Compatible basal area models for live and dying trees using diffusion processes

Abstract

Models that predict a forest stand’s evolution are essential for developing plans for sustainable management. A simple mathematical framework was developed that considers the individual tree and stand basal area under random resource competition and is based on two assumptions: (1) a sigmoid-type stochastic process governs tree and stand basal area dynamics of living and dying trees, and (2) the total area that a tree may potentially occupy determines the number of trees per hectare. The most effective method to satisfy these requirements is formalizing each tree diameter and potentially occupied area using Gompertz-type stochastic differential equations governed by fixed and mixed-effect parameters. Data from permanent experimental plots from long-term Lithuania experiments were used to construct the tree and stand basal area models. The new models were relatively unbiased for live trees of all species, including silver birch (Betula pendula Roth) and downy birch (Betula pubescens Ehrh.), [spruce (Picea abies), and pine (Pinus sylvestris)]. Less reliable predictions were made for the basal area of dying trees. Pines gave the highest accuracy prediction of mean basal area among all live trees. The mean basal area prediction for all dying trees was lower than that for live trees. Among all species, pine also had the best average basal area prediction accuracy for live trees. Newly developed basal area growth and yield models can be recommended despite their complex formulation and implementation challenges, particularly in situations when data is scarce. This is because the newly observed plot provides sufficient information to calibrate random effects.

Keywords

Basal area / Occupied area / Stochastic process / Probability distribution

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