We prove the existence of traveling fronts in diffusive Rosenzweig–MacArthur and Holling–Tanner population models and investigate their relation with fronts in a scalar Fisher-KPP equation. More precisely, we prove the existence of fronts in a Rosenzweig–MacArthur predator-prey model in two situations: when the prey diffuses at the rate much smaller than that of the predator and when both the predator and the prey diffuse very slowly. Both situations are captured as singular perturbations of the associated limiting systems. In the first situation we demonstrate clear relations of the fronts with the fronts in a scalar Fisher-KPP equation. Indeed, we show that the underlying dynamical system in a singular limit is reduced to a scalar Fisher-KPP equation and the fronts supported by the full system are small perturbations of the Fisher-KPP fronts. We obtain a similar result for a diffusive Holling–Tanner population model. In the second situation for the Rosenzweig–MacArthur model we prove the existence of the fronts but without observing a direct relation with Fisher-KPP equation. The analysis suggests that, in a variety of reaction–diffusion systems that rise in population modeling, parameter regimes may be found when the dynamics of the system is inherited from the scalar Fisher-KPP equation.
Singular Limit Approach to Stability and Bifurcation for Bistable Reaction Diffusion Systems
