Dynamics of a predator–prey model with Allee effect on prey and ratio-dependent functional response

Abstract

We analyze a ratio-dependent predator–prey model with Allee effect on the prey by making a parametric analysis of the stability properties of the dynamics of the system in which the functional response is a function of the ratio of prey to predator abundance. An important mathematical feature of these type of models is that while the functional response is undefined at the origin, the origin is singular equilibrium. We present the different types of system behaviors for different parameter values, showing the existence of separatrix curves in the phase plane determining that the long-term system's dynamic is dependent on the initial conditions. The model is studied analytically as well as numerically, including stability and bifurcation analysis. We also discuss the biological relevance of the model regarding both coexistence (conservation) and extinction (biological control) issues.