Asymptotic behaviors of a delay difference system of plankton allelopathy

The main purpose of this paper is to study a general delay difference system of the growth of two species of plankton having competing and allelopathic effects on each other. We first show that the system is permanent under some suitable assumptions. Next, by using the continuous theorem of coincidence degree theory and constructing a Lyapunov function, a set of sufficient conditions which guarantee the existence and global attractivity of positive periodic solutions are obtained. Finally, two examples together with their numerical simulations are presented to illustrate the feasibility of our main results.