Some Challenging Mathematical Problems in Evolution of Dispersal and Population Dynamics

Abstract

We discuss the effects of dispersal (either random or biased) and spatial heterogeneity on population dynamics via reactionadvection-diffusion models. We address the question of determining optimal spatial arrangement of resources and study how advection along resource gradients affects the extinction of species. The effects of dispersal and spatial heterogeneity on the total population size of single species are carefully investigated, along with some other properties of species. These properties have important applications to invasions of rare species. Some interesting connection between the evolution of unconditional dispersal and diffusion-driven extinction is revealed. We also investigate the outcome of competition for two similar species, and show how invasion and coexistence are affected by resource utilization, inter-specific competition, and dynamics of habitat edges. In particular, interesting effects of intermediate values of dispersal rates are found. The evolution of conditional dispersal is also addressed, and we illustrate that the geometry of a habitat can play an important role in the evolution of conditional dispersal and that strong directed movement of species can induce the coexistence of competing species. If both species disperse by random diffusion and advection along environmental gradients and one species has much stronger biased movement than the other one, then at least two scenarios can occur: either both species can coexist or the “smarter” species is always the loser. These results seem to suggest that selection is against large advection along resource gradient and that an intermediate biased movement rate may