The Role of Advection in a Two-Species Competition Model: A Bifurcation Approach

Abstract

© 2016 American Mathematical Society. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. The effects of weak and strong advection on the dynamics of reaction-diffusion models have long been studied. In contrast, the role of intermediate advection remains poorly understood. For example, concentration phenomena can occur when advection is strong, providing a mechanism for the coexistence of multiple populations, in contrast with the situation of weak advection where coexistence may not be possible. The transition of the dynamics from weak to strong advection is generally difficult to determine. In this work we consider a mathematical model of two competing populations in a spatially varying but temporally constant environment, where both species have the same population dynamics but different dispersal strategies: one species adopts random dispersal, while the dispersal strategy for the other species is a combination of random dispersal and advection upward along the resource gradient. For any given diffusion rates we consider the bifurcation diagram of positive steady states by using the advection rate as the bifurcation parameter. This approach enables us to capture the change of dynamics from weak advection to strong advection. We will determine three different types of bifurcation diagrams, depending on the difference of diffusion rates. Some exact multiplicity results about bifurcation points will also be presented. Our results can unify some previous work and, as a case study about the role of advection, also contribute to our understanding of intermediate (relative to diffusion) advection in reaction-diffusion models.