A case study of multiple wave solutions in a reaction-diffusion system using invariant manifolds and global bifurcations

TL;DR

This paper thoroughly analyzes multiple wave solutions in a reaction-diffusion predator-prey model using invariant manifolds and bifurcation theory, revealing complex dynamics including chaos and diverse traveling waves.

Contribution

It introduces a comprehensive approach combining stability analysis, bifurcation theory, and invariant manifold computations to identify various traveling wave solutions in a four-component reaction-diffusion system.

Findings

Identification of homoclinic and heteroclinic connections

Discovery of periodic and chaotic wave solutions

Global invariant manifold computations near bifurcations

Abstract

A thorough analysis is performed to find traveling waves in a qualitative reaction-diffusion system inspired by a predator-prey model. We provide rigorous results coming from a standard local stability analysis, numerical bifurcation analysis, and relevant computations of invariant manifolds to exhibit homoclinic and heteroclinic connections, and periodic orbits in the associated traveling wave system with four components. In so doing, we present and describe a zoo of different traveling wave solutions. In addition, homoclinic chaos is manifested via both saddle-focus and focus-focus bifurcations as well as a Belyakov point. An actual computation of global invariant manifolds near a focus-focus homoclinic bifurcation is also presented to unravel a multiplicity of wave solutions in the model.