A systematic approach to evaluate sustained stochastic oscillations

Although the populations of biological systems are inherently discrete and their dynamics are strongly stochastic, it is usual to consider their limiting behaviour for large environments in order to study some of their features. Such limiting behaviour is described as the solution of a set of ordinary differential equations, i.e., a continuous and deterministic trajectory. It will be shown that this trajectory does not always average correctly the system behaviour, such as sustained oscillations, in the neighbourhood of deterministic equilibrium points. In order to overcome this mismatch, an alternative set of differential equations based on spherical coordinates is proposed. This set of equations can be used to easily compute the average amplitude and frequency of stochastic oscillations