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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

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

Pages: 489 - 509, Volume 24, Issue 2, June 2009      doi:10.3934/dcds.2009.24.489

 
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Kousuke Kuto - Department of Intelligent Mechanical Engineering, Fukuoka Institute of Technology, 3-30-1 Wajiro-Higashi, Higashi-ku, Fukuoka 811-0295, Japan (email)

Abstract: 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.

Keywords:  SKT model, heterogeneous environment, coexistence states, stability, Hopf bifurcation, Lyapunov-Schmidt reduction, limiting system.
Mathematics Subject Classification:  Primary: 35J65, 35B32; Secondary: 92D25.

Received: June 2008;      Revised: October 2008;      Available Online: March 2009.