Hydra effect and global dynamics of predation with strong Allee effect in prey and intraspecific competition in predator

Abstract

We consider a predator-prey system with strong Allee effect in prey and intraspecific competition in predator. We divide the parameter space into several mutually exclusive regions, and for each region we give a rather complete description of the global dynamics including equilibrium stability, Hopf bifurcation and its directions, existence of a heteroclinic orbit loop, a homoclinic orbit loop and limit cycles. In comparison with systems without the Allee effect in prey and systems without the intraspecific competition in predator, we find that the Allee effect in prey and the over-exploitation of the prey by the predator may drive the extinction of two species, but the intraspecific competition in the predator can generate “hydra effect” that regulates the population size of both predator and prey by suppressing the over-exploitation of prey, improving the probability of survivals of two species.