Coexistence-segregation dichotomy in the full cross-diffusion limit of the stationary SKT model

TL;DR

This paper analyzes the asymptotic behavior of positive steady-states in a Lotka-Volterra competition model with cross-diffusion, revealing a dichotomy between coexistence and segregation as diffusion coefficients grow large, supported by bifurcation analysis and numerical verification.

Contribution

It characterizes the full cross-diffusion limit of the stationary SKT model, identifying coexistence and segregation solutions, and explores their bifurcation structure both analytically and numerically.

Findings

Positive steady-states bifurcate into coexistence and segregation types.

Bifurcation branches are numerically traced using pde2path.

Theoretical results are supported by numerical bifurcation analysis.

Abstract

This paper studies the Lotka-Volterra competition model with cross-diffusion terms under homogeneous Dirichlet boundary conditions. We consider the asymptotic behavior of positive steady-states as equal two cross-diffusion coefficients tend to infinity (so-called the full cross-diffusion limit). The first result shows that, at the full cross-diffusion limit, the set of positive steady-state solutions can be classified into two types: the small coexistence or the complete segregation. The small coexistence can be characterized by the set of positive solutions of a nonlinear elliptic equation, while the complete segregation can be characterized by the set of nodal solutions of another nonlinear elliptic equation. The second result concerns the global bifurcation structure of the 1d model with large cross-diffusion terms to show that the branch of small coexistence bifurcates from the trivial solution and many branches of complete segregation bifurcate from solutions on the branch of small coexistence. Finally, in order to verify the theoretical results, we chase some bifurcation branches numerically via the continuation software pde2path.