Dynamics for a Single Species Reaction‐Diffusion Model With Seasonal Michaelis–Menten Type Harvesting and Free Boundaries

ABSTRACT Seasonal harvesting is a common strategy in industries such as fishing and animal husbandry, aimed at balancing resource utilization and ecological sustainability. This work focuses on a seasonal harvesting reaction‐diffusion model with free boundaries, studying the dynamics of species expansion and persistence under ecological and harvesting constraints. To study the free boundary problem, we first analyze periodic reaction‐diffusion problems in both unbounded and bounded domains. In unbounded domains, the existence of spreading speeds and traveling waves is established, while in bounded domains, the critical domain size necessary for species persistence is determined. Building on these results, we derive the spreading‐vanishing dichotomy, establish criteria for species persistence, and provide estimates for spreading speeds under free boundary conditions. Our findings demonstrate that spreading and vanishing phenomenon is influenced by the harvesting duration, initial region size, and the expansion ability of the species. In particular, the spreading speed is significantly slower in free boundary scenarios, highlighting the influence posed by free boundaries in seasonal harvesting.