Analytical and numerical detection of traveling wave and wave-train solutions in a prey–predator model with weak Allee effect

Abstract

The traveling wave and wave-train fronts correspond to the transition zones from a trivial steady state to a coexistence steady state and a limit cycle around the coexistence steady state, respectively. In this study, we establish the existence of both types of fronts for a reaction–diffusion system of prey–predator interactions with weak additive Allee effect in prey growth, Holling type II functional response and density-dependent death rate for predators. For analytical simplicity, we consider immobile prey population. Under this consideration, the existence of both the traveling fronts is mathematically equivalent to the existence of point-to-point and point-to-cycle heteroclinic connections in $$\mathbb {R}^{3}$$. The proof for the existence of traveling fronts relies on the construction of an unbounded wedged region, a shooting argument and an appropriate Lyapunov function. Also, by employing the Hopf bifurcation theorem we show the existence of traveling wave-train solution. In both the cases, we find successful invasion by the predator species. Further, we provide adequate numerical illustrations in order to corroborate our theoretical predictions. Our numerical simulations also suggest that the theoretically obtained minimum wave speed serves as an asymptotic speed of traveling wave propagation.