Wave train selection behind invasion fronts in reaction–diffusion predator–prey models

Abstract: Wave trains, or periodic travelling waves, can evolve behind invasion fronts in oscillatory reaction–diffusion models for predator–prey systems. Although there is a one-parameter family of possible wave train solutions, in a particular predator invasion a single member of this family is selected. Sherratt (1998) has predicted this wave train selection, using a system that is a valid approximation near a supercritical Hopf bifurcation in the corresponding kinetics and when the predator and prey diffusion coefficients are nearly equal. Away from a Hopf bifurcation, or if the diffusion coefficients differ somewhat, these predictions lose accuracy. We develop a more general wave train selection prediction for a two-component reaction–diffusion predator–prey system that depends on linearizations at the unstable homogeneous steady states involved in the invasion front. This prediction retains accuracy farther away from a Hopf bifurcation, and can also be applied when the predator and prey diffusion coefficients are unequal. We illustrate the selection prediction with its application to three models of predator invasions. [Copyright &y& Elsevier]