Memory, Delay, and Adaptive Feedback in Evolutionary Rescue: A Bellman–Harris–Volterra Approach
Abstract
Abstract Evolutionary rescue is typically studied within stochastic population models that assume exponential waiting times and memoryless dynamics. While analytically convenient, such models may overlook important biological mechanisms associated with delayed reproduction, maturation, and adaptive responses to environmental stress. In this work, we investigate whether temporal delay can fundamentally alter the conditions under which evolutionary rescue becomes possible. To address this question, we develop an age-dependent branching framework in which reproduction times follow a Gamma distribution and mutation intensity depends on population size. The resulting process belongs to the class of Bellman–Harris branching models and incorporates both temporal memory and adaptive feedback. We show that the expected population dynamics satisfy a Volterra integral equation and that state-dependent mutation transforms this formulation into a nonlinear feedback system linking population decline with adaptive variation. Numerical investigations reveal that delayed reproduction and mutation interact in a nontrivial manner, producing a structured survival landscape characterized by extinction, evolutionary rescue, and growth-dominated regimes. The model further exhibits a phase-transition boundary separating survival and extinction outcomes, whose location depends on both the strength of adaptive mutation and the temporal structure of reproduction. Comparisons with the classical…
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