Stability and Bifurcations in a Discrete-Time Eco-Evolutionary Logistic Model

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.