Extinction and Persistence of Species in Discrete Competitive Systems with a Safe Refuge

We analyze a two species discrete competition model in which one species diffuses between two patches, A and B. In this model, two species, species 1 and 2, compete in patch A with species 1 being the sedentary species. Thus, patch B is a safe refuge for species 2. We obtain sufficient conditions for the extinction of species 1. Species 2 is the superior competitor whenever a linear combination of its growth rates always exceeds the growth rate of the sedentary species 1. By using a specific example, we demonstrate that providing a safe refuge does not always make a species a superior competitor. In fact, without diffusion, species 2 drives species 1 to extinction. However, with the addition of diffusion, there is stable coexistence of the two species. If the safe refuge is not suitable for its growth and reproduction, species 2 may go extinct. We obtain sufficient conditions for the extinction of species 2. We also show that a species persists whenever all of its carrying capacities are sufficiently large. This result rules out the possibility of a population becoming arbitrarily close to zero and therefore risking extinction