On the structure of the permanence region for competing species models with general diffusivities and transport effects

In this work we study the problem of the coexistence of two competing species in an inhabited region by analyzing the shape of the region where the species exhibit permanence. To make this analysis we first obtain a singular perturbation result for an elliptic boundary value problem associated to a logistic equation with a general differential operator. Then, we analyze how varies the principal eigenvalue of the operator at its singular limit in the case when the operator admits a reduction to the selfadjoint case. These results are new and of a great interest by themselves. Finally, we shall apply them to the problem of the permanence.