How generalist predation shapes a predator-prey model: A bifurcation analysis

The sigmoidal Holling III functional response is particularly suited for generalist predators, which exhibit 'prey-switching' behavior at low prey densities. In this paper, we consider a Leslie-type predator-prey model incorporating generalist predation and Holling III functional response. We first establish the existence of a nilpotent cusp or focus of codimension up to 3, or a weak focus of order at most 3. As parameters vary, the system can undergo a nilpotent cusp or focus bifurcation of codimension 3, or a Hopf bifurcation of codimension 2. Moreover, sufficient conditions for global asymptotic stability of the unique positive equilibrium are derived. Our results demonstrate that the Holling III functional response not only prompts the coexistence of both populations, but also induces richer bifurcation phenomena and dynamics, such as tristability, or three limit cycles. Furthermore, our results highlight the crucial role of generalist predators and their 'prey switching' behavior in regulating intricate dynamics and bifurcation thresholds in ecological systems.