Spatiotemporal dynamics of Leslie–Gower predator–prey model with Allee effect on both populations

Abstract

In the present paper, we consider a reaction–diffusion system, based on the Leslie–Gower predator–prey model with Allee effect in both the predator and prey populations. The model is a generalization of the case where the individual population is subject to the Allee effect and it is flexible such that one can easily obtain the scenario with the Allee effect only on the prey population or, predator population. The non-spatial model was analyzed by using the positivity of solutions, uniformly boundedness and local stability of the interior equilibrium. The sufficient conditions for the instability with zero-flux boundary conditions are obtained for the spatial model. We have derived the analytical conditions for Hopf and Turing bifurcation on the spatial domain. We also observe that the system exhibits complex dynamics like Bogdanov–Takens (BT) bifurcation and generalized Hopf (GH) bifurcation. Our numerical simulations reveal that depending on the strength of Allee effects, the model dynamics exhibit stripes, spots, cold–hot spots–stripes coexistence and cold–hot spots patterns. We obtain very interesting characteristics of the reaction–diffusion model which reflects the fact that the system can develop patterns both inside and outside of the Turing parameter domain. Our study may enrich the field of the Allee effect and will help us to acquire a better understanding of the predator–prey interaction in a real environment.