A Bayesian statistical evaluation of the competition indices used in eucalyptus tree growth modelling

Abstract

Eucalyptus is the most frequently planted genus tree in the world which is grown in more than 100 different countries. It is a subfamily of the Myrtaceae family with approximately 800 species of flowering evergreen trees and bushes. Since it offers wood for applications such as manufacturing pulp and solid wood, eucalyptus tree farming is especially significant in South Africa, where it is one of the most important sources of income. The study aims to evaluate the individual tree growth as well as competition present in eucalyptus plantations measured as a function of the growth rate during a particular growth period. The study used confidential secondary data, which were collected from a eucalyptus grandis cross with eucalyptus urophylla seedlings planted in the Kwa-Zulu Natal midlands in South Africa. The data were collected from the year 2000 to 2010 by the Institute for Commercial Forestry Research (ICFR) with trial number W184/03. For this study, one plot was randomly selected from all the plots planted for trail number W184/03. This randomly selected plot was plot 9 with a specified planting density of 1959 stems planted per hectare. A re-purposed Bayesian mixed effects variance component model, known as the Sire-Model in animal breeding problems, was used to determine the marginal posterior distributions of unknown parameters and provided the estimates. The re-purposing and subsequent use of the Bayesian Sire Model in individual tree growth modelling, was originally proposed by Kepe, Little and Hugo during the 61st Annual Conference of the South African Statistical Association, held from 27 – 29 November 2019 at the Nelson Mandela University in Port Elizabeth, South Africa (Programme and Abstracts, SASA 2019). Estimated tree growth indices were determined and used to make probability statements in order to rank the individually selected trees based on the amount of growth observed. A tree growth index is a measurement of a tree's average growth performance in relation to the average growth performance of all trees on the same plot. For this study specifically, the growth index was not calculated in relation to all the trees on the same plot, it was calculated in relation to the trees in the selected buffer. The marginal posterior densities were observed using a sampling-based approach known as the Gibbs sampler. MATLAB R2022b was used to obtain the results. Since the DBH is taken at a standard height of 1.37 meters above the ground and is therefore constant across measurements. The study used descriptive statistics analysis using the tree DBH to determine average growth instead of using the circumference of the trees. The findings of a descriptive statistics revealed that plot 9 had maximum DBH of 23.4 cm. According to literature, competition starts to set in after canopy closure (Morley et al., 2008 ). By considering the previously mentioned growth curve, it can be seen that after growth periods 3 (after year 3), the gradient of the growth curve reduced, suggesting that canopy closure took place during, or just after, period 3. The marginal posterior densities of four trees namely tree 182, tree 184, tree 214, and tree 216 for the variance components and random effects were estimated using the Gibbs Sampler where competition between the trees was assumed, as well as for the case where it was assumed that no competition takes place. An identity matrix was utilised in Gibbs sampling when it was assumed that there is no competition between the trees. A distance independent competition index called Lorimer (1983), was used to generate a matrix that was used in Gibbs sampling when it was assumed that there is competition between the trees. Results assuming no competition, revealed that the estimated marginal posterior densities of the error variance and tree variance, were slightly positively skewed and severely positively skewed, respectively. The estimated marginal posterior density for the growth index for trees 182, 184, 214 and 216 were symmetrical and all the equal tails credibility intervals contained zero. This indicated that there was no significant difference in the average growth of these selected trees from plot 9 with a density of 1959. However, the marginal posterior densities of the fixed effects indicated that there was a significant difference in the average growth rates of the selected trees from plot 9 since their equal tails credibility intervals did not contain zero. This therefore indicated that the specific treatment applied, had a significant effect. Results when competition was assumed, revealed that the estimated marginal posterior densities of the error variance as well as tree variance, were again positively and severely positively skewed. The estimated marginal posterior density for the growth indices for tree 182, tree 184, tree 214 and tree 216 were also symmetrical and, similar to the case when no competition was assumed, their equal tails credibility intervals also contained zero. However, when competition between the trees was considered, the growth indices for tree 182, tree 184, tree 214 and tree 216 however appeared to be lower than the case when no competition was assumed. Given that the results were based on a distance-independent competition index, it is advised that an investigation be conducted using a distance-dependent index as well.