Bogdanov-Takens Bifurcation of Codimension 4 in a Leslie-Gower Predator-Prey Model With Allee Effect and Intraspecific Competition

Predator-prey models, such as the Leslie-Gower model, are essential for understanding population dynamics and stability within ecosystems. These models help explain the balance between species under natural conditions, but the inclusion of factors like the Allee effect and intraspecific competition adds complexity and realism to these interactions, enhancing our ability to predict system behavior under stress. To detect early indicators of population collapse, this study investigates the intricate dynamics of a modified Leslie-Gower predator-prey model with both Allee effect and intraspecific competition. We analyze the existence and stability of equilibria, as well as bifurcation phenomena, including saddle-node bifurcations of codimension 2, Hopf bifurcations of codimension 2, and Bogdanov-Takens bifurcations of codimension at least 4. Detailed transitions between bifurcation curves–specifically saddle-node, Hopf, homoclinic, and limit cycle bifurcations–are also examined. We observe a novel transition phenomenon, where a system jumps from saddle-node bifurcation to homoclinic and limit cycle bifurcations. This suggests that burst oscillations may serve as an early warning of system collapse rather than simply a tipping point. Our findings indicate that moderate levels of intraspecific competition or Allee effect support coexistence of both populations, while excessive levels may destabilize the entire biological system, leading to collapse. These insights offer valuable implications for ecological management and the early detection of risks in population dynamics.