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

5 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 (US)
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

ASJC Scopus subject areas

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


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