Eco-Evolutionary Feedback and Basin-Dependent Survival in a Ricker Model with a Strong Allee Effect

Trait evolution can be both a population's rescue and its ruin. This raises a central eco-evolutionary question: under a strong Allee effect, can the evolution of a quantitative trait help a sparse population escape extinction, or can it instead destabilize an otherwise viable population and increase extinction risk? To address this, we study a discrete-time Darwinian extension of the Ricker model coupling population density to the evolution of a mean trait: a low-density reproductive limitation term generates the strong Allee effect, and the trait evolves along the fitness gradient. The model is a two-dimensional map with eco-evolutionary feedback: density shapes selection, while the evolving trait alters reproduction and competition. After determining the equilibrium structure and the threshold separating extinction from positive-density survival, we analyze how the upper positive equilibrium loses stability. Period-doubling (PD) and Neimark-Sacker (NS) bifurcations replace steady survival by alternating-generation and oscillatory dynamics. Codimension-two phenomena-the $1:2$ point where the PD and NS curves meet, together with the $1:3$ and $1:4$ strong resonances-organize transitions toward two-, three-, and four-generation cycles, with quasiperiodic and more complex fluctuations. These mechanisms arise from the non-monotone Ricker-type recruitment term and are absent from the monotone one-dimensional Beverton-Holt recruitment map. We show that evolutionary response and trait-mediated competition can either favor persistence or promote destabilization, depending on parameters. Basin computations reveal coexistence between positive-density attractors and extinction, so long-term survival depends on both parameters and initial conditions. Overall, the model clarifies how Darwinian trait evolution and low-density demographic limitation jointly shape extinction thresholds, oscillatory survival, and population fate.