Non-linear determinacy of minimum wave speed for a Lotka–Volterra competition model

Abstract

For a reaction–diffusion system that serves as a 2-species Lotka–Volterra diffusive competition model, suppose that the corresponding reaction system has one stable boundary equilibrium and one unstable boundary equilibrium. Then it is well known that there exists a positive number c ⁎ , called the minimum wave speed, such that, for each c larger than or equal to c ⁎ , the reaction–diffusion system has a positive traveling wave solution of wave speed c connecting these two equilibria if and only if c ⩾ c ⁎ . It has been shown that the minimum wave speed for this system is identical to another important quantity – the asymptotical speed of population spread towards the stable equilibrium. Hence to find the minimum wave speed c ⁎ not only is of the interest in mathematics but is of the importance in application. It has been conjectured that the minimum wave speed can be determined by studying the eigenvalues of the unstable equilibrium, called the linear determinacy. In this paper we will show that the conjecture on the linear determinacy is not true in general.