The Relationships Between the Diameter Growth and Distribution Laws

Abstract

The processes of growth play an important role in different fields of science, such as biology, medicine, forestry, ecology, economics. Usually, in applied sciences the averaged trend kinetics is represented by means of logistic laws (Verhulst, Gompertz, Mitscherlich, von Bertalanffy, Richards etc.). We used a generalized stochastic logistic model for predicting the tree diameter distribution of forestry stands. The purpose of this paper was to develop a diameter probability density function for even-aged and uneven-aged stands using the stochastic logistic law of diameter's growth. The parameters of stochastic logistic growth law were estimated by the maximum likelihood procedure using a large dataset on permanent sample plots provided by Lithuanian National Forest Inventory. Subsequently, we numerically simulated the probability density function of diameter distribution for the Verhulst, Gompertz, Mitscherlich, von Bertalanffy, Richards stochastic growth laws. The exact solution (transition probability density function of diameter size) of the Fokker-Planck equation (the partial differential equation for evolving distribution of diameter size) was derived exclusively for the Gompertz stochastic growth law. The comparison of the goodness of fit among probability density functions was made by the normal probability plot and the p-value of the Kolmogorov-Smirnov and Cramer -von Mises tests. To model the diameter distribution, as an illustrative experience, is used a real data set from repeated measurements on permanent sample plots of pine stands in Dubrava district. The results are implemented in the symbolic computational language MAPLE.