Periodic Traveling Waves in Integrodifferential Equations for Nonlocal Dispersal

Periodic traveling waves (wavetrains) have been extensively studied for reaction-diffusion equations. One important motivation for this work has been the identification of periodic traveling wave patterns in spatiotemporal data sets in ecology. However, for many ecological populations, diffusion is no more than a rough phenomenological representation of dispersal, and spatial convolution with a dispersal kernel is more realistic. This paper concerns periodic traveling wave solutions of differential equations with nonlocal dispersal terms, and with local dynamics of lambda–omega form. These kinetics include the normal form near a standard supercritical Hopf bifurcation and are therefore significant for a wide range of applications. For general dispersal kernels, an explicit family of periodic traveling wave solutions is derived, as well as the condition for waves to be stable to perturbations of arbitrarily small wavenumber. Three specific kernels are then considered in detail: Laplace, Gaussian, and top hat. For Laplace and Gaussian kernels, it is shown that stability to perturbations of arbitrarily small wavenumber implies stability, a result that also applies for reaction-diffusion equations with lambda–omega kinetics. However, for the top hat kernel it is shown that periodic traveling waves may be stable to perturbations with small wavenumber but not to those with larger wavenumber. The wave family for the top hat kernel also shows significant qualitative differences from those for the Laplace and Gaussian kernels, and for reaction-diffusion equations with the same kinetics.