Stability and Hopf bifurcations for a delayed diffusion system in population dynamics

Xiang-Ping Yan; Wan-Tong Li

doi:10.3934/dcdsb.2012.17.367

Article Contents

2012, Volume 17, Issue 1: 367-399. Doi: 10.3934/dcdsb.2012.17.367

This issue Previous Article Exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with non-local delay Next Article Global stability of a five-dimensional model with immune responses and delay

Stability and Hopf bifurcations for a delayed diffusion system in population dynamics

Xiang-Ping Yan^1, and
Wan-Tong Li^2,

1.
Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, Gansu 730070
2.
School of Mathematic and Statistics, Lanzhou University, Lanzhou, Gansu 730000

Received: November 30, 2010

Revised: January 31, 2011

Published: December 2011

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Abstract

A generalized two-species Lotka-Volterra reaction-diffusion system with a discrete delay and subject to homogeneous Dirichlet boundary conditions is considered. By regarding the delay as the bifurcation parameter and analyzing in detail the spectrum of the associated linear operator, the stability of the positive steady state bifurcating from the zero solution is studied. In particular, it is shown that the system can undergo a forward Hopf bifurcation at the positive steady state solution when the delay take a sequence of critical values via the implicit function theorem. To verify the obtained theoretical results, some numerical simulations are also included.

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Mathematics Subject Classification: Primary: 35K57, 35B32, 35B10, 92D25, 92D40.

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