Math Biology Seminar

Speaker: Wenzhang Huang, University of Alabama, Huntsville

Title: The Minimum Wave Speed of Traveling Waves for a Lotka-Volterra Competition Model

Abstract: Consider a reaction-diffusion system that serves as a 2-species Lotka-Volterra competition model with each species having logistic growth in the absence of the other. Suppose that the corresponding reaction system has one unstable boundary equilibrium E_1 and one stable boundary equilibrium E_2. 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 E_1 and E_2, and the system has no nonnegative traveling wave with wave speed less than C_*. It has been shown that the minimum wave speed for this system is identical to another important quantity - the speed of the population spread towards to 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. Although much research work has been done to give an estimate of C_* and some partial results have been obtained, the problem on finding an algebraic or analytic expression for the minimum wave speed remains unsolved in general. In this talk we will introduce a new, more efficient approach that enable us to determine precisely the minimum wave speed algebraically under conditions weaker than those given previously. We also show that the minimum wave speed in general cannot be determined by the linearization at the unstable equilibrium point. The conjecture on the precise minimum wave speed is also given.