Darwinian dynamics of a juvenile-adult model

The bifurcation that occurs from the extinction equilibrium in a basic discrete time, nonlinear juvenile-adult model for semelparous populations, as the inherent net reproductive number R0 increases through 1, exhibits a dynamic dichotomy with two alternatives: an equilibrium with overlapping generations and a synchronous 2-cycle with non-overlapping generations. Which of the two alternatives is stable depends on the intensity of competition between juveniles and adults and on the direction of bifurcation. We study this dynamic dichotomy in an evolutionary setting by assuming adult fertility and juvenile survival are functions of a phenotypic trait u subject to Darwinian evolution. Extinction equilibria for the Darwinian model exist only at traits u• that are critical points of R0(u). We establish the simultaneous bifurcation of positive equilibria and synchronous 2-cycles as the value of R0(u•) increases through 1 and describe how the stability of these dynamics depend on the direction of bifurcation, the intensity of between-class competition, and the extremal properties of R0(u) at u•. These results can be equivalently stated in terms of the inherent population growth rate r(u).