On the long term spatial segregation for a competition-diffusion system

TL;DR

This paper studies the long-term behavior of a competition-diffusion system modeling two species, showing that as competition intensifies, the species spatially segregate under certain conditions.

Contribution

It proves convergence to a segregated state for low-regularity data in a coupled Lotka-Volterra system as competition rate increases.

Findings

Solutions converge to spatially segregated states

Segregation occurs in the limit of infinite competition rate

Results hold under low regularity assumptions

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 equation 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 Dirichlet 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.