Traveling wave solutions of diffusive Lotka-Volterra equations: a heteroclinic connection in $\textbf {R}^{4}$

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 ${c^\ast } > 0$ such that for $c > {c^\ast }$ there is a traveling wave moving with speed $c$. 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 $4$-dimensional phase space.