Allee effect and hunting-induced bifurcation inquisition and pattern formation in a modified Leslie–Gower interacting species system

This article extensively explores the intricacies and complexities of dynamics in a modified Leslie–Gower predator–prey model, enriched with the additive Allee effect and hunting cooperation. The study commences by establishing the well-posedness of the problem and meticulously validating the dynamical system. A thorough theoretical investigation is conducted to explore the feasible equilibria of the model system, followed by an analysis of their stability, instability and all possible bifurcation scenarios. The model exhibits bistability and global asymptotic stability, along with a diverse range of local and global bifurcations, encompassing saddle node, Hopf, Bogdanov–Takens, transcritical, cusp, homoclinic bifurcations, as well as limit point cycle (LPC). These bifurcations serve to exemplify the intricate dynamical nature of the model system. In conjunction with the study of the one-parameter bifurcation diagram, by selectively varying two significant system parameters at a time, the entire parametric space is divided into distinct regions, allowing for a more comprehensive understanding of system dynamics within each region. A quantitative analysis based on numerical simulation is carried out to verify all the analytical findings and the application of the model. Our observations also encompass the manifestation of the bubbling effect and a scenario leading to the total extinction of the prey population. The evolution of diffusion-driven pattern generation in spots, stripes, labyrinthines, mixtures of stripes and holes, and hole replication is demonstrated in a two-dimensional (2D) plane. We have observed that the hunting cooperation and the Allee effect both have an impact on these spatial patterns. Numerical simulations are utilized to validate the theoretical findings and evaluate their biological implications, with the results consistently corroborating and strengthening the theoretical observations.