Singular Perturbation Analysis for a Holling–Tanner Model with Additive Allee Effect

Abstract

In this article, a reaction–diffusion predator–prey equation with additive Allee effect acting on prey is investigated by twice singular perturbation analysis. Our focus is to study the traveling fronts of predator invasion under the assumption that the diffusion ability of the predator is much greater than that of the prey. Our results exhibit two kinds of traveling fronts for a Holling–Tanner system in two different limit cases. And these traveling fronts correspond to the heteroclinic connections between a saddle and either a stable node or a stable focus. In addition, we show that heteroclinic connections are formed in different ways for different limit cases. One is formed on the slow manifold, which has one time scale. While the other is formed by the intersection of the slow manifold and the fast manifold, which has two time scales. Furthermore, the existence of traveling front solutions in different limit cases is demonstrated through theoretical analysis and numerical simulation. The main tools are geometric singular perturbation theory and Bendixson’s criteria.