Qualitative analysis of a cooperative reaction-diffusion system in a spatiotemporally degenerate environment
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In this paper, we are concerned with the cooperative system in which ∂tu-Deltau = muu + alfa(x, t)v-a(x, t)up and ∂tv-Deltav = muv + beta(x, t)u-b(x, t)vq in Omega × (0,∞); (∂nu u, ∂nuv) = (0, 0) on ∂Omega×(0,∞); and (u(x, 0), v(x, 0)) = (u0(x), v0(x)) > (0, 0) in Omega, where p, q > 1, Omega ⊂ RN (N ≥ 2) is a bounded smooth domain, alfa, beta > 0 and a, b ≥ 0 are smooth functions that are T-periodic in t, and mu is a varying parameter. The unknown functions u(x, t) and v(x, t) represent the densities of two cooperative species. We study the long-time behavior of (u, v) in the case that a and b vanish on some subdomains of Omega × [0, T]. Our results show that, compared to the nondegenerate case where a, b > 0 on Omega × [0, T], such a spatiotemporal degeneracy can induce a fundamental change to the dynamics of the cooperative system. © 2014 Society for Industrial and Applied Mathematics.
cooperative reaction-diffusion system; dynamical behavior; positive periodic solutions; principal eigenvalue; spatiotemporal degeneracy; dynamical behaviors; positive periodic solution; principal eigenvalues; reaction diffusion systems; spatiotemporal degeneracy; eigenvalues and eigenfunctions; dynamic models