifurcated equilibrium is stable under some conditions, and Hopf bifurcation cannot occur. Guo and Yan [24] investigated the following diffusive Lotka–Volterra type population model with nonlocal delay effect:. Through analyzing the distribution of eigenvalues of the infinitesimal generator associated with the linearized system, we show the stability of spatially nonhomogeneous steady-state solutions and the existence of Hopf bifurcation with the changes of the time delay. We investigate the following diffusive Gompertz population model with nonlocal delay effect:.

The existence of the positive spatially nonhomogeneous equilibrium

Eigenvalue problems

Stability and Hopf bifurcations

The direction of the Hopf bifurcation

Numerical simulations