Role of Allee effect on prey–predator model with component Allee effect for predator reproduction

Abstract

In the context of prey–predator interaction, Allee effect can have a significant impact and capture the complex dynamics in ecology. In this work, we modify the predator–prey model with component Allee effect for predator reproduction by incorporating the strong Allee effect in prey growth function. We explore the system dynamics in two aspects. Firstly, we study the system dynamics of the model without Allee effect through a comprehensive bifurcation structure and perform the sensitivity analysis of model parameters for fixed coexistence extensively; ii) we analyze the impact of Allee effect on the system dynamics. We determine the number of fixed coexistence points through graphical representation of non-trivial prey and predator nullclines. We study the stability analysis of the fixed coexistence point with the help of the graphical Jacobian method. Interestingly, we observe that initially, a low concentration of prey drives the system toward total extinction and the system will be settled to predator extinction for initially high prey concentration. This system behavior supplements the existence of bi-stability involving trivial and predator extinction equilibria independent of parametric conditions. The inclusion of the Allee effect enhances the stability behavior of the proposed model i.e. tetra stable equilibrium points are deduced. We demonstrate the system dynamics through co-dimension one and two bifurcations structure and also show possible phase portraits. Model with Allee effect generates all possible local and global bifurcations namely Hopf bifurcation, saddle–node bifurcation, B-T bifurcation, Bautin bifurcation and homoclinic bifurcation respectively. We observe that low predator reproduction growth rate provides oscillations with low prey densities and high predator reproduction growth rate results in oscillations with high prey densities. We investigate that the low impact of Allee always promotes the persistence of the coexistence. For a model with the Allee effect, we perform sensitivity analysis of model parameters for fixed coexistence points. We demonstrate results analytically and make them more comprehensive, we perform numerical simulation. Moreover, to show the vast applicability of our results, we compare it with the model without Allee effect.