Effect of dispersal in single-species discrete diffusion systems with source-sink populations

A multi-patch source-sink model with and without intraspecific competition in the sink patches is considered. First, we study the dynamics of the model when the matrix of migration is irreducible and reducible. We show that, there is a threshold number of source patches such that the population potentially becomes extinct below the threshold and established above the threshold. Next, used the theory of perturbation singular and theorem of Tikhonov, in the case of perfect mixing, i.e. when the diffusion rate tends to infinity, we calculate the equilibrium of the model and we give a good approximations of the solutions in this case. Second, we determine, in some particular cases, the conditions under which fragmentation and the existence of sinks patches can lead to a total equilibrium population greater or smaller than the sum of the carrying capacities of the source patches. Finally, we study the effect of the rapid growth source population and rapid death sink population on the dynamics of the total equilibrium population and on the coexistence of the species