Realization of prescribed patterns 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 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 E > 0, the competition model A figure is presented. under natural boundary conditions on ∂Ω, possesses an asymptotically stable positive steady-state solution (uE, vE) that has pattern D, that is, roughly speaking, as E → 0, uE converges to a positive function over D, while it converges to 0 over the rest of Ω; on the other hand, vE converges to 0 over D but converges to some positive function in the rest of Ω. In other words, the two competing species uE and vE become spatially segregated as E → 0, with uE concentrating on D and vE concentrating on Ω\D. © 2003 Elsevier Science (USA). All rights reserved.