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Traveling Wave Solutions of Diffusive Lotka-Volterra Equations: A Heteroclinic Connection in R4

Steven R. Dunbar
Transactions of the American Mathematical Society
Vol. 286, No. 2 (Dec., 1984), pp. 557-594
DOI: 10.2307/1999810
Stable URL: http://www.jstor.org/stable/1999810
Page Count: 38
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Traveling Wave Solutions of Diffusive Lotka-Volterra Equations: A Heteroclinic Connection in R4
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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.

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