Large Speed Traveling Waves for the Rosenzweig-MacArthur Model with Spatial Diffusion

TL;DR

This paper investigates the existence and uniqueness of traveling wave solutions, including periodic wave trains, in the spatially diffusive Rosenzweig-MacArthur ecological model, especially at large wave speeds.

Contribution

It provides new analytical results on the existence and uniqueness of traveling waves and periodic wave trains in the model, using invariant manifold and global attractor theories.

Findings

Existence of traveling wave solutions connecting equilibria.

Uniqueness of periodic wave trains under certain conditions.

Behavior of solutions at large wave speeds.

Abstract

This paper focuses on traveling wave solutions for the so-called Rosenzweig-MacArthur model with spatial diffusion. The main results of this note are concerned with the existence and uniqueness of traveling wave solution as well as periodic wave train solution in the large wave speed asymptotic. Depending on the model parameters we more particularly study the existence and this uniqueness of a traveling wave connecting two equilibria or connecting an equilibrium point and a periodic wave train. We also discuss the existence and uniqueness of such a periodic wave train. Our analysis is based on ordinary differential techniques by coupling the theories of invariant manifolds together with those of global attractors.