Periodic and Chaotic Traveling Wave Patterns in Reaction-Diffusion/ Predator-Prey Models with General Nonlinearities

TL;DR

This paper analyzes traveling wave patterns in reaction-diffusion and predator-prey models, revealing how bifurcations lead to stable periodic waves or chaotic regimes, depending on nonlinearities and bifurcation type.

Contribution

It demonstrates the occurrence of Hopf bifurcations in generalized predator-prey models and characterizes the resulting complex or chaotic wave dynamics in these systems.

Findings

Supercritical bifurcations produce stable periodic wavetrains.

Subcritical bifurcations can lead to chaotic wave behavior.

Numerical analysis confirms chaotic regimes via spectral and fractal measures.

Abstract

Traveling wavetrains in generalized two-species predator-prey models and two-component reaction-diffusion equations are considered. The stability of the fixed points of the traveling wave ODEs (in the usual "spatial" variable) is considered. For general functional forms of the nonlinear prey birthrate/prey deathrate or reaction terms, a Hopf bifurcation is shown to occur at two different critical values of the traveling wave speed. The post-bifurcation dynamics is investigated for five different functional forms of the nonlinearities. In cases where the bifurcation is supercritical, the post-bifurcation behaviour yields stable periodic orbits of the traveling-wave ODEs in the spatial variable. These correspond to stable periodic wavetrains of the full PDEs. Subcritical Hopf bifurcations yield more complex post-bifurcation dynamics in the PDE wavetrains. In special cases where the subcritical bifurcation marks the end of the regime of stability, the post-bifurcation behavior in the spatial ODEs is chaotic, corresponding to wavetrains of the original PDEs which are spatially coherent, but have chaotic temporal dynamics. All the models are integrated numerically to investigate the post-bifurcation dynamics and chaotic regimes are characterized by computing power spectra, autocorrelation functions, and fractal dimensions.