### Abstract

Population dynamics with demographic variability is frequently studied using discrete random variables with continuous-time Markov chain (CTMC) models. An approximation of a CTMC model using continuous random variables can be derived in a straightforward manner by applying standard methods based on the reaction rates in the CTMC model. This leads to a system of Itô stochastic differential equations (SDEs) which generally have the form dy=μdt+GdW, where y is the population vector of random variables, μ is the drift vector, and G is the diffusion matrix. In some problems, the derived SDE model may not have real-valued or nonnegative solutions for all time. For such problems, the SDE model may be declared infeasible. In this investigation, new systems of SDEs are derived with real-valued solutions and with nonnegative solutions. To derive real-valued SDE models, reaction rates are assumed to be nonnegative for all time with negative reaction rates assigned probability zero. This biologically realistic assumption leads to the derivation of real-valued SDE population models. However, small but negative values may still arise for a real-valued SDE model. This is due to the magnitudes of certain problem-dependent diffusion coefficients when population sizes are near zero. A slight modification of the diffusion coefficients when population sizes are near zero ensures that a real-valued SDE model has a nonnegative solution, yet maintains the integrity of the SDE model when sizes are not near zero. Several population dynamic problems are examined to illustrate the methodology.

Original language | English (US) |
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Pages (from-to) | 487-515 |

Number of pages | 29 |

Journal | Journal Of Mathematical Biology |

Volume | 81 |

Issue number | 2 |

DOIs | |

State | Published - Aug 1 2020 |

### Keywords

- Demographic variability
- Population dynamics
- Stochastic differential equation

### ASJC Scopus subject areas

- Modeling and Simulation
- Agricultural and Biological Sciences (miscellaneous)
- Applied Mathematics

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## Cite this

*Journal Of Mathematical Biology*,

*81*(2), 487-515. https://doi.org/10.1007/s00285-020-01516-8