Oscillations in two-species models: tying the stochastic and deterministic approaches

We analyze general two-species stochastic models, of the kind generally used for the study of population dynamics. We show that the conditions for the stochastic (microscopic) model to display approximate sustained oscillatory behavior are governed by the parameters of the corresponding deterministic (macroscopic) model. We provide a quantitative criterion for the quality of the stochastic oscillation, using a dimensionless parameter that depends only on the deterministic model. When this parameter is small, the oscillations are clear, and the frequencies of the stochastic and deterministic oscillations are close, for all stochastic models compatible with the same deterministic one. On the other hand, when it is large, the oscillations cannot be distinguished from a noise.