Complex pattern formation driven by the interaction of stable fronts in a competition-diffusion system

TL;DR

This paper investigates how the interaction of stable traveling waves in a three-species competition-diffusion system leads to complex spatio-temporal patterns, including spirals and dynamic structures, driven by bifurcations and wave interactions.

Contribution

It uncovers the mechanism behind pattern formation from stable wave interactions, highlighting bifurcation structures and the transition from reflection to merging regimes in a three-species model.

Findings

Stable traveling waves can interact to produce complex patterns.

Bifurcation analysis reveals a transition from reflection to merging.

Complex patterns include oscillating spirals and breakup into dynamic structures.

Abstract

The ecological invasion problem in which a weaker exotic species invades an ecosystem inhabited by two strongly competing native species is modelled by a three-species competition-diffusion system. It is known that for a certain range of parameter values competitor-mediated coexistence occurs and complex spatio-temporal patterns are observed in two spatial dimensions. In this paper we uncover the mechanism which generates such patterns. Under some assumptions on the parameters the three-species competition-diffusion system admits two planarly stable travelling waves. Their interaction in one spatial dimension may result in either reflection or merging into a single homoclinic wave, depending on the strength of the invading species. This transition can be understood by studying the bifurcation structure of the homoclinic wave. In particular, a time-periodic homoclinic wave (breathing wave) is born from a Hopf bifurcation and its unstable branch acts as a separator between the reflection and merging regimes. The same transition occurs in two spatial dimensions: the stable regular spiral associated to the homoclinic wave destabilizes, giving rise first to an oscillating breathing spiral and then breaking up producing a dynamic pattern characterized by many spiral cores. We find that these complex patterns are generated by the interaction of two planarly stable travelling waves, in contrast with many other well known cases of pattern formation where planar instability plays a central role.