Coexistence and segregation for strongly competing species in special domains

Abstract

We deal with strongly competing multispecies systems of Lotka-Volterra type with homogeneous Dirichlet boundary conditions. For a class of nonconvex domains composed of balls connected by thin corridors, we show the occurrence of pattern formation (coexistence and spatial segregation of all the species) as the competition grows indefinitely. As a result we prove the existence and uniqueness of solutions for a remarkable system of differential inequalities involved in segregation phenomena and optimal partition problems.

Uniqueness and least energy property for solutions to strongly competing systems

“…A wide literature is devoted to this topic, mainly for the case of competition models of Lotka-Volterra type (see e.g. [1,7,8,9,10,11,12,13,14,15,16]). As a prototype for the study of this phenomenon, in [4] we consider the competition-diffusion system of k differential equations:…”

Coexistence Solutions for a Periodic Competition Model with Singular–Degenerate Diffusion

“…Most of the works in the earlier literature were devoted to studying the linear diffusion case: that is, m = n = 1, p = q = 2 (see, for example, [1,7,13,16,20,21]). We also mention that, as a special case, the coexistence steady state for biological community system has also been investigated in some references: see [3,5,6,8,18]. The biological background makes it very interesting to study the coexistence periodic solutions of a nonlinear diffusion system of the same type as (1.1).…”

Global minimizers of coexistence for competing species

“…Nonetheless, a mechanism to avoid extinction can be found in the spatial inhomogeneity of the territory. Indeed, working in a special class of non-convex domains close to a union of k disjoint balls, the existence of local minima of E where all the species are present is proven in [5], (see also [4]).…”

“…It is worth pointing out that the investigation of positive solutions to competitive systems in the case of k ≥ 3 densities is a challenging task and only partial results are known, see e.g. [4,5,6,9,10,12] and the discussions therein for more references.…”