Optimal Protected Area Design for Allee Effect Mitigation in Spatial Predator-Prey Systems

Endangered populations often experience limited growth ability at low densities, a phenomenon described by the Allee effect. In this thesis, we investigate a predator-prey model incorporating the Allee effect within a two-dimensional nonlinear reaction-diffusion framework, with the aim of understanding how local spatial refuges can promote the persistence of low-density populations by enabling them to surpass recovery thresholds. We first simulate an extinction-prone scenario in which initial densities fall below the Allee threshold, demonstrating that most populations tend toward extinction. We then introduce protected areas together with positive growth terms to facilitate survival. To assess the role of diffusion-reaction dynamics, we construct an objective function based on the shape and location of protected areas, and employ a bi-objective optimization approach. Our results reveal that as the weights of the objective function vary, the optimal protectedarea configuration shifts between fragmented and contiguous patterns. We begin with a single-species prey analysis and subsequently extend the model to include predators, where we use mathematical analysis to investigate the steady states of the two-species system.