The role of protection zones in the persistence of aquatic species: A reaction–diffusion–advection model for fishery management

To ensure the sustainable use of fisheries resources and preserve endangered aquatic species, the establishment of aquatic protection zones is essential. In this paper, we propose a reaction–diffusion–advection model with a protection zone, which describes a single-species population in rivers with a local no-harvesting zone. We investigate the persistence and extinction of populations under three different scenarios: strong-flow, strong-diffusion, and weak-diffusion. It turns out that in strong-flow or weak-diffusion scenarios, if the downstream river environment is closed, then only the boundary-downstream protection zone enables the population to survive under any harvesting intensity. In strong-diffusion scenarios, hostile boundary conditions always put the population at risk of extinction, irrespective of how long and where the protection zone is. Furthermore, for populations satisfying logistic or weak Allee growth in homogeneous environments with general boundary conditions, we derive the optimal protection zones, namely, the critical lengths of the internal protection zone, the boundary-upstream protection zone, and the boundary-downstream protection zone. Finally, numerical simulations show that incorporating advection dynamics can lead to a complex relationship between the spatial location of protection zones and species survival thresholds, whereas the influence of protection zone length exhibits a straightforward monotonic trend. These results may provide valuable insights for the rational design of aquatic protection zones, which can contribute to more effective fisheries management and the long-term conservation of aquatic species.