Advantage and Disadvantage of Dispersal in Two-Species Competition Models

We consider a two-species competition model in which both populations are identical except their movement strategies: One species moves upward along the fitness gradient, while the other does not diffuse. While both species can coexist in homogeneous environment, we show that the species with directed movement has some advantage over the non-diffusing species in certain measurement. In contrast, if one species moves by random dispersal while the other does not diffuse, then the non-diffusing population could have advantage. Understanding the full dynamics of these ODE-PDE hybrid systems poses challenging mathematical questions. AMS subject classifications: 35K57, 35Q92, 92D25