About Deterministic Extinction in Ratio-dependent Predator–Prey Models

Ratio-dependent predator–prey models set up a challenging issue regarding their dynamics near the origin. This is due to the fact that such models are undefined at (0, 0). We study the analytical behavior at (0, 0) for a common ratio-dependent model and demonstrate that this equilibrium can be either a saddle point or an attractor for certain trajectories. This fact has important implications concerning the global behavior of the model, for example regarding the existence of stable limit cycles. Then, we prove formally, for a general class of ratio-dependent models, that (0, 0) has its own basin of attraction in phase space, even when there exists a nontrivial stable or unstable equilibrium. Therefore, these models have no pathological dynamics on the axes and at the origin, contrary to what has been stated by some authors. Finally, we relate these findings to some published empirical results.