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

Abstract

Abstract 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 E 0 at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:math> but needs not to connect the coexistence equilibrium <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mi>E</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:math> . 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 n 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mi>E</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:math> . 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.

Topics & Concepts