Global structure of steady-states to the full cross-diffusion limit in the Shigesada-Kawasaki-Teramoto model

TL;DR

This paper analyzes the solution structure of a limiting system derived from the Shigesada-Kawasaki-Teramoto model, providing conditions for solution existence and a bifurcation diagram for nonconstant solutions.

Contribution

It offers new insights into the solution set of the limiting system, including existence conditions and bifurcation analysis, extending previous asymptotic studies.

Findings

Sufficient conditions for existence/nonexistence of nonconstant solutions

Bifurcation diagram of solutions in one dimension

Analysis using topological and nonlocal methods

Abstract

In a previous paper(2021), the author studied the asymptotic behavior of coexistence steady-states to the Shigesada-Kawasaki-Teramoto model as both cross-diffusion coefficients tend to infinity at the same rate. As a result, he proved that the asymptotic behavior can be characterized by a limiting system that consists of a semilinear elliptic equation and an integral constraint. This paper studies the set of solutions of the limiting system. The first main result gives sufficient conditions for the existence/nonexistence of nonconstant solutions to the limiting system by a topological approach using the Leray-Schauder degree. The second main result exhibits a bifurcation diagram of nonconstant solutions to the one-dimensional limiting system by analysis of a weighted time-map and a nonlocal constraint.