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 \( d \mathbf {y} = \varvec{\mu } \, dt + G \, d \mathbf {W},\) where \(\mathbf {y}\) is the population vector of random variables, \(\varvec{\mu }\) 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.

经常使用具有连续时间马尔可夫链（CTMC）模型的离散随机变量来研究具有人口变异性的种群动态。通过使用基于CTMC模型中反应速率的标准方法，可以通过直接方式得出使用连续随机变量的CTMC模型的近似值。这导致了一个Itô随机微分方程（SDE）系统，其形式通常为\（d \ mathbf {y} = \ varvec {\ mu} \，dt + G \，d \ mathbf {W}，\）其中\（\ mathbf {y} \）是随机变量的总体向量，\（\ varvec {\ mu} \）是漂移向量，而G是扩散矩阵。在某些问题中，派生的SDE模型可能并非始终具有实值或非负解。对于此类问题，可以宣布SDE模型不可行。在这项研究中，新的SDE系统是由实值解决方案和非负解决方案得出的。为了推导实值SDE模型，假定反应速率在所有时间内均为非负值，负反应率分配为零。这种生物学上的现实假设导致了实值SDE种群模型的推导。但是，对于实值SDE模型，仍然可能出现较小但为负的值。这是由于人口数量接近零时某些与问题相关的扩散系数的大小所致。当总体大小接近零时，对扩散系数进行轻微修改可确保实值SDE模型具有非负解，而当大小不接近零时仍可保持SDE模型的完整性。研究了几个人口动态问题，以说明该方法。