On the dynamics of a class of Darwinian matrix models

Using the methodology of evolutionary game theory (EGT), I study a class of Darwinian matrix models which are derived from a class of nonlinear matrix models for structured populations that are known to possess stable (normalized) distributions. Utilizing the limiting equations that result from this ergodic property, I prove extinction and stability results for the limiting equations of the EGT versions of these kinds of structured population models. This is done in a bifurcation theory context. The results provide conditions sufficient for a branch of non-extinction equilibria to bifurcate from the branch of extinction equilibria. When this bifurcation is supercritical (explicit criteria are given), these equilibria are stable and represent stable non-extinction equilibria (which are also candidate ESS equilibria). These kinds of matrix models are motivated by applications to size structured populations, and I give an application of this type. Besides illustrating the formal theory, this application shows the importance of trade-offs among life history parameters that are necessary for the existence of an evolutionarily stable equilibrium.