Abstract
Physical systems described by deterministic differential equations represent idealized situations since they ignore stochastic effects. In the context of biomathematical modeling, we distinguish between environmental or extrinsic noise and demographic or intrinsic noise, for which it is assumed that the variation over time is due to demographic variation of two or more interacting populations (birth, death, immigration, and emigration). The modeling and simulation of demographic noise as a stochastic process affecting units of populations involved in the model is well known in the literature, resulting in discrete stochastic systems or, when the population sizes are large, in continuous stochastic ordinary differential equations and, if noise is ignored, in continuous ordinary differential equation models. The inverse process, i.e., inferring the effects of demographic noise on a natural system described by a set of ordinary differential equations, is still an issue to be addressed. With this paper, we provide a technique to model and simulate demographic noise going backward from a deterministic continuous differential system to its underlying discrete stochastic process, based on the framework of chemical kinetics, since demographic noise is nothing but the biological or ecological counterpart of intrinsic noise in genetic regulation. Our method can, thus, be applied to ordinary differential systems describing any kind of phenomena when intrinsic noise is of interest.
Highlights
Physical systems are usually modeled by differential equations, either ordinary (ODEs) or delay (DDEs) or partial (PDEs)
Our technique is based on the theory developed in the field of chemical kinetics, according to which an ODE model can be seen as the deterministic counterpart of the stochastic differential system arising from the ∆t → 0 limit of the discrete Markov process modeling the interaction between species, assuming them to be expressed by number of units instead of by concentration of populations
We provided a method to offer insights into deterministic differential models when intrinsic noise is of interest
Summary
Physical systems are usually modeled by differential equations, either ordinary (ODEs) or delay (DDEs) or partial (PDEs). These models represent idealized situations, as they ignore stochastic effects. By incorporating random elements in a differential equation model, a system of stochastic differential equations (SDEs), either stochastic ordinary (SODEs) or stochastic delay (SDDEs) or stochastic partial differential equations (SPDEs), respectively, arises. Stochastic models arise if we assume that a physical system operates in a noisy environment or if we wish to study the noisy behavior in the systems themselves, for example the intrinsic variability of interaction between individuals of two or more competing species. External noise can arise from random fluctuations of one or more model parameters around some given mean value or Mathematics 2019, 7, 1204; doi:10.3390/math7121204 www.mdpi.com/journal/mathematics
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