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Critical thresholds in a relaxation system with resonance of characteristic speeds
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 |
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.
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