Journal of Mathematical Biology

, Volume 45, Issue 3, pp 183–218

Analysis of linear determinacy for spread in cooperative models


  • Hans F. Weinberger
    • School of Mathematics, University of Minnesota, 514 Vincent Hall, 206 Church Street S.E., Minneapolis, Minnesota 55455, USA
  • Mark A. Lewis
    • Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA
  • Bingtuan Li
    • Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA

DOI: 10.1007/s002850200145

Cite this article as:
Weinberger, H., Lewis, M. & Li, B. J Math Biol (2002) 45: 183. doi:10.1007/s002850200145


 The discrete-time recursion system $\u_{n+1}=Q[\u_n]$ with $\u_n(x)$ a vector of population distributions of species and $Q$ an operator which models the growth, interaction, and migration of the species is considered. Previously known results are extended so that one can treat the local invasion of an equilibrium of cooperating species by a new species or mutant. It is found that, in general, the resulting change in the equilibrium density of each species spreads at its own asymptotic speed, with the speed of the invader the slowest of the speeds. Conditions on $Q$ are given which insure that all species spread at the same asymptotic speed, and that this speed agrees with the more easily calculated speed of a linearized problem for the invader alone. If this is true we say that the recursion has a single speed and is linearly determinate. The conditions are such that they can be verified for a class of reaction-diffusion models.

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© Springer-Verlag Berlin Heidelberg 2002