Invariant manifolds and global bifurcations to find multiple wave solutions in a reaction-diffusion system

Abstract

A thorough analysis is performed in a predator-prey reaction-diffusion model which includes three relevant complex dynamical ingredients: (a) a strong Allee effect; (b) ratio-dependent functional responses; and (c) transport attributes given by a diffusion process. As is well-known in the specialized literature, these aspects capture adverse survival conditions for the prey, predation search features and non-homogeneous spatial dynamical distribution of both populations. We look for traveling-wave solutions and provide rigorous results coming from a standard local analysis, numerical bifurcation analysis, and relevant computations of invariant manifolds to exhibit homoclinic and heteroclinic connections and periodic orbits in the associated dynamical system in R 4 . In so doing, we present and describe a diverse zoo of traveling wave solutions; and we relate their occurrence to the Allee effect, the spreading rates and propagation speed. In addition, homoclinic chaos is manifested via both saddle-focus and focus-focus bifurcations as well as a Belyakov point. An actual computation of global invariant manifolds near a focus-focus homoclinic bifurcation is also presented to enravel a multiplicity of wave solutions in the model. A deep understanding of such ecological dynamics is therefore highlighted.