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![Growth definitions for the state variable y obtained from function f. Growth curve (blue), state variable at time 0 (f(t0)) and time 1 (f(t0)), increment (Δy) between t0 and t1, and growth-rate or derivative (dy/dt) at t0 (f′(t0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f^{\prime }(t_{0})$\end{document}) and t1 (f′(t1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f^{\prime }(t_{1})$\end{document}). The derivative, or instantaneous growth-rate, is the slope of the tangent lines (black) at points t0 and t1](profile/Christian-Salas-Eljatib/publication/357163700/figure/fig1/AS:11431281119691015@1676175666249/Growth-definitions-for-the-state-variable-y-obtained-from-function-f-Growth-curve.png)
Growth definitions for the state variable y obtained from function f. Growth curve (blue), state variable at time 0 (f(t0)) and time 1 (f(t0)), increment (Δy) between t0 and t1, and growth-rate or derivative (dy/dt) at t0 (f′(t0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f^{\prime }(t_{0})$\end{document}) and t1 (f′(t1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f^{\prime }(t_{1})$\end{document}). The derivative, or instantaneous growth-rate, is the slope of the tangent lines (black) at points t0 and t1
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Purpose of Review
Growth equations have been widely used in forest research, commonly to assess ecosystem-level behavior and forest management. Nevertheless, the large number of growth equations has obscured the growth-rate behavior of each of these equations and several different terms for referring to common phenomena. This review presents a unif...
Citations
... Secondly, the potential biological interpretation of the parameters and the models providing accurate predictions for the H-D relationships were ensured. Each model presented in Table 2 can be expressed in a more generalized format as follows, as described in Equation 1 (Salas-Eljatib et al., 2021) ( ) ...
Tree attributes, such as height (H) and diameter at breast height (D), are essential for predicting forest growth, evaluating stand characteristics and developing yield models for sustainable forest management. Measuring tree H is particularly challenging in uneven-aged forests compared to D. To overcome these difficulties, the development of updated and reliable H-D models is crucial. This study aimed to develop robust H-D models for Larix gmelinii forest by incorporating stand variables. The dataset consisted of 7,069 Larix gmelinii trees sampled from 96 plots at Northeast China, encompassing a wide range of stand densities, age classes, and site conditions. Fifteen widely recognized nonlinear functions were assessed to model the H-D relationship effectively. Model performance was assessed using root mean square error (RMSE), mean absolute error (MAE), and the coefficient of determination (R2). Results identified the Ratkowsky model (M8) as the best performer, achieving the highest R2 (0.74), the lowest RMSE (16.47%) and MAE (12.50%), at statistically significant regression coefficients (p < 0.05). Furthermore, M8 was modified into 5 generalized models (GMs) by adding stand-variables (i.e., mean height, mean diameter and volume and their combination), the results indicate that GM2 was the best model achieving R2 of 0.82% and RMSE of 13.7%. We employed generalized nonlinear mixed-effects modeling approach with both fixed and random effects to account for variations at the individual plot level, enhancing the predictive accuracy. The model explained 71% of variability with significant trends in the residuals. The model was calibrated using response calibration method, through EBLUP theory. Our findings suggest that incorporating stand-level variables representing plot-specific characteristics can further improve the fit of mixed- effects models. These advancements provide forest authorities with enhanced tools for supporting sustainable forest management.
... In addition, this statistical approach can deal with heteroskedasticity and autocorrelation of errors by allowing for different variance and covariance structures (Mehtätalo and Lappi, 2020;Pinheiro and Bates, 2000). The use of a nonlinear mixed-effects model requires a mathematical base model to represent the shape of any cumulative growth curve and the corresponding observed growth phases: acceleration, intermediate, and deceleration (Salas-Eljatib et al., 2021). For this purpose, a mathematical base model with a theoretical basis is more appropriate to better understand the complexity of plant growth (Burkhart and Tomé, 2012;Pretzsch, 2020). ...
Growth depensation, the variation of size with age exhibited by populations, is attributed to biological, ecological, and environmental factors, as well as autocorrelation. Several studies have focused on the study of diameter growth of tropical trees to obtain information on ontogenic traits and silvicultural metrics of interest for ecology and forest management. However, few studies have rigorously and adequately considered autocorre-lation as a primary factor contributing to growth depensation. The aim of this study was to investigate diameter growth in tree species from the Chocó biogeographic region. We used tree-ring data corresponding to 38 trees and 5 species. Our modeling approach included von Bertalanffy type equations to estimate diameter growth trajectories for each species using mixed effects models. ARIMA specifications were included in the residual terms to account for autocorrelation. The estimated parameters allowed us to calculate ontogenic traits and silvicultural metrics for each species. The results indicate that autocorrelation was a critical factor in growth depensation for all species studied, and was satisfactorily accounted for by the proposed modeling approach. Autocorrelation patterns on residuals showed a stochastic trend and were investigated by correlation structures of ARIMA(1,1,0) and ARIMA(2,1,0). Ontogenic traits and silvicultural metrics obtained for these species were biologically consistent, providing reliable and useful information to understand the population ecology of tropical trees and to inform management and conservation strategies of natural forests.
... Growth curves are expressed in various sigmoidlike models such as logistic, Chapman-Richards, and Gompertz functions, and they help predict the stem size at certain ages and characterize growth patterns [15][16][17]. Zeltiņš et al. [17] pointed out that growth curve patterns varied among clones in Picea abies. Based on the asymptote and rate parameters of growth curves obtained from the logistic function applied to growth curves, Nagamitsu et al. [15] found that Larix kaempferi trees originated from different provenances exhibited the stable radial growth patterns across different provenance test sites in Japan. ...
The objectives of the present study are to clarify the effect of macro- and micro-environment on the radial growth patterns and radial variation patterns of basic density in hinoki cypress ( Chamaecyparis obtusa (Sieb. et Zucc.) Endl.). We evaluated the radial variation patterns of cumulative annual ring width (as radial growth pattern) and basic density by modeling methods using hinoki cypress 36 families planted at two progeny test sites. In addition, narrow-sense heritability and correlation between sites for annual ring width and basic density were investigated. As the results of modeling for radial growth patterns, radial growth patterns slightly differed between sites. In addition, the stem diameter reaching the plateau might be varied among blocks in a site. On the other hand, radial variation of basic density was affected by genetic factors rather than blocks in the site. However, the radial growth rate may somewhat affect the radial variation of basic density. The heritability and correlation coefficients between sites in basic density were higher than those of annual ring width. Therefore, although radial growth in hinoki cypress varies by the effects of micro- and macro-environmental factors and has some influence on the radial variation of basic density, basic density is more strongly affected by genetic factors than by these influences, allowing for effective improvement for wood density by tree breeding program.
... With the development and progress of science and statistical technology, data Bayesian methods, machine learning, and deep learning [93] can be useful under various conditions. However, a greater emphasis on biological and ecological mechanisms is also required to meet the challenges of estimating the effects of climate change and human disturbance [94]. ...
Accurate estimates of tree height (H) are critical for forest productivity and carbon stock assessments. Based on an extensive dataset, we developed a set of generalized mixed-effects height–DBH (H–D) models in a typical natural mixed forest in Northeastern China, adding species functional traits to the H–D base model. Functional traits encompass diverse leaf economic spectrum features as well as maximum tree height and wood density, which characterize the ability of a plant to acquire resources and resist external disturbances. Beyond this, we defined expanded variables at different levels and combined them to form a new model, which provided satisfactory estimates. The results show that functional traits can significantly affect the H–D ratio and improve estimations of allometric relationships. Generalized mixed-effects models with multilevel combinations of expanded variables could improve the prediction accuracy of tree height. There was an 82.42% improvement in the accuracy of carbon stock estimates for the studied zone using our model predictions. This study introduces commonly used functional traits into the H–D model, providing an important reference for forest growth and harvest models.
... To avoid confusion with the forestry application of CMAI, we call this point "peak average carbon increment." As others have used periodic annual increment and CMAI to identify three "phases" of forest development (Assmann, 1970;Salas-Eljatib et al., 2021), we use these objectively defined transitions in carbon accumulation to identify four stages of forest maturation: (1) "early seral" from stand initiation to peak periodic carbon increment; (2) "young forest" from peak periodic carbon increment to peak average carbon increment; ...
... Douglas-fir/hemlock or Engelmann spruce/subalpine fir forest), even as they add biomass with stand age. Nevertheless, there are alternative functional forms akin to the sigmoidal shape of Chapman-Richards (e.g., Weibull, logistic, Gompertz) that warrant future evaluation (Salas-Eljatib et al., 2021). For example, a more flexible function could model the post-disturbance drop in total forest carbon that the Chapman-Richards function cannot. ...
Because of its importance to both carbon storage and biodiversity, old forest has regained currency as a focus of forest management and policy. However, absence of a systematic approach to classifying stages of forest development across all forest types precludes our understanding of the current distribution of the mature and old-growth forest estate. Here, we propose functional definitions of mature and old-growth forests consistent with theories of forest stand development and evaluate the implications for assessing their spatial distribution nationwide. Using plot data from a national forest inventory and assuming space-for-time substitution, we modeled forest carbon accumulation over time using saturating, non-linear growth models. We define the onset of old-growth characteristics as occurring at the age when the density of total forest carbon stored in live and dead biomass reaches 95% of the empirically derived maximum, and the mature forest stage as occurring between the peak average carbon increment and the age of onset of old growth. We fit models within unique forest type-groups and, where possible, accounted for differences in site productivity. Population-level estimates of the mature and old-growth forest estate were calculated using sample design-based estimators. Across forest type-groups, the age at onset of old growth varied from 34 to 577 years, and the onset of mature forest conditions ranged between 16 and 313 years. Within forest type-groups, the effect of site quality on the age at onset of mature and old-growth forest varied but generally supported our hypothesis that high site quality accelerates forest development and increases forest carbon storage in old forests. We classified 6.3% of current forested lands in the United States as old growth and almost one-third as mature. Of the current old-growth forest estate, approximately 46% is found on federal public lands, and 11% is currently in congressional reserves. We posit that continued improvements to modeling the dynamic process of forest development and integration with structural definitions of old growth will be needed to ensure targets for old-growth retention and development are achieved.
... Radial growth increments can be represented by a sigmoid model, such as the Gomperz function [11]. Based on the sigmoid model, the current annual increment (CAI), which is the difference in radial growth at the beginning and end of the year, can be estimated. ...
Promoting wood utilization from fast-growing tree species is one solution to address supply and demand issues relating to wood resources while sequestering carbon dioxide in large quantities. Information on the quality of wood from fast-growing tree species and its relationship with changes in stem size is essential for promoting the establishment of plantations and wood utilization of fast-growing tree species. To explore the relationship between the xylem maturation process and radial growth increments of stems in fast-growing tree species, we examined radial variations in annual ring widths and wood properties in Liriodendron tulipifera in Japan. The cambial ages at which current annual increment and mean annual increment values were greatest were 4.9 years and 7.4 years, respectively. Based on radial variations evaluated by mixed-effects modeling of wood properties, all properties increased or decreased near the pith before becoming stable towards the cambium. Changing ratios of multiple wood properties at 1-year intervals became stable after a cambial age of 9 years. These results point to an ecological strategy in L. tulipifera , in which there is a tradeoff between radial growth increments and wood properties. As part of this strategy, in response to competition among individual trees within a stand, the tree produces a large volume of xylem with lower physical and mechanical properties, allowing it to increase its volume faster than that of the surrounding trees. Subsequently, it produces xylem that is more stable, with greater physical and mechanical properties. This wood forms at a slower growth rate compared to the xylem that forms at the time of initial tree growth. Based on the ecological strategy adopted by L. tulipifera , wood that forms before a cambial age of 9 years can be used for utility applications, and wood that forms after a cambial age of 9 years can be used for structural applications.
... Rather, parameter instability in the asymptotic growth limit estimate is a greater issue for multi-species and multi-aged natural forests, as well as using the parameterized functions to model growth beyond the age range of the data. Further, others have concluded that the biological basis of growth functions have been overstated and users should focus instead on plausible growth functions that best meet their needs, as well as parameter estimation strategies 29 . ...
... Given our objectives of comparing aboveground carbon accumulation across plantation types, we followed this procedure and fixed b at 1 [effectively stating that AGC (t=0) = 0] and fixed m at 0.67. Fixing m at 0.67 is a common practice within the literature and produces the von Bertalanffy special case of the Chapman-Richards function, which is the original function that Richards generalized 19,29,61 . We therefore fixed m at 0.67; however, we also considered how alternative values of m affected our parameter estimates (see Supplementary Information). ...
Restoring forest cover is a key action for mitigating climate change. Although monoculture plantations dominate existing commitments to restore forest cover, we lack a synthetic view of how carbon accumulates in these systems. Here, we assemble a global database of 4756 field-plot measurements from monoculture plantations across all forested continents. With these data, we model carbon accumulation in aboveground live tree biomass and examine the biological, environmental, and human drivers that influence this growth. Our results identify four-fold variation in carbon accumulation rates across tree genera, plant functional types, and biomes, as well as the key mediators (e.g., genus of tree, endemism of species, prior land use) of variation in these rates. Our nonlinear growth models advance our understanding of carbon accumulation in forests relative to mean annual rates, particularly during the next few decades that are critical for mitigating climate change.
Nowadays, many forests are intensely exposed to the adverse effects of climate change such as drought. To increase the resilience of the forests against climate change, prescribing appropriate silvicultural treatments is of great importance. At this point, closely scrutinizing the simultaneous effects of competition and climate on the forests has become a very important issue. In the present study, a climate-based individual-tree diameter increment model based on the Weibull growth equation was developed using the mixed-effects framework. The data were collected from naturally established and managed Crimean pine (Pinus nigra subsp. pallasiana (Lamb.) Holmboe) stands located in three separate climate regions, ranging from southwest to north of Türkiye. The measurements were conducted in a total of 108 randomly selected sample plots from these regions. The results of the current study showed that there were significant differences in the diameter increment between climate regions, ranging from 13 to 30%. The growing season total precipitation (GSTP) and the warmest month maximum temperature (WaTMax) were identified as significant climate variables in explaining the variation in the diameter increment. In addition, GSTP had a stronger effect on the diameter increment than WaTMax. The quantitative analysis demonstrated that a 10% decline in GSTP together with a 1 °C increase in WaTMax resulted in a considerable reduction in the diameter increment, varying from 10 to 25% depending on the competition levels. The proposed climate-based diameter increment model may be useful for the forest managers and practitioners to determine optimal thinning cycle under various climate change scenarios.
The growth traits (stem diameter at 1.3 m above the ground and tree height), stress-wave velocity of the stems, and log characteristics (taper and dynamic Young’s modulus of logs) were examined for 54 trees from 18 half-sib families planted in a seedling seed orchard of the first-generation Neolamarckia macrophylla (11-year-old) in Wonogiri, Central Java, Indonesia. The mean values for stem diameter and tree height were 20.2 cm and 20.0 m, respectively. The stress-wave velocity of the stems was 3.76 km s－1. Meanwhile, the taper and dynamic Young’s modulus of logs were 0.57 cm m－1 and 8.13 GPa, respectively. The heritability values of each trait were 0.412, 0.365, 0.101, <0.001, and 0.092 for the stem diameter, tree height, stress-wave velocity of stems, taper of logs, and dynamic Young’s modulus of logs, respectively, suggesting that the improvement of all traits is possible for the next generation, with the exception of the log taper. The 18 half-sib families could be classified into three groups for different potential uses based on the principal component analysis and cluster analysis results.
