Traveling wave solutions of a class of multi-species non-cooperative reaction–diffusion systems

Abstract

<p>In this paper we establish a sharp existence result on weak traveling wave solutions for a general class of multi-species reaction–diffusion systems. Moreover, the minimal speed of the traveling waves is explicitly determined. Such a weak traveling wave solution connects the predator-free equilibrium point <i>E</i>0 at x = −∞ but needs not to connect the coexistence equilibrium <i>E</i>^∗ at x = ∞. We apply this result to three important non-cooperative systems: the classical diffusive SIS system for the spread of infectious disease, a predator–prey system with age structure and a generalised Lotka–Volterra predator–prey system of one predator species feeding on <i>n</i> prey species, and prove with the aid of Lyapunov functions and the LaSalle invariance principle that their weak traveling wave solutions are actually traveling wave solutions that connect <i>E</i> ^∗ at <i>x</i> = ∞. For the SIS system and the generalised Lotka–Volterra predator–prey system, we develop additional techniques to establish the boundedness of their weak traveling wave solutions before applying the LaSalle's invariance principle.</p>