Resilience in a modified Leslie-Gower model with dual Allee effects and cooperative hunting

The present study is devoted to the precise characterization of dynamic transitions within a continuous two-dimensional ecological framework, induced by a bifurcation module. This investigation aims to enhance our understanding of ecological evolution by disentangling the individual and interactive influences of the double Allee effect and hunting cooperation. Dynamics of all non-negative equilibria are investigated to disclose the degenerate nature of them. Bifurcation points are systematically identified, as their determination is critical for devising strategies aimed at stabilizing ecological systems or averting species extinction, particularly within networks influenced by dual Allee effects and cooperative predation. Both local and global bifurcation analyses, distinguished by the number of relevant parameters and the qualitative behaviour have been explored to capture the system's intricate dynamics. A comprehensive overview of current numerical bifurcation analysis techniques and their ecological applications is provided. Emphasis is placed on the computational challenges encountered in detecting extinction-driven bifurcations and in identifying diverse attractor landscapes and phase transitions. Particular attention has been given to the numerical difficulties posed by both weak and strong forms of the Allee effect, and potential resolutions to these methodological bottlenecks are proposed. Through this structured analysis, an in-depth understanding of species interaction dynamics within bifurcation-driven ecological models is sought, aiming to elucidate complex behaviours that are often obscured in conventional studies of large bifurcating ecological networks.