Discrete-Time Eco-Evolutionary Dynamics: Stability, Bifurcations, and Co-evolutionary Trajectories

In this talk, we explore the mathematical modeling of discrete-time eco-evolutionary systems, focusing on the fundamental interplay between population dynamics and the evolution of phenotypic traits. Eco-evolutionary models have become an important framework for understanding how demographic processes and natural selection interact across generations, particularly in species with non-overlapping generations or seasonal reproduction. The emphasis is placed on deterministic models, both autonomous and periodic, where the coupling between demography and selection is described through systems of nonlinear difference equations.

The analysis begins with the study of the local stability of equilibria (fixed points) using the Jacobian formalism together with the localization of eigenvalues relative to the unit circle through Jury stability conditions. These techniques provide a natural framework for identifying persistence thresholds and qualitative transitions in the long-term dynamics of populations. We then discuss the principal bifurcations governing these systems, namely the transcritical bifurcation, associated with changes in persistence and extinction states, and the period-doubling bifurcation, which signals the emergence of oscillatory and potentially chaotic regimes. Particular attention is devoted to the Neimark--Sacker bifurcation, describing the transition from stationary eco-evolutionary equilibria to quasi-periodic oscillations, reflecting persistent fluctuations in both population abundance and evolving traits.

In the context of seasonal environments, we introduce the analysis of periodic systems via map composition and discrete Floquet theory. Finally, we discuss how trait diversity and coadaptation mechanisms may influence the robustness of eco-evolutionary systems, potentially stabilizing or destabilizing long-term population dynamics.