A bifurcation theorem for Darwinian matrix models

Abstract

Matrix models are widely used to describe the discrete time dynamics of structured populations (i.e., biological populations in which individuals are classified into discrete categories such as age, size, etc.). A fundamental biological question concerns population extinction and persistence, i.e., the stability or instability of the extinction state versus the existence of stable positive equilibria. A fundamental bifurcation theorem provides one general answer to this question, using the inherent population growth rate $r$ as a bifurcation parameter, by asserting the existence a continuum of positive equilibria that bifurcates from the extinction state at $r=1$. Moreover, stability of the bifurcating non-extinction equilibria is determined by the direction of bifurcation (at least near the bifurcation point). Evolutionary game theoretic methods generalize structured population dynamic models so as to include the dynamics of (mean) phenotypic traits subject to natural selection. The resulting Darwinian matrix model describes both the structured population dynamics and the evolutionary trait dynamics and the way in which they interplay. Here we generalize the fundamental bifurcation theorem for structured population dynamics to Darwinian matrix models. We give two applications.