Invasion Generates Periodic Traveling Waves (Wavetrains) in Predator-Prey Models with Nonlocal Dispersal

Periodic Traveling waves (wavetrains) have been studied extensively in systems of reaction-diffusion equations. An important motivation for this work is the identification of periodic Traveling waves of abundance in ecological data sets. However, for many natural populations diffusion is a poor representation of movement, and spatial convolution with a dispersal kernel is more realistic because of its ability to reflect rare long-distance dispersal events. In marked contrast to the literature on reaction-diffusion systems, there has been almost no previous work on periodic Traveling waves in models with nonlocal dispersal. In this paper the author considers the generation of such waves by the invasion of the unstable coexistence state in cyclic predator-prey systems with nonlocal dispersal for which the dispersal kernel is thin-tailed (exponentially bounded). The main result is formulae for the wave period and amplitude when the parameters of the local population dynamics are close to a Hopf bifurcation point. This result is tested via detailed comparison of the dependence on parameters of the stability of the periodic Traveling waves generated by invasion. The paper concludes with a comparison between the predictions of the nonlocal model and the corresponding reaction-diffusion model. Specifically, the parameter regions giving stable and unstable waves are shown to be the same to leading order close to a Hopf bifurcation point, irrespective of the choice of dispersal kernel.