In this paper I study a two-dimensional discrete-time evolutionary logistic-type model describing the coupled dynamics of population density and a continuously evolving trait. I provide a local bifurcation analysis of the equilibria, deriving explicit conditions for their existence and local stability. In particular, I show that the boundary and interior equilibria exchange stability through a transcritical bifurcation, and I characterize analytically the subsequent loss of stability of the interior equilibrium via period-doubling and Neimark–Sacker bifurcations. Transversality is established in all cases, and the criticality of the bifurcations is determined through normal form and Lyapunov coefficient computations. I show that the period-doubling bifurcation can be supercritical or subcritical, while the Neimark–Sacker bifurcation is generically nondegenerate and may be either supercritical or subcritical, depending on parameter values.
Rafael Luís (Tue,) studied this question.