Effects of a protection zone in a reaction-advection-diffusion model with strong Allee effect

We consider a reaction-advection-diffusion equation that models a population in a bounded habitat with a strong Allee effect and a protection zone. We assume that the population growth function exhibits the strong Allee effect and has a positive integral within the protection zone (indicating population persistence) and a negative integral in the surrounding patches (indicating population decay). We prove that the existence of positive steady-state solutions to this system depends on the length of the protection zone. It is demonstrated that there exists a threshold value $ H^* $ such that, for a protection zone of size $ H^* $, there exists one positive steady-state solution and, for a larger protection zone, there exist multiple positive steady-state solutions. For smaller protection zones, we prove there exists no positive steady-state solution. The dynamics of the equation are further examined via numerical simulations.