Traveling wave solutions for Lotka-Volterra system re-visited

Anthony W. Leung; Xiaojie Hou; Wei Feng

doi:10.3934/dcdsb.2011.15.171

Article Contents

2011, Volume 15, Issue 1: 171-196. Doi: 10.3934/dcdsb.2011.15.171

This issue Previous Article Periodic and quasi--periodic orbits of the dissipative standard map Next Article Dynamic phase transition for binary systems in cylindrical geometry

Traveling wave solutions for Lotka-Volterra system re-visited

1.
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45219
2.
Department of Mathematics and Statistics, University of North Carolina at Wilmington, Wilmington, NC 28403
3.
Department of Mathematics and Statistics, UNC Wilmington, Wilmington, NC 28403

Received: December 2009

Revised: September 2010

Published: July 2011

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Abstract

Using a new method of monotone iteration of a pair of smooth lower- and upper-solutions, the traveling wave solutions of the classical Lotka-Volterra system are shown to exist for a family of wave speeds. Such constructed upper and lower solution pair enables us to derive the explicit value of the minimal (critical) wave speed as well as the asymptotic decay/growth rates of the wave solutions at infinities. Furthermore, the traveling wave corresponding to each wave speed is unique up to a translation of the origin. The stability of the traveling wave solutions with non-critical wave speed is also studied by spectral analysis of a linearized operator in exponentially weighted Banach spaces.

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Mathematics Subject Classification: Primary 35B35, Secondary 91B18, 35K57, 35B40.

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