Dynamical behavior of solutions of a reaction-diffusion model in river network

In this paper, we study the long-time behavior of solutions of a reaction-diffusion model in a one-dimensional river network, where the river network has two branches, and the water flow speeds in each branch are the same constant beta. We show the existence of two critical values c(0) and 2 with 0<c(0)<2, and prove that when -c(0)<=beta<2, the population density in every branch of the river goes to 1 as time goes to infinity; when -2<beta<-c(0), then, as time goes to infinity, the population density in every river branch converges to a positive steady state strictly below 1; when |beta|>= 2, the species will be washed down the stream, and so locally the population density converges to 0. Our result indicates that only if the water-flow speed is suitably small (i.e., |beta|<2), the species will survive in the long run.