Predator-prey systems with a variable habitat for predators in advective environments

Community composition in aquatic environments can be shaped by a broad array of factors, encompassing habitat conditions in addition to abiotic conditions and biotic interactions. This paper pertains to reaction-diffusion-advection predator-prey model featuring a variable predator habitat in advective environments, governed by a unidirectional flow. First, we establish the near-complete global dynamics of the system. In instances where the functional response to predation conforms to Holling-type I or II, we explore the uniqueness and stability of positive steady-state solutions via the application of particular auxiliary techniques, the comparison principle for parabolic equations, and perturbation analysis. Furthermore, we obtain the critical positions at the upper and lower boundaries of the predator's habitat, which determine the survival of the prey irrespective of the predator's growth rate. Finally, we show how the location and length of the predator's habitat affect the persistence and extinction of predators and prey in the event of a small population loss rate at the downstream end. From the biological point of view, these results contribute to our deeper understanding of the effects of habitat on aquatic populations and may have applications in aquaculture and the establishment of protection zones for aquatic species.