On the Conjecture of the Role of Advection in a Two-Species Competition-Diffusion Model

Abstract

We consider a reaction-diffusion-advection model for two competing species in a heterogeneous environment where the two species are ecologically identical except that they adopt different dispersal strategies: one is assumed to disperse randomly while the other is “smarter,” dispersing by random diffusion together with advection upward along the resource gradient. In the work by Averill, Lam, and Lou [Mem. Amer. Math. Soc., 245 (2017), no. 1161], among other things, the authors conjectured the following: (i) if the species without advection is a slower diffuser, then it will exclude its competitor when the advection rate is sufficiently small and lose competitive advantage when the advection rate passes some critical value; (ii) the species without advection will always be invaded by its competitor if it adopts a faster diffusion rate. In this paper, we partially solve this conjecture under mild assumptions on the resource function and the diffusion rates of the two competing species.