Hopf bifurcation in a delayed reaction diffusion predator-prey model with weak Allee effect on prey and fear effect on predator

In this work, a Leslie-Gower model with a weak Allee effect on the prey and a fear effect on the predator is proposed. By using qualitative analyses, the local stability of the coexisting equilibrium and the existence of Turing instable are discussed. By analyzing the distribution of eigenvalues, the existence of a Hopf bifurcation is studied by using the gestation time delay as a bifurcation parameter. By utilizing the normal form method and the center manifold theorem, we calculate the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. We indicate that both the weak Allee effect on the prey and fear effect on the predator have an important impact on the dynamical behaviour of the new Leslie-Gower model. We also verify the obtained results by some numerical examples.