Qualitative analysis of a predator–prey system with double Allee effect in prey

Abstract

Growing biological evidence from various ecosystems (Berec et al., 2007) suggests that the Allee effect generated by two or more mechanisms can act simultaneously on a single population. Surprisingly, prey–predator system incorporating multiple Allee effect is relatively poorly studied in literature. In this paper, we consider a ratio dependent prey–predator system with a double Allee effect in prey population growth. We have taken into account the case of the strong and the weak Allee effect separately and study the stability and complete bifurcation analysis. It is shown that the model exhibits the bi-stability and there exists separatrix curve(s) in the phase plane implying that dynamics of the system is very sensitive to the variation of the initial conditions. Generic normal forms at double zero singularity are derived by a rigorous mathematical analysis to show the different types of bifurcation of the proposed model system including Bogdanov–Takens bifurcation, saddle–node bifurcation curve, a Hopf bifurcation curve and a homoclinic bifurcation curve. The complete analysis of possible topological structures including elliptic, parabolic, or hyperbolic orbits, and any combination of them in a neighborhood of the complicated singular point (0,0) in the interior of the first quadrant is explored by using a blow up transformation. These structures have important implications for the global behavior of the model. Numerical simulation results are given to support our theoretical results. Finally, the paper concludes with a discussion of the ecological implications of our analytic and numerical findings.