STABILITY OF POSITIVE STEADY-STATE SOLUTIONS IN A DELAYED LOTKA-VOLTERRA DIFFUSION SYSTEM

- Journal title : Journal of the Korean Mathematical Society
- Volume 49, Issue 4, 2012, pp.715-731
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2012.49.4.715

Title & Authors

STABILITY OF POSITIVE STEADY-STATE SOLUTIONS IN A DELAYED LOTKA-VOLTERRA DIFFUSION SYSTEM

Yan, Xiang-Ping; Zhang, Cun-Hua;

Yan, Xiang-Ping; Zhang, Cun-Hua;

Abstract

This paper considers the stability of positive steady-state solutions bifurcating from the trivial solution in a delayed Lotka-Volterra two-species predator-prey diffusion system with a discrete delay and subject to the homogeneous Dirichlet boundary conditions on a general bounded open spatial domain with smooth boundary. The existence, uniqueness and asymptotic expressions of small positive steady-sate solutions bifurcating from the trivial solution are given by using the implicit function theorem. By regarding the time delay as the bifurcation parameter and analyzing in detail the eigenvalue problems of system at the positive steady-state solutions, the asymptotic stability of bifurcating steady-state solutions is studied. It is demonstrated that the bifurcating steady-state solutions are asymptotically stable when the delay is less than a certain critical value and is unstable when the delay is greater than this critical value and the system under consideration can undergo a Hopf bifurcation at the bifurcating steady-state solutions when the delay crosses through a sequence of critical values.

Keywords

Lotka-Volterra competition-cooperation system;discrete delay;diffusion effect;positive steady-state solution;stability;

Language

English

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