On the Relationship Between r and R0 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 R0 = R0(u) are functions of u. For a general matrix model, we show that 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 nonlinear Darwinian matrix models from 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 ESS and an application to an experimental study of the evolution to a genetic polymorphism.