NONLINEAR ASYMMETRIC DISPERSAL IN TWO-PATCH SOURCE-SINK HETEROGENEOUS ENVIRONMENT

Abstract

This paper analyzes a source-sink system with nonlinear asymmetric dispersal between two patches, where populations grow logistically in the source patch and decays logistically in the sink patch. First, using the theory of cooperative differential systems, we prove the global stability of the model. Next, applying the theory of singular perturbations and Tikhonov's theorem, we study the case of perfect mixing, i.e., when the diffusion rate tends to infinity. In this regime, we compute the equilibrium of the model, which differs from the carrying capacity of the source patch, and provide accurate approximation of the solutions. Furthermore, a complete analysis of the model demonstrates a mechanism by which nonlinear asymmetric dispersal may lead to: (i) an increase in the total population size across the two patches, (ii) a decrease in the total population size with persistence in both patches, or (iii) extinction in both patches when the migration rate becomes infinite. We also perform numerical comparisons, for specific parameter values, between the total equilibrium populations of models with linear and nonlinear asymmetric dispersal. These results show that the two populations are generally different, although they may coincide for certain migration rates. Finally, we investigate the effect of rapid growth in the source patch and rapid decline in the sink patch on the dynamics of the total equilibrium population and on the coexistence of populations in both patches.