Spreading speeds as slowest wave speeds for cooperative systems

Bingtuan Li, Hans F. Weinberger, Mark A. Lewis

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230 Scopus citations


It is well known that in many scalar models for the spread of a fitter phenotype or species into the territory of a less fit one, the asymptotic spreading speed can be characterized as the lowest speed of a suitable family of traveling waves of the model. Despite a general belief that multi-species (vector) models have the same property, we are unaware of any proof to support this belief. The present work establishes this result for a class of multi-species model of a kind studied by Lui [Biological growth and spread modeled by systems of recursions. I: Mathematical theory, Math. Biosci. 93 (1989) 269] and generalized by the authors [Weinberger et al., Analysis of the linear conjecture for spread in cooperative models, J. Math. Biol. 45 (2002) 183; Lewis et al., Spreading speeds and the linear conjecture for two-species competition models, J. Math. Biol. 45 (2002) 219]. Lui showed the existence of a single spreading speed c* for all species. For the systems in the two aforementioned studies by the authors, which include related continuous-time models such as reaction-diffusion systems, as well as some standard competition models, it sometimes happens that different species spread at different rates, so that there are a slowest speed c* and a fastest speed c f*. It is shown here that, for a large class of such multi-species systems, the slowest spreading speed c* is always characterized as the slowest speed of a class of traveling wave solutions.

Original languageEnglish (US)
Pages (from-to)82-98
Number of pages17
JournalMathematical Biosciences
Issue number1
StatePublished - Jul 2005

Bibliographical note

Copyright 2008 Elsevier B.V., All rights reserved.


  • Cooperative systems
  • Discrete-time systems
  • Reaction-diffusion
  • Spreading speed
  • Traveling waves


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