On the relationship between r and R0 and its role in the bifurcation of stable equilibria of Darwinian matrix models

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16 Scopus citations


If the demographic parameters in a matrix model for the dynamics of a structured population are dependent on a parameter u, then the population growth rate r = r(u) and the net reproductive number R0 = R0(u) are functions of u. For a general matrix model, we showthat r and R0 share critical values and extrema at values u = u* for which r(u*) = R0(u*) = 1. This allows us to re-interpret, in terms of the more analytically tractable quantity R0, a fundamental bifurcation theorem for non-linear Darwinian matrix models from the evolutionary game theory that concerns the destabilization of the extinction equilibrium and creation of positive equilibria. Two illustrations are given: a theoretical study of trade-offs between fertility and survivorship in the evolution of an evolutionarily stable strategies and an application to an experimental study of the evolution to a genetic polymorphism.

Original languageEnglish (US)
Pages (from-to)277-297
Number of pages21
JournalJournal of biological dynamics
Issue number3
StatePublished - May 2011


  • Bifurcation
  • Darwinian matrix models
  • Equilibria
  • Net reproductive number
  • The evolutionary game theory

ASJC Scopus subject areas

  • Ecology, Evolution, Behavior and Systematics
  • Ecology


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