PERIODIC AND CHAOTIC TRAVELING WAVE PATTERNS IN REACTION-DIFFUSION/PREDATOR-PREY MODELS WITH GENERAL NONLINEARITIES

Stefan C. Mancas (USA) and Roy S. Choudhury (USA)

Received August 4, 2008

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 birth rate/prey death rate 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. These correspond to wavetrains of the original PDEs which exhibit chaos in the spatial variable, and thus include those which are of physical interest as information carriers in that they are spatially coherent but with complex 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.

Keywords and phrases: Hopf bifurcation, reaction-diffusion/predator prey systems with arbitrary nonlinearities, periodic and chaotic patterns.