Maximum Size-density Relationship Research Articles

Effects of species mixing on maximum size-density relationships in Chinese fir (Cunninghamia lanceolata (Lamb.) Hook.)-dominated mixed forests converted from even-aged pure stands.

Density management is a key silvicultural tool in management programs that enhances compositional and structural diversity and hence forest growth during the conversion of even-aged pure stands into mixed forests. To determine the optimum stand density, a model of maximum size-density relationships was employed to explore the relationship of the self-thinning trajectory with growth, species mixing, latitude, and site conditions during the transition of even-aged pure Chinese fir stands to Chinese fir-dominated mixed forests using stochastic frontier analysis. Data were obtained from a total of 591 permanent plots located in Fujian, Jiangxi, Zhejiang, and Anhui provinces in southern China. The results showed that (1) the slope of the maximum size-density relationship of Chinese fir-dominated mixed forests increased and plateaued over time; (2) the slope of the maximum size-density relationship of Chinese fir-dominated mixed forests did not deviate from Reineke's assumed universal slope of -1.605; and (3) mixing proportion had a positive effect on maximum size-density relationships, and latitude and site conditions had the opposite effect on maximum size-density relationships. Our findings will provide valuable guidance for the forest management of areas in which even-aged pure stands are being converted to mixed forests (i.e., when broadleaved tree species are planted after thinning to improve overall stand density and promote stand growth).

A Density Management Diagram for Pitch Pine to Illustrate Tradeoffs between Carbon and Wildfire Risk

Abstract Pitch pine (Pinus rigida Mill.) can be found across a broad range in eastern North America but assumes local dominance only on poor soils in the northeastern United States. Contemporary management goals in the Northeast for areas dominated by pitch pine are focused on noncommercial benefits of forests, such as carbon density, reduced wildfire risk, habitat for rare species, and water provisioning. We present a density management diagram that empirically articulates the size-density limits of even-aged pitch pine stands. Included in the diagram are wildfire risk and carbon density, which are inversely related for most stand sizes. Maximum possible aboveground live tree carbon begins to decline at a quadratic mean diameter greater than 9 in., while crown fire risk remains high along the size-density limit until a quadratic mean diameter above 12 in. is achieved. Study Implications: Modern silvicultural tools that illustrate forest stand conditions have not been developed for pitch pine, but this species occurs in a region with much public attention on forests. We develop and present a density management diagram to show the interplay of different social goals for the forest and how they relate to the maximum size-density relationship. Pitch pine stands with high levels of aboveground live carbon are at high risk of crown fire, particularly in the smaller size classes.

The Importance of Using Permanent Plots Data to Fit the Self-Thinning Line: An Example for Maritime Pine Stands in Portugal

Density-dependent mortality occurs in the evolution of even-aged populations when these approach crown closure age. This density-dependent mortality is regulated by the so-called “3/2 power law of self-thinning” that assumes a constant slope for the line relating the log of stand density with the log of the average tree size, the self-thinning line or maximum size–density relationship, MSDR. A good estimate of the self-thinning line is therefore an essential component to any forest growth model. Two concepts for the MSDR have emerged: (1) a static upper limit for the species; and (2) a dynamic self-thinning line influenced by several factors (e.g., management techniques, site quality and/or genetics). The objective of this study was to estimate a new static self-thinning line based on the quadratic mean diameter at breast height (Reineke’s self-thinning line) for the generalized use in maritime pine growth models in Portugal. Data from 41 observations obtained in nine long-term permanent experimental trials of maritime pine species were carefully selected from a data set of 186 plots as being under self-thinning. Two methods were used: OLS and mixed linear models. An exploratory analysis on the impact of each environmental variable on the slope and intercept of the self-thinning line led to the selection of a subset of environmental variables later used in an all possible regressions algorithm to find the subsets leading to the lowest values of Akaike information criterion (AIC). The OLS procedure showed that the differences between the plots could be explained by site index, by climate variables (e.g., evaporation or climatic indices) and the use of more than one covariable slightly improved the fit. Nevertheless, the best MSDR line fitted with mixed linear models (ln N = 12.97158 − 1.83926 ln dg) having the plot random effect in the intercept, largely outperformed the best OLS model and is therefore recommended for generalized use in forest growth models.

Aridity index and quantile regression influences on the maximum size-density relationship for coniferous and broad-leaved mixed forests

The maximum size-density relationship (MSDR) of mixed stands is of great significance for optimizing stand structural components and increasing stand productivity. However, model structure, climatic factors and parameter estimation methods affect the prediction accuracy of MSDR for multi-species mixed stands with complex structures, emphasizing the need to improve MSDR prediction methods. This study applied likelihood analysis to 333 sample plots of larch (Larix principis-rupprechtii Mayr.) and birch (Betula platyphylla Suk.) coniferous and broad-leaved mixed forests in Saihanba, Hebei Province, China to determine the structure of MSDR. Nonlinear least squares and nonlinear quantile regression were used to construct the MSDR of mixed stands, including the Martonne aridity index (M) which is a composite index that responds to temperature and precipitation. The results indicate that the optimal model form for predicting the MSDR of mixed stands is a nonlinear additive error structure. When considering M and using a quantile q = 0.95, the MSDR can accurately describe the variation in stand density of mixed larch-birch coniferous and broad-leaved forests.

Climate, site conditions, and stand characteristics influence maximum size-density relationships in Korean red pine (Pinus densiflora) and Mongolian oak (Quercus mongolica) stands, South Korea

This study examines the effects of climate, site index and structural diversity on maximum size-density relationships (MSDRs) for Korean red pine (Pinus densiflora Siebold & Zucc.) and Mongolian oak (Quercus mongolica Fisch. ex Ledeb.) which are widely distributed and abundant across South Korea. We used data collected between 2006 and 2015 from the Korean National Forest Inventory and stochastic frontier function regression to develop static self-thinning lines (STLs) for pure pine, pure oak, and pine-oak mixed-species stands. Climate variables, site condition (e.g., site index), and stand characteristics (e.g., structural diversity and pine proportion) were applied to the STLs to examine the factors affecting MSDRs and the effects of climate on MSDRs. Our results indicate that the slope and intercept values differed significantly between the three stand types, with the pine STL being steeper than other STLs. Degree days above 18 °C in spring (DD18_sp), degree days above 5 °C in spring (DD5_sp), and annual heat moisture index (AHM) mainly influenced the position of STLs. Increases in these climate variables (e.g., warmer and drier conditions) reduce maximum densities owing to increased evapotranspiration and water stress in stands. Increasing site index (SI) was associated with an upward shift in the position of the STL for pine, and the result shows that better site quality can improve carrying capacity and increase stand stocking. However, high stand stocking can encourage tree competition with increasing dryness and reduce maximum stocking. Higher stand structural diversity caused an upward shift in all STLs, and the effect of climate on the STLs of different structural diversities varied with stand type. Structurally diverse stands can stock more trees in their stands owing to efficient space use, especially with respect to shade tolerant species. Nonetheless, under drier conditions, trees in dense stands experience severe competition, and their carrying capacity decreases. Our results suggest that it is important to include stand density control in forest management plans to prepare for global warming in South Korea.

Site carrying capacity of Norway spruce and Scots pine stands has increased in Germany and northern Europe

The maximum size-density relationship describes site carrying capacity, i.e., the maximum number of trees of a given size that can be stocked per unit area (self-thinning line). We analysed whether the self-thinning lines of Norway spruce (Picea abies (L.) Karst.) and Scots pine (Pinus sylvestris L.) have remained unchanged over time in South Germany, Norway and Finland, i.e., over a wide climatic gradient from Central Europe up to the Arctic circle. The analyses are based on long-term growth and yield experiments measured on individual tree basis over several decades, the oldest experiments established during the early 20th century. The stochastic frontier analysis was used to analyse changes in the species-specific self-thinning lines. The results show that the self-thinning lines have shifted upwards over time in all the regions. Thus, currently stands sustain higher stand densities than in the past. The increase of the maximum density for a given average stem size was more pronounced for pine than for spruce, but similar in all studied geographical regions. In addition, increasing site index was associated with increasing site carrying capacity for spruce and pine in all regions. The results imply that environmental changes have altered site properties in similar fashion across the whole study region. In practical forestry, increased site carrying capacity will reduce mortality and loss of growing stock.

Rethinking maximum stand basal area and maximum SDI from the aspect of stand dynamics

The maximum stand basal area (BA) and maximum stand density index (SDI) are often used to express stand carrying capacity in forestry. But their definitions are inconsistent in the literature. The maximum stand BA in most previous studies is associated with an upper boundary for a population of stands and is commonly derived from maximum size-density relationships (MSDRs). The maximum stand BA implied by MSDRs, under the assumption that stands attain maximum BA while attaining maximum SDI, is non-decreasing and divergent. We defined the maximum stand BA and maximum SDI for individual stands based on their BA and SDI development behaviors. Based on the mathematical relationships among stand BA, SDI, thinning, and quadratic mean diameter trajectories and their observed full trajectories in 551 unthinned plots selected from six long-term loblolly pine experimental studies, we theoretically and empirically proved that stands would achieve their maximum BA only after reaching their maximum SDI. The general mean maximum stand BA and maximum SDI for loblolly pine in the southern US were 46.2 m2 ha−1 and 1002 trees ha−1, respectively, and both showed significant variation (30.2–61.7 m2 ha−1 and 600–1410 trees ha−1, respectively). With the information of site quality, initial density and silvicultural treatments, our developed models could estimate the magnitude and timing of maximum stand BA for loblolly pine stands. Insights into individual stand BA and SDI trajectories provide more information for developing site specific silvicultural prescriptions.

Potential climatic influence on maximum stand carrying capacity for 15 Mediterranean coniferous and broadleaf species

Climate change projections for the Mediterranean basin predict a continuous increase in extreme drought and heat episodes, which will affect forest dynamics, structure and composition. Understanding how climate influences the maximum size-density relationship (MSDR) is therefore critical to designing adaptive silvicultural guidelines based on the potential stand carrying capacity of tree species. With this aim, data from the Third Spanish National Forest Inventory (3NFI) and WorldClim databases were used to analyze climate-related variations of the maximum stand carrying capacity for 15 species from the Pinus, Fagus and Quercus genera. First, basic MSDR were fitted using linear quantile regression and observed size-density data from monospecific 3NFI plots. Reference values for maximum stocking, expressed in terms of the Maximum Stand Density Index (SDImax), were estimated by species. Then, climate-dependent MSDR models including 35 annual and seasonal climatic variables were fitted. The best climate-dependent models, based on the Akaike Information Criteria (AIC) index, were used to determine the climatic drivers affecting MSDR, to analyze general and species-specific patterns and to quantify the impact of climate on maximum stand carrying capacity. The results showed that all the selected climate-dependent models improved the goodness of fit over the basic models. Among the climatic variables, spring and summer maximum temperatures were found to be key drivers affecting MSDR for the species studied. A common trend was also found across species, linking warmer and drier conditions to smaller SDImax values. Based on projected climate scenarios, this suggests potential reductions in maximum stocking for these species. In this study, a new index was proposed, the Q index, for evaluating the impact of climate on maximum stand carrying capacity. Our findings highlight the importance of using specific climatic variables to better characterize how they affect MSDR. The models presented in this study will allow us to better explain interactions between climate and MSDR while also providing more precise estimates concerning maximum stocking for different Mediterranean coniferous and broadleaf tree species.

Evaluating maximum stand density and size–density relationships based on the Competition Density Rule in Korean pines and Japanese larch

Understanding the relationship between size and density is an important issue for stand density management in terms of self-thinning in even-aged stands. This study was conducted to determine the maximum stand density and evaluate the size–density relationship based on the Competition Density Rule (C-D rule) for Korean red pine, Korean white pine, and Japanese larch in Korea. The study area covered permanently monitored plots of even-aged plantations established for investigating and modeling long-term growth changes through silviculture. Data were applied using the first inventory within the permanent plots before implementing artificial thinning according to research design. To predict the maximum stem number by species, the C-D rule was applied, and the three-parameter method was shown to be better than the four-parameter method. According to the results based on the C-D rule with three parameters, stand density index by species was 1041 trees (Korean red pine, Pinus densiflora), 1089 trees (Korean white pine, P. koraiensis), and 1005 trees (Japanese larch, Larix kaempferi) in 25 cm of quadratic mean diameter. In the relationship between the maximum stem number and quadratic mean diameter with log–log scale, the intercept for Korean red pine, Korean white pine, and Japanese larch was 10.317, 10.901, and 11.331, and the slope by species was −1.047, −1.214, and −1.373, respectively. To analyze the size–density relationship, the sample plots were classified based on relative spacing and relative density. In regression analysis with log–log scale using the censored sample plot data, maximum size–density relationship was explained the most with quadratic mean diameter rather than mean stem volume and dominant height.

THE GROWING SPACE UTILIZATION OF MAIN TREE SPECIES IN NORTHERN TURKEY

ABSTRACT Relationships between tree size and density are important to define the growing space utilization in a stand. Although a universal slope for the maximum size-density relationships (MSDRs) has been previously suggested, recent research have highlighted that these relationships are species-specific. Oriental beech (Fagus orientalis Lipsky), Trojan fir (Abies nordmanniana subsp.equi-trojani), black pine (Pinus nigra Arnold) and Scots pine (Pinus sylvestris L.) represent different crown architecture, growth rate and shade tolerance; however, MSDRs have not been developed for these tree species in northern Turkey. In this study, average maximum density (DAM) slopes for these tree species were determined. Results suggested that MSDRs varied among the species, and that their slope differed from the universal slope of -1.605. The MSDRs described in this study are useful for managing stand density in natural stands of the species analyzed.

Maximum size-density relationship in bamboo forests: Case study of Phyllostachys pubescens forests in Japan

Bamboo forests are likely to be subjected to self-thinning due to their rapid growth rate. In fully stocked bamboo forests, the stand density (ρ) was suggested to be inversely proportional to the square of mean diameter at breast height (D) (ρ ∝ D−2), differing from the conventional power-law for trees, i.e., Reineke equation (ρ ∝ D−1.6). Nevertheless, the validity and mechanism of the inverse-square law remained unclear, despite its significance for managing bamboo forests. In this study, we derived an allometric model that predicts the slope of the maximum size-density relationship (MSDR) between D and ρ on double logarithmic coordinates based on well-known ecological laws. The model indicates that the slope of −2 for bamboo is theoretically valid, with the difference in the slope between bamboo and trees being caused by their differences in inner-culm (or inner-stem) structure. We also determined the MSDR empirically by compiling data of 415 Phyllostachys pubescens Mazel ex Houz. pure stands across Japan. The obtained MSDR describes the upper boundary of the D-ρ relationship well, with a slope of −1.996 that is very close to − 2. We further established a stand density management diagram of P. pubescens stands based on the MSDR. This diagram allows forest managers to regulate ρ depending on D as well as management purposes (production of bamboo shoot, bamboo charcoal and culm wood).

Intra- and inter-specific variation of the maximum size-density relationship along an aridity gradient in Iberian pinewoods

Abstract The diverse applications of the maximum size-density relationship (MSDR) in monospecific and mixed forests, such as in ecological and economic aspects, lead to continuous advances in our knowledge on this issue. One of the most recent advances was the inclusion of climatic variables in these studies, revealing the variation in MSDR depending on environmental conditions. However, the importance of climatic conditions on the intra- and inter-specific variation of MSDR is still poorly understood. The aim of this paper is to explore the dependence of MSDR on climatic conditions for the five principal pine species in the Iberian Peninsula (P. halepensis Mill., P. nigra Arn., P. pinea L., P. pinaster Ait., and P. sylvestris L.) and to analyse the importance of this dependence on the relative carrying capacities in mixed stands. Data from the Third Spanish National Forest Inventory (NFI) were used together with four simple climatic indices calculated from raster maps. Using a quantile regression, a MSDR basic model, relating the maximum number of trees and the quadratic mean diameter, was fitted to the data from the plots in monospecific stands. In a second step, the coefficients of basic model were parameterized as a function of climatic variables. The resulting climate-dependent model was also fitted to the plots. Competition equivalence coefficients (CEC) in mixed stands (i.e. ratio between maximum stand density indices of the two species) were calculated from the resulting models. The differences among species’ MSDRs confirm the inter-specific variability of maximum densities and the need for species-specific models. According to the Akaike information criteria, the climate-dependent models, and particularly those dependent on Martonne aridity index, were always better than basic models. Although the higher the aridity the lower the maximum stand density, the influence of climate on the MSDR also varies according to the species considered, this influence being more evident for P. pinaster and P. halepensis, which also display high ecological plasticity. The CEC derived from the basic model for pine-pine mixtures range from 1.10 to 1.70. However, when aridity is considered, these coefficients almost always decrease. Our results highlight the importance of considering environmental variables to better describe and compare the potential density in monospecific and mixed stands and therefore, the utility of species-specific climate dependent models for management decision support.

Effects of climate on maximum size-density relationships in Western Canadian trembling aspen stands

Abstract Maximum size-density relationships (MSDR) are examined in stands of trembling aspen (Populus tremuloides Michx.) in the boreal forest region of Alberta and Saskatchewan, Canada. Stochastic frontier function regression was used to develop the static self-thinning line, and a linear mixed-effects model was used to estimate the average dynamic self-thinning line from repeated measurements. Climate variables obtained from Climate WNA were included in our models. The static self-thinning line was steeper than Reineke’s slope (−1.605), while the average slope value of the dynamic self-thinning line was not different from −1.605. Increasing frost free period, summer heat moisture index, and mean coldest month temperature move the static self-thinning line downwards because these climate variables are linked to increases in evapotranspiration and drought exposure. For the average dynamic self-thinning line, degree days above 5 °C positively increases the intercept and summer heat moisture index has a negative effect on the intercept, but no effect on the slope for both climate variables. Results suggest that increasing summer dryness related to climate change may decrease the carrying capacity and productivity of aspen stands in portions of this region.

Climate influences on the maximum size-density relationship in Scots pine (Pinus sylvestris L.) and European beech (Fagus sylvatica L.) stands

The maximum size-density relationship (MSDR) reflects the boundary site occupancy and the self-thinning line for a given species, being a useful tool in forestry. Studies focusing on the MSDR often do not cover the whole distribution of the studied species, which results in different boundaries for a given species in different regions. A common MSDR is lacking for the increasingly demanded large-scale studies. However, this information is important where silvicultural responses must be prioritized among monospecific stands or where comparisons among maximum and relative stand densities between and within species are required.For the purposes of this study, we used data from 9911 sample plots located in Scots pine and European beech monospecific stands. Both of these species are of considerable importance and widely distributed throughout Europe. The data came from National or Regional Forest Inventories of five European countries (Austria, Germany, France, Spain and Poland) and therefore were distributed across a wide range of climatic conditions.The main aim of this study was to determine whether the MSDR of these species depends on environmental variables and to develop a MSDR model for each species that explain this variability along a climate gradient.The resulting models showed that both parameters of species boundary lines were climate-dependent, but that the pattern of variation differed between species. Hence, the higher the humidity, the steeper the MSDR (more negative exponent) and the higher the intercept for beech, while in the case of pine, the higher the humidity, the straighter the MSDR and the lower the intercept. According to these models, the stand density indices, for a reference diameter of 25cm, varied with the humidity in a different way for each species. Consequently, the ratio between the two species increases with humidity, although it also depends on stand diameter.These results are in accordance with the yield level theory and could contribute to the development of more precise silvicultural guidelines and growth models based on the self-thinning line. Moreover, they are of particular importance in the discussion of growth and the effects of mixing on mixed species stands.

Estimation of carrying capacity in loblolly pine (Pinus taeda L.)

For plant populations, carrying capacity is the maximum number or biomass of a species that can be sustained under finite site resources on a long-term basis. This study was aimed at estimating the carrying capacity in planted stands of loblolly pine. Maximum stand basal area (BA) that can be sustained over a long period of time can be regarded as a measure of carrying capacity. To quantify and project stand BA carrying capacity, one approach is to use the estimate from a fitted cumulative BA-age equation; another approach is to obtain BA estimates implied by maximum size-density relationships (MSDRs), denoted implied maximum stand BA. The efficacy of three diameter-based MSDR measures: Reineke’s self-thinning rule, competition-density rule and Nilson’s sparsity index, were evaluated. Estimates from these three MSDR measures were compared with estimates from the Chapman-Richards (C-R) equation fitted to the maximum stand BA observed on plots from spacing trials. The spacing trials, established in two physiographic regions (Piedmont and Coastal Plain) in the southeastern United States, and at two different scales (operational and miniature) were examined to provide an empirical basis for potential carrying capacity of loblolly pine.Results showed that the value of implied maximum stand BA converted from three diameter-based MSDR measures was similar to the maximum stand BA curve obtained from the C-R equation. All three MSDR measures provided similar estimates of stand BA carrying capacity, with Nilson’s sparsity index yielding the most stable and reliable estimate. The potential stand BA carrying capacity between the two physiographic regions was significantly different. The maximum stand BA associated with planting density developed similarly at the operational and miniature scales.

Maximum size-density relationship between stem surface area and stand density

Maximum size-density relationship between stem surface area and stand density

Evaluating a single tree-based growth model for even-aged stands against the maximum size–density relationship: Some insights from balsam fir stands in Quebec, Canada

In this study, we addressed the issue of model evaluation when long-term monitoring data are unavailable or inappropriate. More specifically, we fitted a single tree-based growth model for pure even-aged balsam fir stands and we compared stochastic predictions with an existing maximum size–density relationship (MSDR). Growth trajectories for plots of different initial densities and diameter distributions were simulated over a 70-year period using 500 realizations for each combination of initial density-diameter distribution. Long-term predictions were consistent with the existing MSDR. The model properly reproduced the senescence phase in which the trajectories diverge from the MSDR. This phase was initiated when the average tree volume reached 0.2-0.3 m3 per tree, which roughly corresponded to a DBH (diameter at breast height, 1.3 m from the ground) between 19 and 23 cm. Although it cannot be generalized, our case study shows that a simple single tree-based growth model with a distance-independent competition index and no stand density index can reproduce an existing MSDR. The match between long-term predictions and an existing MSDR strengthens the confidence in the biological behaviour of the model.

Static and dynamic maximum size–density relationships for mixed trembling aspen and white spruce stands in western Canada

We examine maximum size–density relationships (MSDR) of pure and mixed stands of trembling aspen (Populus tremuloides Michx.) and white spruce (Picea glauca (Moench) Voss.) in the Boreal Forest Natural Region of Alberta, Canada. Stochastic frontier function regression was used to estimate the MSDR species boundary or static line and mixed models were used to investigate how individual stands self-thin (dynamic thinning line). Effects of age, stand composition, soil nutrient regime and soil moisture regime were also evaluated. A steeper slope was obtained for the dynamic than for the static thinning line, and both MSDR lines slopes are statistically different from the theoretical value proposed by Reineke (1933). The deciduous component (percent of stand total basal area that is deciduous) has a negative effect on the slope and a positive effect on the intercept of the static line. Composition (increasing aspen basal area) also has a negative effect on the intercept of the dynamic line although no effect was detected on the slope. Soil nutrient regime has a positive effect on the intercept and a negative effect on the slope of the dynamic thinning line. Results suggest that local differences such as site quality and stand composition are important factors in determining maximum size–density relationships for these mixedwoods stands and how individual stands develop and self-thin.

Comparison of maximum size–density relationships based on alternate stand attributes for predicting tree numbers and stand growth

Quantification of site occupancy or stand density is essential for modeling forest stand mortality, growth and yield. A variety of measures of stand density have been proposed. Diverse tree and stand attributes have been employed, but direct comparisons of the effectiveness of various measures for estimating number of trees for fully stocked stands and for predicting growth for stands at different stages of development are not possible due to the varying forms of the measures.Maximum size–density relationships are a widely-used type of stand density measure. Reineke’s stand density index (SDI), which is based on number of stems per unit area and quadratic mean stem diameter, is the most commonly-applied stand density measure of this type. In the study reported here, stand density indices were developed by employing the structure of Reineke’s index but using the stand attributes of mean stem volume and mean stand height, as well as mean stem diameter. When estimating number of trees and periodic volume growth per unit area using data from sample plots in planted stands of loblolly pine, the SDI based on mean diameter performed best with mean stem volume being second best and mean stand height third.

Development of Planting Density‐Specific Density Management Diagrams for Loblolly Pine

For density management diagrams (DMDs), it is usually assumed the relative value between Reineke’s stand density index (SDI) where self-thinning is expected to begin and a maximum size-density relationship species boundary line is constant regardless of planting density. Using estimated size-density trajectories of loblolly pine plantations in the southeastern United States, the quadratic mean diameter at which the onset of self-thinning occurs was determined. An equation was then fitted to estimate the quadratic mean diameter at which self-thinning is expected to begin relative to planting density. DMDs that accounted for planting density showed that self-thinning began at 40 –72% of the species maximum SDI for planting densities ranging from 605 to 2,722 seedlings per acre.