Existence of traveling pulses for a diffusive prey–predator model with strong Allee effect and weak distributed delay

Abstract

In this article, we investigate the existence of traveling pulses in a diffusive prey–predator model with a strong Allee effect and weak distributed delay. Assuming that the growth and death rates of the predator are much smaller than those of the prey and that the average delay is small, we transform the model into a singularly perturbed problem with a three-time scale structure. Using generalized rotated vector field theory, geometric singular perturbation theory, and the phase plane method, we demonstrate the existence of a homoclinic orbit depending on the intermediate-slow system. In particular, we obtain the positions of the transverse heteroclinic orbit in the intermediate layer problem and provide explicit expressions for the associated heteroclinic orbit and wave speed. Additionally, we construct possible singular homoclinic orbits of the model and show the persistence of these orbits for two different small perturbation parameters.