Dynamics of predator-prey models with a strong Allee effect on the prey and predator-dependent trophic functions

The complex dynamics of a two-trophic chain are investigated. The chain is described by a general predator-prey system, in which the prey growth rate and the trophic interaction functions are defined only by some properties determining their shapes. To account for undercrowding phenomena, the prey growth function is assumed to model a strong Allee effect; to simulate the predator interference during the predation process, the trophic function is assumed predator-dependent. A stability analysis of the system is performed, using the predation efficiency and a measure of the predator interference as bifurcation parameters. The admissible scenarios are much richer than in the case of prey-dependent trophic functions, investigated in Buffoni et al. (2011). General conditions for the number of equilibria, for the existence and stability of extinction and coexistence equilibrium states are determined, and the bifurcations exhibited by the system are investigated. Numerical results illustrate the qualitative behaviours of the system, in particular the presence of limit cycles, of global bifurcations and of bistability situations.