Global dynamics of a reaction-diffusion switching model with multiple stable states and linear harvesting rate

The closed fishing season policy plays a crucial role in fishery management by contributing to the restoration and protection of fishery resources, maintaining ecological balance and promoting sustainable development. The population dynamics of fish, particularly marine species, are highly complex. Under the combined effects of ecological mechanisms (such as predation, resource limitations, and competition), fish populations can exhibit multiple stable states. Overfishing increases the vulnerability of fish populations, making them prone to shift from a high-density stable state to a low-density one, and in some cases, leading to the risk of extinction. In this context, developing effective closed fishing season policies to ensure the sustainable development of fishery resources has become a pressing issue. In this paper, we propose a reaction-diffusion model consisting of two sub-equations with multiple stable states and a linear harvesting rate to describe the continuous switching between closed and open fishing seasons. We define a threshold value $\overline{T}_{}^{\ast }$ for the duration of the fishing ban $\overline{T}$ . When $\overline{T}\le \overline{T}_{}^{\ast }$ , the trivial stable state is globally asymptotically stable. Uniqueness of periodic solutions is generally a mathematically challenging problem. However, employing the comparison theorem, we find that conditions on the uniqueness of periodic solutions to the associated ODE system are also applicable to our model. Specifically, under certain conditions, when $\overline{T}>\overline{T}_{}^{\ast }$ , we provide sufficient conditions on the existence of a globally asymptotically stable periodic solution. Finally, we offer discussion and numerical simulations to illustrate our findings.