Dynamics of a Predator–Prey Model with Fear and Its Carryover Effects, Allee in Predator and Diffusion-Driven Patterns

Recent research suggests that the presence of predator may affect the physical characteristics of prey species to the level that it is more effective compared to direct predation in reducing the prey biomass. Furthermore, such nonlethal impacts might persist across seasons or generations. This research work centers on a predator–prey interaction considering the predator’s fear and its carryover effects (COEs) on prey as well as the Allee effect on predator species. This model incorporates both self- and cross-diffusions to explore the species distribution due to the movement of the species subject to the Neumann boundary condition. First, we study the positivity, boundedness, feasible equilibria and their local stability; different bifurcations appear in the nonspatial system. Global dynamics of the system are discussed, showing that coexistence equilibrium can never be globally stable. Both fear and its COEs destabilize the system, but the Allee effect in predators, once destabilized, again stabilizes the system. COEs of fear can influence the effect of Allee and destabilize the system even when the Allee effect is high, but fear has no such impact on the system dynamics, coupling with the Allee parameter. The system exhibits bistability, and the combined influence of fear and its COEs results in the “paradox of enrichment”. For the spatial model, Turing instability conditions, wave number range for instability and different instability regions are derived. The system dynamics permits mainly the spot stationary patterns. Diffusion causes an increase in gatherings with rising COEs and the Allee effect in predators, but an increase in the fear effect diminishes the prey gatherings in specific spots. Hopf instability leads to oscillation in the system. The delicate dynamics of fear and its COEs, the predator’s Allee effect along with diffusion reveal insights into the harmony in nature.