Dynamics of a predator-prey system with foraging facilitation and group defense

Foraging facilitation and group defense are widespread phenomena in population ecosystem, which promote and prevent the predation in opposite processes respectively. In this work, a predator–prey system with foraging facilitation among predators and group defense in prey is proposed and investigated. To demonstrate the impact of both the foraging facilitation and the group defense on the system dynamics, the analysis primarily focuses on the qualitative properties of equilibria and possible bifurcations that the system may exhibit. It is shown that there are various non-hyperbolic equilibria for various parameter values such as the saddle–node, the weak focus of multiplicity at most three and the nilpotent cusp of codimension at most two as well as the system exhibits numerous kinds of bifurcation phenomena as the parameter values vary including the saddle–node bifurcation, the transcritical bifurcation, the pitchfork bifurcation, the Hopf bifurcation, the homoclinic bifurcation and the Bogdanov–Takens bifurcation. The results reveal that both foraging facilitation and group defense mechanisms play an important role in enriching the dynamics of predator–prey system and promoting the balance and diversity of ecosystem.