HOW DOES NONLOCAL DISPERSAL AFFECT THE SELECTION AND STABILITY OF PERIODIC TRAVELING WAVES?

In ecology a number of spatiotemporal datasets on cyclic populations reveal periodic traveling waves of abundance. This calls for studies of periodic traveling wave solutions of ecologically realistic mathematical models. For many species, such models must include long-range dispersal. However, mathematical theory on periodic traveling waves is almost entirely restricted to reactiondi ffusion equations, which assume purely local dispersal. I study integrodifferential equation models in which dispersal is represented via a convolution. The dispersal kernel is assumed to be of either Gaussian or Laplace form; in either case it contains a parameter scaling the width of the kernel. I show that as this parameter tends to zero, the integrodifferential equation asymptotically approaches a reaction-diffusion model. I exploit this limit to determine the effect of a small degree of nonlocality in dispersal on periodic traveling wave properties and on the selection of a periodic traveling wave solution by localized perturbation of an unstable steady state. My analysis concerns equations of λ-ω" type, which are the normal form of a large class of oscillatory systems close to a Hopf bifurcation point. I finish the paper by showing how my results can be used to determine the effect of nonlocal dispersal on spatiotemporal dynamics in a predator-prey system. [ABSTRACT FROM AUTHOR]