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Critical thresholds in a relaxation system with resonance of characteristic speeds
Stability and Hopf bifurcation of coexistence steadystates to an SKT model in spatially heterogeneous environment
1.  Department of Intelligent Mechanical Engineering, Fukuoka Institute of Technology, 3301 WajiroHigashi, Higashiku, Fukuoka 8110295, Japan 
u_{t} = Δ[(1+kρ(x) v)u] +u(auc(x)v)
in Ω Χ (0, ∞),
τv_{t} = Δv +v(b+d(x)uv) in Ω Χ (0, ∞)
in a bounded domain Ω ⊂ R^{N} with Neumann boundary conditions δ_{v}u = δ_{v}v = 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.
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