The Partial Migration Evolution in a Time‐Periodic Environment

This work focuses on the phenomenon of partial migration in time‐periodic environments. Time periodicity refers to cyclic variations in environmental conditions, such as seasonal changes, significantly influencing an organism’s habitat and resources. Partial migration, observed in numerous species, including birds, fish, and mammals, involves a fraction of the population undertaking migratory movements while others remain sedentary. We have developed a mathematical framework for understanding evolutionarily stable strategies (ESSs) and ideal free distributions (IFDs) in environments that change periodically over time. By focusing on a range of periodic Beverton–Holt functions, we have established a criterion involving environmental functions that is both necessary and sufficient to determine the existence of ESSs and IFDs. This criterion assesses environmental variations across both spatial and temporal dimensions throughout a periodic cycle, thereby broadening the application of IFDs to encompass general time‐periodic contexts. These strategies are evolutionarily stable and act as neighborhood invaders within the framework of evolutionary game theory. Our results build upon previous work that primarily considered temporally constant environments. Using a stage‐structured time periodic matrix model, we show the existence and stability of the k ‐cycle. In this study, we demonstrated the existence of ESS and IFD through a series of numerical examples, which supports the theoretical findings.