Complex dynamics near extinction in a predator-prey model with ratio dependence and Holling type III functional response

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