Modeling Demographic Noise in Discrete and Continuous Settings

Deterministic differential equations describe dynamical systems in idealized states, neglecting any random influences. Within biomathematical modeling, incorporating stochasticity requires a clear distinction between environmental (extrinsic) noise and demographic (intrinsic) noise. The latter framework assumes that temporal fluctuations arise strictly from the demographic dynamics of interacting populations rather than environmental variability. The literature thoroughly documents how demographic noise can be modeled and simulated as a stochastic process acting on individual members of a population, which yields discrete stochastic systems. For large population sizes, these discrete processes approximate continuous ones, leading to stochastic differential equations (SODEs). If random effects are omitted, these SODEs simplify back to standard ordinary differential equations (ODEs). Conversely, deducing how demographic noise impacts a natural system previously modeled by ODEs represents a major challenge. In this paper, we present an initial comparison of two distinct methodologies for reconstructing demographic noise by working backward from a deterministic, continuous differential system to its discrete stochastic process: the traditional Allen’s method and the backward approach recently introduced by Carletti and Banerjee (2019).