Dynamics of adding variable prey refuge and an Allee effect to a predator–prey model

Prey refuge from predators can play an important role in stabilising an ecological system by reducing interactions between species, while Allee effects can arise from a range of biological phenomena, such as anti-predator vigilance, genetic trends and feeding deficiencies. We develop a predator–prey model that combines these phenomena, considering variable prey refuge with additive Allee effect on the prey species, with a Holling type II response function for the prey growth function. We use the predator and prey nullclines to determine the existence and stability of interior equilibria. We also investigate all possible local and global bifurcations that the system could undergo, showing that prey refuge and a strong Allee effect can lead to saddle-node bifurcations, Hopf bifurcations or Bogdanov–Takens bifurcation. We have investigated the appearance of Hopf bifurcations in a neighborhood of the unique interior equilibrium point of the dynamical system. The rich behaviour of the dynamics suggests that both prey refuge and a strong Allee affect are important factors in ecological complexity.