This study proposes a general bivariate stochastic differential equation model of population growth which includes random forces governing the dynamics of the bivariate distribution of size variables. The dynamics of the bivariate probability density function of the size variables in a population are described by the mixed-effect parameters Vasicek, Gompertz, Bertalanffy, and the gamma-type bivariate stochastic differential equations (SDEs). The newly derived bivariate probability density function and its marginal univariate, as well as the conditional univariate function, can be applied for the modeling of population attributes such as the mean value, quantiles, and much more. The models presented here are the basis for further developments toward the tree diameter–height and height–diameter relationships for general purpose in forest management. The present study experimentally confirms the effectiveness of using bivariate SDEs to reconstruct diameter–height and height–diameter relationships by using measurements obtained from mountain pine tree (Pinus mugo Turra) species dataset in Lithuania.

中文翻译：

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树高-直径连接的可约化随机微分方程系统的构造

这项研究提出了人口增长的一般双变量随机微分方程模型，该模型包括控制大小变量双变量分布动态的随机力。人口中大小变量的双变量概率密度函数的动力学由混合效应参数Vasicek，Gompertz，Bertalanffy和伽马型双变量随机微分方程（SDE）描述。新推导的双变量概率密度函数及其边际单变量以及条件单变量函数可用于总体属性（例如平均值，分位数等）的建模。这里介绍的模型是进一步发展树木直径，高度和高度，直径之间关系的基础，以实现森林管理中的通用目的。立陶宛的Pinus mugo Turra）物种数据集。