Hopf bifurcations in the full SKT model and where to find them

TL;DR

This paper investigates how cross-diffusion affects bifurcation structures in the SKT model, revealing conditions for Hopf bifurcations and multi-stability in competitive systems.

Contribution

It provides a detailed analysis of bifurcation changes due to cross-diffusion in the SKT model, including predictions of bifurcation types and stability regions.

Findings

Cross-diffusion can destabilize or stabilize homogeneous equilibria.

Increasing cross-diffusion coefficients shifts bifurcations from super-critical to sub-critical.

Stable time-periodic spatial patterns can emerge via Hopf bifurcations.

Abstract

In this paper, we consider the Shigesada-Kawasaki-Teramoto (SKT) model, which presents cross-diffusion terms describing competition pressure effects. Even though the reaction part does not present the activator-inhibitor structure, cross-diffusion can destabilise the homogeneous equilibrium. However, in the full cross-diffusion system and weak competition regime, the cross-diffusion terms have an opposite effect and the bifurcation structure of the system modifies increasing the interspecific competition pressure. The major changes in the bifurcation structure, the type of pitchfork bifurcations on the homogeneous branch, as well as the presence of Hopf bifurcation points are here investigated. Through weakly nonlinear analysis, we can predict the type of pitchfork bifurcation. Increasing the additional cross-diffusion coefficients, the first two pitchfork bifurcation points from super-critical become sub-critical, leading to the appearance of a multi-stability region. The interspecific competition pressure also influences the possible appearance of stable time-period spatial patterns appearing through a Hopf bifurcation point.