The stochastic Lotka-Volterra predator-prey model: it keeps cycling on!

The marginal stability of the oscillations exhibited by the Lotka-Volterra rate equations has been blamed for the apparent failure of stochastic predator-prey dynamics to replicate a stationary regime of stable oscillations; either of the prey or predator populations typically go extinct after a finite time. I will show that a simple modification of the master equation which adequately accounts for the conservation law underlying the rate equations is sufficient to prevent extinction. In the large system-size limit, the observed regime of oscillations is distributed according to the canonical equilibrium distribution of the thermostatted rate equations. Applications to standard time series arising in population ecology will be discussed.