The dynamics of a temporally discrete diffusive competition model with free boundaries

To describe the evolution of an invasive population with discrete Stefan free boundaries and a native competitor that occupies the whole space, this paper investigates a temporally discrete competition system with diffusion and free boundaries. The well-posedness of the problem is established, which is largely different from the one for continuous time counterpart. Furthermore, the long time behavior of the solutions in certain cases is obtained. More precisely, when the invasive species is an inferior competitor, it is shown that the invader can only expand to a bounded region and eventually go extinct. When the invader is a superior competitor, a dichotomous conclusion about spreading or vanishing is demonstrated. Several sufficient conditions are also provided to ensure the invader's successful spreading (or failed invasion). Finally, we present numerical simulations to substantiate the theoretical results.