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<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<-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.

Automated summary

In this paper, we investigate a reaction-diffusion model in a one-dimensional river network and find that the long-term survival of a species is closely related to the water flow speed. The species can survive when the water flow speed is small, but it will be washed away when the speed is large.