The existence and instability of blowing-up steady states for the Shigesada-Kawasaki-Teramoto competition model with cross-diffusion

TL;DR

This paper studies the existence, detailed structure, and instability of large-amplitude steady states in a two-species competition model with cross-diffusion, revealing their blow-up behavior and spectral instability in both classical and perturbed settings.

Contribution

It introduces a new approach using transformations and Lyapunov-Schmidt reduction to analyze blow-up steady states and their instability in the SKT competition model.

Findings

Blow-up steady states are characterized and their asymptotic structure is detailed.

All large-amplitude steady states are spectrally unstable.

Existence and instability of steady states are established for the perturbed SKT model with large cross-diffusion.

Abstract

We investigate the existence and instability of a class of blowing-up positive steady states arising in a shadow system of the Shigesada-Kawasaki-Teramoto (SKT) two-species competition model, as well as in the corresponding perturbed SKT model with a sufficiently large cross-diffusion coefficient and bounded random diffusion parameters. In their classical work (Lou-Ni, 1999), it was shown that, under the limit where one cross-diffusion parameter tends to infinity, coexistence steady states of the SKT model are characterized by three types of shadow systems. In a previous study (Kuto, 2015), the first author analyzed one of these shadow systems in one space dimension (the second shadow system) and proved that a component of a bifurcating branch blows up as the bifurcation parameter approaches the first positive Neumann eigenvalue of $-\Delta$. In the present paper, using a new approach based on suitable transformations and the Lyapunov-Schmidt reduction method, we derive the existence and detailed asymptotic structure of several branches of positive steady states near blow-up points for shadow systems with one or two cross-diffusion terms, in both one- and multi-dimensional domains. We further show that all such large-amplitude steady state branches are spectrally unstable. Moreover, by means of perturbation arguments, we establish the existence and instability of corresponding branches of perturbed positive steady states for the original SKT model when one cross-diffusion coefficient is sufficiently large.