Dynamical behavior of a general reaction-diffusion-advection model for two competing species

It is well known that the studies of the evolution of biased movement along a resource gradient could create very interesting phenomena. This paper deals with a general two-species Lotka-Volterra competition model for the same resources in an advective nonhomogeneous environment, where the individuals are exposed to unidirectional flow (advection) but no individuals are lost through the boundary. It is assumed that the two species have the same population dynamics but different diffusion and advection rates. It is shown that at least five scenarios can occur (i) If one with a very strong biased movement relative to diffusion and the other with a more balanced approach, the species with much larger advection dispersal rate is driven to extinction; (ii) If one with a very strong biased movement and the other is smaller compare to its diffusion, the two species can coexist since one species mainly pursues resources at places of locally most favorable environments while the other relies on resources from other parts of the habitat; (iii) If both of the species random dispersal rates are sufficiently large (respectively small), two competing species coexist; (iv) If one with a sufficiently large random dispersal rate and the other with a sufficiently small one, two competing species still coexist; (v) If one with a sufficiently small random dispersal rate and the other with a suitable diffusion, which causes the extinction of the species with smaller random movement. Where (iii), (iv) and (v) show the global dynamics of (5) when both of the species dispersal rates are sufficiently large or sufficiently small. These results provide a new mechanism for the coexistence of competing species, and they also imply that selection is against excessive advection along environmental gradients (respectively, random dispersal rate), and an intermediate biased movement rate (respectively, random dispersal rate) may evolve. Finally, we also apply a perturbation argument to illustrate the evolution of these rates.