About deterministic extinction in ratio-dependent predator-prey models

Ratio-dependent predator-prey models set up a challenging issue regarding t heir 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 comm on ratio-dependent model and demonstrate that this equilibrium can be eithe r a saddle point or an attractor for certain trajectories. This fact has im portant implications concerning the global behavior of the model, for examp le 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 bas in of attraction in phase space, even when there exists a nontrivial stable or unstable equilibrium. Therefore, these models have no pathological dyna mics on the axes and at the origin, contrary to what has been stated by som e authors. Finally, we relate these findings to some published empirical re sults. (C) 1999 Society for Mathematical Biology.