Modeling Number of Trees per Hectare Dynamics for Uneven-Aged, Mixed-Species Stands Using the Copula Approach

For the monitoring and management of forest resources, the main index is the stand volume, which is determined on the basis of the tree diameter, height, and number of trees per hectare of three-dimensional distribution. The development of trees in the forest stand is dynamic and is driven by random phenomena. In this study, the tree diameter, the potentially available area, and the height are described by the mixed-effect parameters of the Gompertz-type diffusion process. A normal copula function is used to connect a three-dimensional distribution to its one-dimensional margins. The newly developed model was illustrated using empirical data from 53 permanent experimental plots (measured for seven cycles), which were characterized as follows: pine forests (Pinus sylvestris), 63.8%; spruce (Picea abies), 30.2%; silver birch (Betula pendula Roth and Betula pubescens Ehrh.), 5.8%; and others, 0.2%. An analysis of the tree diameter and height of growth, including current and mean increments and inflection points, is presented. The models for the change in the number of trees per hectare with age are presented on the basis of the probabilistic density functions of the solutions of stochastic differential equations and the copula function. The dynamics of the number of trees per hectare are visualized graphically, and the goodness of fit of the newly developed models is evaluated using standard statistical measures.