# American Institute of Mathematical Sciences

January  2012, 17(1): 367-399. doi: 10.3934/dcdsb.2012.17.367

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

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

Received  December 2010 Revised  February 2011 Published  October 2011

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.
Citation: Xiang-Ping Yan, Wan-Tong Li. Stability and Hopf bifurcations for a delayed diffusion system in population dynamics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 367-399. doi: 10.3934/dcdsb.2012.17.367
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