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On the relationship between r and R 0 and its role in the bifurcation of stable equilibria of Darwinian matrix models

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 R 0=R 0(u) are functions of u. For a general matrix model, we show that r and R 0 share critical values and extrema at values u=u* for which r(u*)=R 0(u*)=1. This allows us to re-interpret, in terms of the more analytically tractable quantity R 0, 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.