  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.
