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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

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


Author: Steven R. Dunbar
Journal: Trans. Amer. Math. Soc. 286 (1984), 557-594
MSC: Primary 35K57; Secondary 58F40, 92A15
MathSciNet review: 760975
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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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DOI: http://dx.doi.org/10.1090/S0002-9947-1984-0760975-3
PII: S 0002-9947(1984)0760975-3
Keywords: Traveling wave solution, diffusive Lotka-Volterra system, heteroclinic orbit, shooting argument
Article copyright: © Copyright 1984 American Mathematical Society