# American Institute of Mathematical Sciences

June  2009, 24(2): 489-509. doi: 10.3934/dcds.2009.24.489

## Stability and Hopf bifurcation of coexistence steady-states to an SKT model in spatially heterogeneous environment

 1 Department of Intelligent Mechanical Engineering, Fukuoka Institute of Technology, 3-30-1 Wajiro-Higashi, Higashi-ku, Fukuoka 811-0295, Japan

Received  June 2008 Revised  October 2008 Published  March 2009

This paper is concerned with the following Lotka-Volterra cross-diffusion system

ut = Δ[(1+kρ(x) v)u] +u(a-u-c(x)v) in Ω Χ (0, ∞),
τvt = Δv +v(b+d(x)u-v) in Ω Χ (0, ∞)

in a bounded domain Ω ⊂ RN with Neumann boundary conditions δvu = δvv = 0 on δΩ. In the previous paper [18], the author has proved that the set of positive stationary solutions forms a fishhook shaped branch Γ under a segregation of $\rho (x)$ and $d(x)$. In the present paper, we give some criteria on the stability of solutions on Γ. We prove that the stability of solutions changes only at every turning point of Γ if τ is large enough. In a different case that $c(x)\ >\ 0$ is large enough, we find a parameter range such that multiple Hopf bifurcation points appear on Γ.

Citation: Kousuke Kuto. Stability and Hopf bifurcation of coexistence steady-states to an SKT model in spatially heterogeneous environment. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 489-509. doi: 10.3934/dcds.2009.24.489
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