A perception-memory PDE framework for seasonal migration dynamics

Seasonally migrating animals must navigate environments where resources shift predictably but are increasingly perturbed by climate change and human activities. Empirical work highlights the importance of cognition for these movements, yet the joint roles of perception and memory in sustaining stable seasonal migration remain poorly understood. We develop and analyze a novel PDE (partial differential equation) model that couples random dispersal with two taxis processes: perception-driven movement toward a nonlocally sensed, periodically shifting resource, and memory-driven movement guided by a spatiotemporal map of past foraging successes over seasonal time windows. We first establish global well-posedness of the system, proving existence, uniqueness, and uniform boundedness of classical solutions. Using Leray-Schauder degree theory, we then show that the model admits at least one time-periodic solution synchronized with the seasonal resource. By constructing a Lyapunov-Krasovskii functional, we further derive sufficient conditions under which this periodic migratory pattern is unique and globally asymptotically stable, revealing a key trade-off between perception- and memory-driven taxis strengths and diffusive spreading. Numerical simulations corroborate the analytical results and demonstrate how the balance of perception and memory, the precision of memory, and its match or mismatch with environmental periodicity jointly govern migration efficiency and persistence. Together, these results provide a rigorous theoretical framework linking individual-level cognitive processes to the emergence, stability, and breakdown of seasonal migration routes in changing environments.