On real-valued SDE and nonnegative-valued SDE population models with demographic variability

E. J. Allen, L. J.S. Allen, H. L. Smith

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


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 languageEnglish
Pages (from-to)487-515
Number of pages29
JournalJournal of Mathematical Biology
Issue number2
StatePublished - Aug 1 2020


  • Demographic variability
  • Population dynamics
  • Stochastic differential equation


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