Nonlocal dispersal and averaging in heterogeneous landscapes

Abstract

While reaction–diffusion equations are the standard modelling framework for many questions in spatial ecology, their nonlocal analogues, integrodifferential equations, have gained in popularity recently. Here we consider integrodifferential equations for population spread and persistence in heterogeneous landscapes, and we develop appropriate averaging methods for these models. We average over landscape and patch scales. While averaging methods for reaction–diffusion equations lead to relatively simple expressions of persistence conditions and invasion speeds, we find that the results are much richer and more complicated for integro-differential equations. We illustrate our results with two dispersal mechanisms: (1) individuals are mobile throughout their lifetime and (2) only offspring move.

Keywords:

• nonlocal dispersal
• averaging
• invasion speed
• persistence condition

AMS Subject Classifications:

• 35B27
• 45C05
• 92D40

Acknowledgement

The author thanks Jim Powell and an anonymous referee for thorough reading of the manuscript and insightful comments.