Continuous-time predator-prey systems with Allee effects in the prey

A continuous-time predator-prey model without spatial variation is proposed to study the impact of Allee effects upon the predator-prey interaction when the prey population is subject to Allee effects. The system exhibits a threshold below which both populations become extinct. The model undergoes a Hopf bifurcation when the unique interior steady state loses its stability. We also examine the notion of Turing instability of the homogeneous interior steady state when both populations can disperse randomly. We prove that the solutions are square integrable and the system does not exhibit Turing instability. It is demonstrated via numerical simulations that the system may exhibit spatial nonhomogeneous steady states.