Modeling Stratified Diffusion in Biological Invasions

Recent data on biological invasion show that range expansion is driven by various modes of dispersal such as neighborhood diffusion and long-distance dispersal that occur side by side within a species. In such a stratified dispersal process, the initial range expansion mainly occurs by neighborhood diffusion. However, as the range of the founder population expands, new colonies created by long-distance migrants increase in number to cause an accelerating range expansion in the later phase. We classify several well-documented examples of geographical expansions into three major types depending on the nonlinearity of the range-versus-time curve. To examine how long-distance dispersal produces accelerating range expansion, we construct a stratified diffusion model, which describes the dynamics of the size distribution of colonies created by long-distance migrants. The model consists of a von Foerster equation combined with a Skellam model. Analyzing the model provides an estimate of range expansion in terms of the rate of expansion due to neighborhood diffusion, the leap distance, and the colonization rate of long-distance migrants. The results explain various types of nonlinear range expansion observed in biological invasions.