Reaction–diffusion models with large advection coefficients

Abstract

The dispersal of organisms in a heterogeneous environment is rarely random as individuals can sense and respond to local environmental cues by moving towards favourable habitats. We study reaction–diffusion–advection models which describe the biased movement of species upward along the resource gradient. Our focus is on the effects of such non-random movement of organisms on the spatial distribution and dynamics of both single and multiple populations. For a single species at equilibrium, previous works show that the species with strong biased movement upward along the resource gradient will overmatch the resource at each global maximum of the resource. In this work, we show that, in fact, the species will overmatch the resource at every local maximum of the resource. As an application, we also study a reaction–diffusion–advection model for two competing species. We assume that the two species are identical except for their dispersal strategies: both species adopt random dispersal combined with the advection upward along the resource gradient. It was shown in previous works that when both advection rates are large, a competitive exclusion phenomenon occurs: the species with relatively weaker advection drives the other to extinction, under the assumption that the resource has a unique maximum. In this work, we extend this competitive exclusion result to general resource functions with finite but arbitrarily many of local maxima.

Keywords:

• Reaction–diffusion–advection
• non-random movement
• competing species

AMS Subject Classifications::

• 35K57
• 92D25

Acknowledgement

This research was partially supported by the National Science Foundation grant DMS-0615845. The authors sincerely thank two anonymous referees for their careful readings and constructive comments.