Realization of prescribed patterns in the competition model

We demonstrate that for any prescribed set of finitely many disjoint closed subdomains D1,…,Dm of a given spatial domain Ω in RN, 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 >0, the competition model under natural boundary conditions on ∂Ω, possesses an asymptotically stable positive steady-state solution (u ,v ) that has pattern D, that is, roughly speaking, as →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 →0, with u concentrating on D and v concentrating on Ω D.