APPROACHING THE IDEAL FREE DISTRIBUTION IN TWO-SPECIES COMPETITION MODELS WITH FITNESS-DEPENDENT DISPERSAL

This paper concerns the global existence and asymptotic behavior of solutions to some reaction-diffusion-advection models for two competing species, where the species have the same population dynamics but different dispersal strategies. When one species possesses a combination of random dispersal and directed movement upward along its fitness gradient whereas the other species adopts random dispersal, the global existence of smooth solutions to the quasi-linear parabolic system is established. When one species adopts the fitness-dependent dispersal but the other species does not disperse at all, we show the global existence of weak solutions to the degenerate parabolic-ODE system and further describe the asymptotic behavior of these weak solutions. In particular, we show that in the latter case the total population density approaches the so-called ideal free distribution in an appropriate sense.