Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Traveling wave solutions for Lotka-Volterra system re-visited

Pages: 171 - 196, Volume 15, Issue 1, January 2011      doi:10.3934/dcdsb.2011.15.171

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Anthony W. Leung - Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45219, United States (email)
Xiaojie Hou - Department of Mathematics and Statistics, University of North Carolina at Wilmington, Wilmington, NC 28403, United States (email)
Wei Feng - Department of Mathematics and Statistics, UNC Wilmington, Wilmington, NC 28403, United States (email)

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.

Keywords:  Traveling Wave, Existence, Monotone Iteration, Asymptotic rates, Uniqueness, Spectrum, Stability.
Mathematics Subject Classification:  Primary 35B35, Secondary 91B18, 35K57, 35B40.

Received: December 2009;      Revised: September 2010;      Available Online: October 2010.