Traveling wave solutions of diffusive Lotka-Volterra equations: a heteroclinic connection in 𝑅⁴

Abstract

We establish the existence of traveling wave solutions for a reaction-diffusion system based on the Lotka-Volterra differential equation model of a predator and prey interaction. The waves are of transition front type, analogous to the solutions discussed by Fisher and Kolmogorov et al. for a scalar reaction-diffusion equation. The waves discussed here are not necessarily monotone. There is a speed c ∗ > 0 {c^\ast } > 0 such that for c > c ∗ c > {c^\ast } there is a traveling wave moving with speed c c . The proof uses a shooting argument based on the nonequivalence of a simply connected region and a nonsimply connected region together with a Liapunov function to guarantee the existence of the traveling wave solution. The traveling wave solution is equivalent to a heteroclinic orbit in 4 4 -dimensional phase space.