Qualitative Analysis of a Time- and Space-discrete Predator–prey Model with Fear Effect and Allee Effect in Predator

The emergence of spatial distributions in predator–prey dynamics is a significant topic. In this study, we respond to this problem by creating an intriguing predator–prey model that is time- and space-discrete. The fear effect of predators on prey’s reproduction, Allee effect of predators, and the species’ self-diffusion is considered. The discrete model is represented using a coupled-map lattice, which assumes a nonlinear link between the predator–prey reaction and dispersal stages. The feasible fixed points and their stability are derived along with the conditions of Flip and Neimark–Sacker bifurcations. Fear effect changes the system’s dynamics from stable fixed point to invariant closed curve to stable fixed point again, whereas time step destabilizes the system by formation of an invariant closed curve. The Allee effect stabilizes the system but a further increase in the Allee effect reveals a critical threshold triggering bi-stability and subsequent predator extinction. To discriminate between chaotic and regular behaviors, maximum Lyapunov exponents are depicted. The condition for Turing instability is described and different instability regions are identified. Simulations reveal that spatial movement of both the species increases the spatial heterogeneity and provides a fascinating diversity of spatial patterns, including regular homogeneous, oscillation, quasi-periodic and irregular chaos. In this discrete system, spatiotemporal chaos even arises in the Turing region, and the initial population influences species dispersion. This study shows how the discrete model’s nonlinear dynamics better represent the pattern generation intricacy of predator–prey systems.