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

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 conjec-tured 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 conjec-ture under mild assumptions on the resource function and the diffusion rates of the two competing species.