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On the long term spatial segregation for a competition-diffusion system
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ON THE LONG TERM SPATIAL SEGREGATION
FOR A COMPETITION-DIFFUSION SYSTEM
MARCO SQUASSINA
Abstract. We investigate the long term behavior for a class of competition-diffusion systems
of Lotka-Volterra type for two competing species in the case of low regularity assumptions on
the data. Due to the coupling that we consider the system cannot be reduced to a single equa-
tion yielding uniform estimates with respect to the inter-specific competition rate parameter.
Moreover, in the particular but meaningful case of initial data with disjoint support and Dirich-
let boundary data which are time-independent, we prove that as the competition rate goes to
infinity the solution converges, along with suitable sequences, to a spatially segregated state
satisfying some variational inequalities.
1. Introduction
Let ? be a bounded, open, connected subset of RN with smooth boundary and let κ be
a positive parameter. The aim of this paper is to investigate the asymptotic behavior of a
competition-diffusion system of Lotka-Volterra type for two competing species of population of
densities u and v, with Dirichlet boundary conditions,
(Pκ)
???????????????????
ut ??u = f(u)? κuv2, in ?× (0,∞),
vt ??v = g(v) ? κvu2, in ?× (0,∞),
u(x, t) = ψ(x, t), on ??× [0,∞),
v(x, t) = ζ(x, t), on ??× [0,∞),
u(x, 0) = u0(x), in ?,
v(x, 0) = v0(x), in ?.
A relevant problem in population ecology is the understanding of the interactions between dif-
ferent species, in particular in the case when the interactions are large and of competitive type.
As the inter-specific parameter κ ruling the mutual interaction of the species gets large, com-
petitive reaction-diffusion systems are expected to approach a limiting configuration where the
populations survive but exhibit disjoint habitats (cf. [7,8,10,19,22,24]). For population dynam-
ics models which require Dirichlet boundary conditions we refer to [8, 22], while for the more
ecological
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