A Geometric Singular Perturbation Approach to Traveling Waves in a Fast-Slow Rosenzweig-MacArthur Model

Applying geometric singular perturbation theory and the qualitative theory of ordinary differential equations, this paper establishes the existence of traveling waves in a fast-slow diffusive Rosenzweig-MacArthur predator-prey model that incorporates a prey refuge. The analysis begins with the spatially homogeneous system, employing fast-slow decomposition to: (i) characterize its equilibria and critical manifolds; (ii) verify the normal hyperbolicity of these manifolds; and (iii) delineate the distinct dynamics of the fast and slow subsystems. The reaction-diffusion equation is then transformed into a four-dimensional, fast-slow ordinary differential equation system, characterized by two distinct singular parameters. Through the systematic application of geometric singular perturbation theory, we rigorously establish the existence of traveling waves via a dimensional reduction approach, progressively analyzing the dynamics of the system from the original four-dimensional formulation to three-dimensional and finally two-dimensional reduced systems. Specifically: two-dimensional reduction involves proving the existence of heteroclinic orbits in the reduced system when both singular parameters vanish, leveraging insights from the Fisher-KPP equation; three-dimensional reduction extends the analysis to the case where one large singular parameter vanishes, demonstrating heteroclinic connections; and full system analysis finally establishes the existence of traveling waves in the original reaction-diffusion equation. Numerical simulations validate these theoretical results, confirming the presence of heteroclinic orbits in the reduced two-dimensional phase space and the predicted wave propagation patterns.