Existence and global attractivity of a periodic solution to a nonautonomous dispersal system with delays

In this paper we consider a nonautonomous periodic dispersal system which models the diffusion of a single species between n patches connected by discrete dispersal in a periodic environment. Using Gaines and Mawhin continuation theorem of coincidence degree (cf. [R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977]), we prove that the system has at least one positive periodic solution. With the help of an appropriately chosen Lyapunov functional it is proved that this periodic solution is globally attractive. We end the paper by some numerical simulations which illustrate the feasibility of our results.