Sharp spatiotemporal patterns in the diffusive time-periodic logistic equation

To reveal the complex influence of heterogeneous environment on population systems, we examine the asymptotic profile (as ϵ→0ϵ→0) of the positive solution to the perturbed periodic logistic equation{∂tu−Δu=au−[b(x,t)+ϵ]upin Ω×[0,T],u(x,0)=u(x,T)in Ω, subject to homogeneous Neumann boundary conditions, where ϵ>0ϵ>0 is a small perturbation parameter, Ω⊂RnΩ⊂Rn (n⩾2n⩾2) is a bounded domain with smooth boundary ∂Ω, a, T   and p>1p>1 are positive constants. The function b∈Cθ,θ/2(Ω¯×R) (0<θ<10<θ<1) is T-periodic in t  , nonnegative, and vanishes (i.e., has a degeneracy) in some subdomain of Ω×RΩ×R. We show that the temporal degeneracy of b induces sharp spatiotemporal patterns of the solution only when spatial degeneracy also exists; but in sharp contrast, whether or not temporal degeneracy appears in b, the spatial degeneracy always induces sharp spatiotemporal patterns, though they differ significantly according to whether temporal degeneracy is also present.