  In this paper, we analyse a recently proposed predator-prey model with ratio
dependence and Holling type III functional response, with particular emphasis
on the dynamics close to extinction. By using Briot-Bouquet transformation we
transform the model into a system, where the extinction steady state is
represented by up to three distinct steady states, whose existence is
determined by the values of appropriate Lambert W functions. We investigate how
stability of extinction and coexistence steady states is affected by the rate
of predation, predator fecundity, and the parameter characterising the strength
of functional response. The results suggest that the extinction steady state
can be stable for sufficiently high predation rate and for sufficiently small
predator fecundity. Moreover, in certain parameter regimes, a stable extinction
steady state can coexist with a stable prey-only equilibrium or with a stable
coexistence equilibrium, and it is rather the initial conditions that determine
whether prey and predator populations will be maintained at some steady level,
or both of them will become extinct. Another possibility is for coexistence
steady state to be unstable, in which case sustained periodic oscillations
around it are observed. Numerical simulations are performed to illustrate the
behaviour for all dynamical regimes, and in each case a corresponding phase
plane of the transformed system is presented to show a correspondence with
stable and unstable extinction steady state.
