Asymptotic and transient approaches of harvested predator–prey models with reaction–diffusion

This article proposes a diffusive predator–prey model with the Allee effect in predator and harvesting efforts on both species. Allee effect and harvesting efforts are taken as control parameters. Due to more than one control parameter, the asymptotic and short-term dynamics are analyzed, dividing the model into three sub-models. The stability analysis and bifurcation of all models’ interior equilibrium are duly reported for the temporal system. The model system’s analytical outcomes are validated numerically through their graphical representations. Besides, in the spatial system, the Turing bifurcation condition is derived. We see that all three systems are always reactive. By observing the changes in reactivity, we see that both the harvesting efforts and the ‘Allee effect constant’ have destabilizing effects. This causes a reduction in the density of the predator species. Various rich spatial patterns, including spots, stripes, and labyrinthine, are detected using numerical simulations. Generally, researchers study long-term dynamics when analyzing spatial systems. With the help of short-term dynamics, we have established the system is always reactive, and it is in increasing mode at turning regions in all subcases. These results are also verified numerically. This resemblance is the main outcome of this work.