Impact of Predator-Driven Allee and Spatiotemporal Effect on a Simple Predator–Prey Model

In this research paper, we consider a Leslie–Gower Reaction–Diffusion (RD) model with a predator-driven Allee term in the prey population. We derive conditions for the existence of nontrivial solutions, uniform boundedness, local stability at co-existing equilibrium points, and Hopf bifurcation criteria from the temporal system. We identify sufficient conditions for Turing instability with no-flux boundary condition for the spatial system. Our investigation delves into the analysis of diffusion-induced Turing instability, incorporating stability conditions for the constant steady-state in the spatial model. We also investigate the conditions for the existence and nonexistence of nonconstant steady states in the diffusion-induced model. During numerical simulations, we observe that the predator-driven Allee term is essential for the model to generate Turing structures. Our findings reveal intriguing properties within the RD system, demonstrating its ability to produce patterns within the Turing domain. The simulation confirms that cold–hot spots and stripes-like patterns (a mixture of spots and strips) arises for different strengths of the predation parameter and Allee parameter. In contrast, we observe that for the above threshold value of the Allee parameter, the above-mentioned patterns may disappear from the system. Interestingly, we also observe that the stationary system produces patterns for both large and small amplitudes of perturbation in the vicinity of the Turing boundary. Our research may contribute valuable insights into the Allee effect and enhance our understanding of predator–prey interactions in naturalistic environments.