Realization of prescribedpatterns in the competition model

Abstract

We demonstrate that for any prescribed set of finitely many disjoint closed subdomains D₁,...,Dm of a given spatial domain Ω in R^N, if d₁,d₂,a₁,a₂,c,d,e are positive continuous functions on Ω and b(x) is identically zero on D:=D₁∪...∪Dm and positive in the rest of Ω, then for suitable choices of the parameters λ, μ and all small ε>0, the competition model u₁(x,t)-d₁(x)Δu(x,t)=λa₁(x)u-[ε⁻¹b(x)+1]u²-c(x)uv... under natural boundary conditions on δΩ, possesses an asymptotically stable positive steady-state solution... that has pattern D, that is, roughly speaking, as ε→0, u... converges to a positive function over D, while it converges to 0 over the rest of Ω; on the other hand, v... converges to 0 over D but converges to some positive function in the rest of Ω. In other words, the two competing species u... and v... become spatially segregated as ε→0, with u... concentrating on D and v... concentrating on ΩD.