Wavetrain Selection Behind Invasion Fronts in Reaction-Diffusion Predator-Prey Models

Wavetrains, also known as periodic travelling waves, are known to evolve behind invasion fronts in oscillatory reaction-diffusion models for predator-prey systems. Mathematical theory predicts that for a given set of parameter values there is in fact a family of possible wavetrain solutions. In a particular predator invasion a single member of this family is somehow selected behind the primary invasion front. Sherratt (1998) has studied this selection mechanism, using the normal form approximation that is valid for such models when the diffusion coefficients are equal and the parameters are near the Hopf bifurcation in the local system. However, away from the Hopf bifurcation or if the predator and prey diffusion coefficients differ the predictions made from the normal form lose accuracy. We develop a more general selection criterion that retrieves the prediction from the normal form system, but that depends on the properties of the wavetrains for the full (non-reduced) predator-prey system and hence retains accuracy away from the Hopf bifurcation. In addition, this criterion can be applied to the case of unequal predator and prey diffusion coefficients. We illustrate this selection criterion with its application to three reaction-diffusion models taken from the literature on predator invasions.