Modelling tree height growth of nothofagus forests in south-central Chile: merging differential equatións and mixed-effects models

Abstract

Growth models are important both in forest management and forest dynamics studies. As such, forest growth modelling have been widely studied. Yet past efforts to model forest growth have focused on the use of equations of either cumulative growth or increment in fixed time periods, and little research has been devoted to the use of differential equations. These mathematical equations are particularly suitable for growth modelling. Tree growth data are hierarchically structured and temporally correlated, challenging traditional statistical assumptions, and must be addressed when building growth models. Mixed-effects models offers a statistical framework for dealing with the hierarchical and temporal nature of tree growth data when fitting growth models. However, research on differential equations being fitted in a mixed-effects model framework is scarce. Chile has a tradition in forest management of forestry plantations of exotic species, where silvicultural research and quantitative models have been developed for these species. Yet, for the Nothofagus obliqua-N. alpina-N. dombeyi forests, which are the most commercially important natural forest type in south-central Chile, mostly botanical and silvicultural research has been conducted, but there is a lack of quantitative tools that support management decision and research for them. This dissertation is an attempt to bring together the use of differential equations and mixed-effects models when building tree growth models. We focus on tree height growth modelling by using the largest available stem analysis data of roble-raul´ı-coig¨ue trees in south-central Chile. Pseudotsuga menziesii data are also used in order to broaden our findings. The objectives of the research were: (a) to assess a stochastic algorithm for reconstructing tree height growth with stem analysis data, (b) to evaluate the suitability of using breast-height age in tree growth models, and (c) to model site productivity of Nothofagus forests in south-central Chile. We found that predictions from a same height growth model fitted with a generated dataset obtained with both a well-known and a proposed stem analysis algorithm are statistically equivalent to the observed height growth, but being the latter algorithm simpler and stochastic. The use of breast-height age was proved to be a better variable for modelling tree height growth than totalage. Both statistical and biological evidence corroborate our findings. We tested both the stem analysis algorithm and the use of breast-height age with data of broadleaved tree Nothofagus species (two of them deciduous, and the other evergreen) growing in Chile, as well as for the coniferous Douglas-fir growing in the Pacific Northwest of USA. We proposed a new type of site index: tree height growth rate at a reference-height. We explain it by developing the autonomous differential equation form from a well-know growth-rate model. This new type of index has the advantages of being applicable to uneven-aged stands, a biometrical device not previously formulated. Finally, we developed a height growth model for dominant trees of N. alpina growing in a region spanning more than 400 km of longitude in south-central Chile. The resulting model of productivity is sensitive to topographic, habitat type, and climate variables.