Dynamical stabilization and traveling waves in integrodifference equations

Integrodifference equations are discrete-time analogues of reaction-diffusion equations and can be used to model the spatial spread and invasion of non-native species. They support solutions in the form of traveling waves, and the speed of these waves gives important insights about the speed of biological invasions. Typically, a traveling wave leaves in its wake a stable state of the system. Dynamical stabilization is the phenomenon that an unstable state arises in the wake of such a wave and appears stable for potentially long periods of time, before it is replaced with a stable state via another transition wave. While dynamical stabilization has been studied in systems of reaction-diffusion equations, we here present the first such study for integrodifference equations. We use linear stability analysis of traveling-wave profiles to determine necessary conditions for the emergence of dynamical stabilization and relate it to the theory of stacked fronts. We find that the phenomenon is the norm rather than the exception when the non-spatial dynamics exhibit a stable two-cycle.