Spreading Phenomena in Integrodifference Equations with Nonmonotone Growth Functions

Abstract

Integrodifference equations are discrete-time cousins of reaction-diffusion equations. Like their continuous-time counterparts, they are used to model spreading phenomena in ecology and other sciences. Unlike their continuous-time counterparts, even scalar integrodifference equations can exhibit nonmonotone dynamics. Few authors studied the existence of spreading speeds and traveling waves in the nonmonotone case; previous numerical simulations indicated the existence of traveling two-cycles. Our numerical observations indicate the presence of several spreading speeds and multiple traveling wave profiles in these equations. We generalize the concept of a spreading speed to encompass this situation and prove the existence of such generalized spreading speeds and associated traveling waves in the corresponding second-iterate operator. Our numerical simulations let us conjecture that these spreading speeds could be linearly determined. We prove the existence of bistable traveling waves in a related second-iterate operator. We relate our results to the existence of stacked waves and to dynamical stabilization.