Self-thinning Law Research Articles

Density-dependent mortality models for mono- and multi-species uneven-aged stands: The role of species mixture

The self-thinning law describes the progressive mortality of growing trees in crowded even-aged stands because of competition. In uneven-aged stands, however, density-dependent mortality is relatively poorly known. We developed density-dependent self-thinning models for mono-species and multi-species uneven-aged stands. Three models for mono-species Picea abies, Abies alba and Fagus sylvatica stands, four models for mixed conifer/broadleaf stands and a model for single tree selection stands as a specific type of uneven-aged forest were parametrized by Quantile Regression (QR) and Stochastic Frontier Analysis (SFA) using data from n = 31,612 forest inventory plots in Slovenia. The average slope of the self-thinning line in uneven-aged stands of −1.422 was statistically significantly lower than Reineke’s slope. Maximum stand density was higher in mono-species stands than in multi-species stands. The higher the proportion of beech in a stand, the higher the intercept and slope of the maximum stand density line. In single tree selection stands, the intercept and the slope of the maximum stand density line were highest and statistically different from all other types of uneven-aged forests. Species mixture had an impact on the slope and intercept of the self-thinning line. The slopes of the self-thinning lines differed between the studied types by up to 36% and 69% in the SFA and QR models, respectively, while the intercepts differed by up to 13% and 24%, respectively. However, maximum stand densities differed significantly only between compositionally very dissimilar stand types, such as Fagus sylvatica- or Picea abies-dominated mixtures versus other broadleaf mixtures (e.g., Quercus petraea, Carpinus betulus). The SFA predictions of maximum stand densities were systematically lower and less varying than the QR predictions, but not statistically different. The models can be used to constrain individual tree growth models in uneven-aged forests and to estimate the onset of inter-tree competition in irregular stands.

Feasible Green Strategy for the Quantitative Bioaccumulation of Heavy Metals by Lemna minor: Application of the Self-Thinning Law.

This study involved the development of mathematical linear regression models to describe the relationships between mean plant biomass (M) and population density (D), M and frond diameter (L), frond numbers (N) and L of Lemna minor under different initial population densities (3200, 4450, and 6400 plants/m2), respectively, from the perspective of the self-thinning law. Our results revealed that the value of the allometric exponents for M and D were - 3/2. Further, the concentrations of Zn, Pb, Cu, Fe, and Ni accumulated in L. minor plants were 0.86, 0.32, 0.36, 0.62, and 0.39mg/kg, respectively. Based on these developed equations and the heavy metal accumulations by L. minor, the phytoremediation capacity of L. minor was quantified via its frond diameters. Overall, the present study provides a cost-effective green method for managing the phytoremediation of heavy metal-contaminated aquatic environments.

Self-thinning Rules at Chinese Fir (Cunninghamia lanceolata) Plantations—Based on a Permanent Density Trial in Southern China

Data selection and methods for fitting coefficients were considered to test the self-thinning law. The Chinese fir (Cunninghamia lanceolata) in even-aged pure stands with 26 years of observation data were applied to fit Reineke's (1933) empirically derived stand density rule (N ∝ –1.605, N = numbers of stems, = mean diameter), Yoda's (1963) self-thinning law based on Euclidian geometry ( ∝ N–3/2, = tree volume), and West, Brown and Enquist's (1997, 1999) (WBE) fractal geometry ( ∝ –8/3). OLS, RMA and SFF algorithms provided observed self-thinning exponents with the seven mortality rate intervals (2%–80%, 5%–80%, 10%–80%, 15%–80%, 20%–80%, 25%–80% and 30%–80%), which were tested against the exponents, and expected by the rules considered. Hope for a consistent allometry law that ignores species-specific morphologic allometric and scale differences faded. Exponents α of N ∝ α, were significantly different from –1.605 and –2, not expected by Euclidian fractal geometry; exponents β of ∝ Nβ varied around Yoda's self-thinning slope –3/2, but was significantly different from –4/3; exponent γ of ∝ γ tended to neither 8/3 nor 3.

Variation in individual biomass decreases faster than mean biomass with increasing density of bamboo stands

The total biomass of a stand is an indicator of stand productivity and is closely related to the density of plants. According to the self-thinning law, mean individual biomass follows a negative power law with plant density. If the variance of individual biomass is constant, we can expect increased stand productivity with increasing plant density. However, Taylor’s power law (TPL) that relates the variance and the mean of many biological measures (e.g. bilateral areal differences of a leaf, plant biomass at different times, developmental rates at different temperatures, population densities on different spatial or temporal scales), affects the estimate of stand productivity when it is defined as the total biomass of large plants in a stand. Because the variance of individual biomass decreases faster than mean individual biomass, differences in individual biomass decline with increasing density, leading to more homogeneous timbers of greater economic value. We tested whether TPL in plant biomass holds for different species and whether the variance of individual biomass changes faster than the mean with increasing stand density. The height, ground diameter and fresh weight of 50 bamboo species were measured in 50 stands ranging from 1 m by 1 m to 30 m by 30 m to ensure more than 150 bamboos in every stand. We separately examined TPL in height, ground diameter, and weight, and found that TPL holds for all three biological measures, with the relationship strongest for weight. Using analysis of covariance to compare the regression slopes of logarithmic mean and variance against the logarithm of density, we found that the variance in individual biomass declined faster than the mean with increasing density. This suggests that dense planting reduced mean individual biomass but homogenized individual biomass. Thus, there exists a trade-off between effective stand productivity and stand density for optimal forest management. Sparse planting leads to large variation in individual biomass, whereas dense planting reduces mean individual biomass. Consequently, stand density for a plantation should be set based on this trade-off with reference to market demands.

Mathematical consideration of the age-related decline in leaf biomass in forest stands under the self-thinning law

In addition to the hypothetical trends proposed by Kira and Shidei (1967), Odum (1969) and Ryan et al. (1997, 2004), Oshima et al. (1958) observed a complicated age-related change in stand leaf biomass in Abies forests. To explain this change in stand leaf biomass theoretically, the age-related change in leaf biomass was modeled based on the following three assumptions after canopy closure: (i) the self-thinning law; (ii) expanded allometric scaling between the mean individual leaf mass and mean individual total mass; and (iii) the formulation of a logistic function for stand density change. The model successfully explained these three trends in forest stand leaf biomass and introduced expanded allometric scaling including the properties of the model based on simple allometric scaling proposed by Ogawa (2017). Therefore, the model developed here can generalize age-related changes in forest stand biomass better than the model proposed by Ogawa (2017).

Hydrological niche segregation of plant functional traits in an individual-based model

Water is one of the major drivers determining distribution and abundance of plant species. Namely, plant species’ presence and location in the landscape can be explained using metrics of soil water because plant species are restricted to a species-specific range of soil water conditions, i.e. their hydrological niche. However, little is known about the specific traits that determine the hydrological niche of a plant species. To investigate the relationship between plant functional traits, community structure and hydrological niche segregation, we developed a new generic individual-based model PLANTHeR which describes plant functional trait abundance as a function solely of soil water potentials and individual behavior. An important innovation is that there are no a priori defined trade-offs so that the model is neither restricted to a certain set of species nor scaled to a specific ecosystem.We show that PLANTHeR is able to reproduce well-known ecological rules such as the self-thinning law. We found that plant functional traits and their combinations (plant functional types − PFTs) were restricted to specific ranges of soil water potentials. Furthermore, the existence of functional trait trade-offs and correlations was determined by environmental conditions. Most interestingly, the correlation intensity between traits representing competitive ability and traits promoting colonization ability changed with water stress intensities in a unimodal fashion.Our results suggest that soil water largely governs the functional composition, diversity and structure of plant communities. This has consequences for predicting plant species’ response to changes in the hydrological cycle due to global change. We suggest that PLANTHeR is a flexible tool that can be easily adapted for further ecological-modelling studies.

Modeling age-related stand respiration changes in forest stands under the self-thinning law

There are three different age-related changes in stand respiration (woody + foliage respiration) in an even-aged pure forest stand after canopy closure: (A) monotonous increase (Kira and Shidei, 1967; Odum, 1969), (B) constancy (Oohata and Shidei, 1974; Ogawa et al., 2010), and (C) monotonous decrease (Ryan et al., 1997; Drake et al., 2011). These contrasting views were based on observational data rather than on theoretical arguments. The present study was performed to model age-related changes in stand respiration to elucidate the mechanism of stand respiration dynamics after canopy closure in compliance with three concepts: (i) the self-thinning law, (ii) power scaling between mean individual annual respiration and mean individual mass, and (iii) the logistic function of stand density changes. Simulations showed that although the three hypotheses were tenable theoretically, that proposed by Kira and Shidei is a mathematically special case. Considering these modeling results, age-related stand respiration changes in plantations of hinoki (Chamaecyparis obtusa) (Ogawa et al., 1985) can be regarded as decreasing as well as constant after canopy closure. The present modeling revealed that there exists possibility of increasing stand respiration in Kira and Shidei’s hypothesis from the theoretical viewpoints, although no observed data support their hypothetical trend. However, from both of the empirical and theoretical viewpoints, we understood that possibilities of stand respiration constancy and decrease in forest stands were regarded to be reasonable after the canopy closure.

Retraction Note: Deriving the mean mass-density trajectory by reconciling the competition-density effect law with the self-thinning law in even-aged pure stands

Retraction Note: Deriving the mean mass-density trajectory by reconciling the competition-density effect law with the self-thinning law in even-aged pure stands

Modeling age-related leaf biomass changes in forest stands under the assumptions of the self-thinning law

Key message The present model indicates that possibility of leaf biomass constancy in forest stands is not so high, and that forest stand leaf biomass declines after canopy closure.

Relação da razão do quociente diâmetro e área basal com o autodesbaste em povoamentos equiâneos de <i>Pinus taeda</i> L.

http://dx.doi.org/10.5902/1980509814584Este trabalho teve por objetivo estudar a relação do diâmetro médio e da área basal no momento do autodesbaste, o crescimento diamétrico e a ocorrência do autodesbaste, o momento de autodesbaste com base na proporção de diâmetro-área basal, a taxa de autodesbaste comparando-a com a lei da potência -3/2 em povoamentos de Pinus taeda L., em vários espaçamentos, manejado em densidade completa. O modelo de Tang foi empregado relacionando o número de árvores por hectare com o diâmetro médio, a partir de dados de parcelas permanentes, medidas anualmente até os 18 anos em experimento de espaçamento. O modelo de autodesbaste mostrou ótima precisão e elevada eficiência, porém, apresentou coeficiente angular de -1,65 com inclinação diferente da preconizada na lei de autodesbaste de -3/2. No momento da ocorrência do autodesbaste, as árvores atingiram diâmetros 14 e 17 cm, nos espaçamento de 2 x 2 e 3 x 2 m, com áreas basais de 38,8 e 47,5 m2/ha, respectivamente. Em todos os espaçamentos a área basal máxima foi de 78 m2/ha, aos 18 anos de idade, apresentando ainda tendência ascendente. Nesses espaçamentos, o quociente do diâmetro pela área basal no momento de ocorrência do autodesbaste foram 0,36 e 0,37, respectivamente.

Deriving the mean mass-density trajectory by reconciling the competition-density effect law with the self-thinning law in even-aged pure stands

The competition–density (C–D) effect law refers to the relationship between mean mass w and density ρ at a particular moment among a set of tree populations grown at a wide range of densities. The self-thinning law refers to the time trajectory of w and ρ in overcrowded stands. Because these two laws have not yet been theoretically harmonised, the aim of this paper is to achieve the unification of the two laws. Under the assumption that the reciprocal equation of the C–D effect in self-thinning stands and the self-thinning equation both hold, the slope of the reciprocal equation becomes the same as that of the self-thinning equation on logarithmic scales as the growth stage progresses. Finally, the reciprocal equation is converted to the w–ρ trajectory, eliminating the biological time from the reciprocal equation. The w–ρ trajectory of stands starting with any initial density has thus been explicitly formulated. Larger values of the relative mortality rate play an important role in relieving the C–D effect and cause the w–ρ trajectory to approach the self-thinning line at an earlier stage of growth. Stands exponentially decreasing in number obey the self-thinning law after a sufficient lapse of biological time, irrespective of their initial densities. Unknown functions, such as the survivorship curve and the ceiling biomass, have been explicitly represented as a function of biological time. The approximate expression for the w–ρ trajectory suitably mimics the time trajectory of mean stem and density in an eastern pine plantation.

Self-Thinning in a Space-Limited Rocky Intertidal Barnacle System

Acorn barnacles were assumed to have a self-thinning exponent close to -3/2 widely acknowledged for plant populations. However, space and food competition mechanisms would lead to a different exponent. Different recruit densities between tidal levels should result in different geometry of space occupation with adult individuals morphologically different and a different biomass-density relationship. Using Tetraclita squamosa, an intertidal barnacle, we investigated the biomass-density relationship along an intertidal gradient to study self-thinning. The self-thinning slope was steeper than the -3/2 power and did not differ among lower tidal levels with an intercept significantly higher in the low intertidal than in the mid-intertidal, whereas there was no significant relationship in the high intertidal. Growth in height of crowded barnacles along with the weaker effect of competition for food may retard mortality and result in a self-thinning slope steeper than -3/2. Our results suggested that self-thinning exponent in barnacles is steeper than -3/2 found for plant populations and allometric growth may not be the only factor causing this deviation from the “self-thinning law”. As one progresses higher in the intertidal, self-thinning processes disappear owing to reduced competition.

Effects of flood frequency and duration on the allometry of community‐level stem size‐density distributions in a floodplain forest

Consistent with the self-thinning law of plant population ecology, Niklas et al. in 2003 proposed that stem size-density distributions (SDDs) of multispecies forest communities should change in very predictable ways as a function of the effects of past disturbances on average tree size. To date, empirical tests of this hypothesis have not been pursued in floodplain settings. SDDs were constructed using tree stem-size and density data from forest plots positioned along a flood frequency and duration gradient in the Upper Mississippi River floodplain. As flooding (both frequency and duration) increased, several small tree species were eliminated from forest plots and the persistent species increased in their size. Consistent with the predictions of Niklas et al., these changes corresponded with overall decreases in stem density, increases in average stem size, and reductions in both the Y-intercept and slope terms of the community-level SDDs. This study adds to a growing list of examples suggesting that theories related to forest community composition and biomass must account for both the broader effects of disturbances as well as the underlying biochemical processes that regulate plant growth. Further study is needed to fully address the role different disturbance frequencies play in determining plant density, diversity, average size, and associated size\frequency distributions.

植物种群自疏过程中构件生物量与密度的关系

不论是在对植物种群自疏规律还是在对能量守衡法则的研究中,个体大小(<em>M</em>)大多针对植物地上部分生物量,地下部分和构件生物量及其动态十分重要又多被忽视。以1年生植物荞麦为材料研究了自疏种群地下部分生物量、包括地下部分的个体总生物量以及各构件生物量与密度的关系。结果表明:平均地上生物量和个体总生物量与密度的异速关系指数(<em>γ</em>_above-ground和<em>γ</em>_individual)分别为-1.293和-1.253,与-4/3无显著性差异(<em>P</em> > 0.05),为-4/3自疏法则提供了有力证据;平均根生物量-密度异速指数<em>γ</em>_root(-1.128)与-1无显著性差异(<em>P</em> > 0.05),与最终产量恒定法则一致;平均茎生物量-密度异速指数<em>γ</em>_stem(-1.263)接近-4/3(<em>P</em> > 0.05),平均叶生物量-密度异速指数<em>γ</em>_leaf(-1.524)接近-3/2(<em>P</em> > 0.05),分别符合-4/3自疏法则与-3/2自疏法则;而繁殖生物量与密度的异速关系指数<em>γ</em>_reproductive(-2.005)显著小于-3/2、-4/3或-1(<em>P</em> < 0.001)。因此,不存在一个对植物不同构件普适的生物量-密度之间的关系。光合产物在地上和地下构件的生物量分配格局以及构件生物量与地上生物量之间特异的异速生长关系导致不同构件具有不同的自疏指数。无论对于地上生物量还是个体总生物量,荞麦种群能量均守衡,而对于地下生物量,荞麦种群能量不守衡。;Body size (<em>M</em>) was specifically referred to above-ground biomass of individual plant for both power law of self-thinning and energy equivalence law for plant populations in most studies. Little is known about the biomass-density relationships with respect to below-ground part and other plant components, such as stem, leaf and reproductive part during the course of self-thinning, although they were very important. The self-thinning law can also be defined as component biomass, below-ground biomass and total biomass when all biomasses of all yield components are measured, such as leaf, stem, reproductive biomass and below-ground biomass. Subsequently, the competition effects of total biomass or each plant component can be observed during the course of self-thinning as required. Since growth in plants is a modular process, it is rational ecologically to examine biomass-density relationships not only at population and individual levels, but also at plant part level. <br> The growth patterns of plant components, such as stems, leaves and reproductive parts, as well as roots, in relationship with plant density were studied with <em>Fagopyrum esculentum</em> populations in a pot experiment. There were 3 plant densities (8000, 24000 and 48000 plants/m²) with 4 replicates in a randomized complete design. There were 6 harvesting times with 4 pots being destructively harvested at each harvesting time. Results showed that the values of self-thinning exponents were -1.293 and -1.253 for mean above-ground biomass and total biomass, respectively, during the course of population development. These values, which were not significantly different from -4/3, can be regarded as evidence in favor of the -4/3 power law of self-thinning. The allometric exponents were -1.128, -1.263, -1.524 and -2.005 for mean biomass of roots, stems, leaves and reproductive parts, respectively. There were no significant difference between <em>γ</em>_roo_tand -1(<em>P</em> > 0.05), <em>γ</em>_stemand -4/3(<em>P</em> > 0.05),<em>γ</em>_leaf and -3/2(<em>P</em> > 0.05),but <em>γ</em>_reproductivewas significantly different from -3/2, -4/3 or -1. It is suggested that there is no universal biomass-density relationship associated with plant components. Competition within the population affects plant development and alters the biomass allocation patterns of plants and may lead to different biomass-density relationships for plant components in the population. Specific allometric relationships between plant component biomass and above-ground biomass can also resulted in slopes of the self-thinning lines which are different from -3/2 or -4/3. Populations of <em>Fagopyrum esculentum</em> obeys the law of energy equivalence during the course of self-thinning for above-ground biomass or total biomass, but for below-ground biomass.

Lianas escape self-thinning: Experimental evidence of positive density dependence in temperate lianas Celastrus orbiculatus and C. scandens

Abstract The neighborhood density of plants strongly affects their growth, reproduction, and survival. In most cases, high density increases competition and negatively affects a focal plant in predictable ways, leading to the self-thinning law. There are, however, situations in which high densities of plants facilitate focal plant performance, resulting in positive density dependence. Despite their importance in forest gap dynamics and distinctive growth form, there have been very few studies of the effect of density on lianas or vines. We grew an invasive (Celastrus orbiculatus) and a native (Celastrus scandens) liana species together in three different density treatments, while also manipulating the light and support availability. We found that across treatment conditions, C. orbiculatus always out-performed C. scandens, showing greater relative growth rate in height and diameter, greater biomass and higher survival. Both species responded similarly to the density treatments: with plants in high density not showing a decrease in relative height growth rate compared to medium density. Aboveground biomass for C. scandens was not affected by density, while for C. orbiculatus, the most massive plants were growing in medium density without support. More surprisingly, survival analysis indicated that the two species both had significantly lower mortality rates in the highest density treatment; this trend held true across the other treatments of light and supports. More generally, this study demonstrates that these lianas can escape the consequences of high density and thus the self-thinning law that affects self-supporting plants. This suggests a broader hypothesis about lianas in general: their greater flexibility in allocating growth resources allows them to grow taller and thinner without collapsing and thereby potentially escape shading and mortality even at high densities.

The Exponent of Self-Thinning Law Varied: a Model Linking Population Spatial Distribution and Individual Growth

The Exponent of Self-Thinning Law Varied: a Model Linking Population Spatial Distribution and Individual Growth

A stand dynamic model for red pine plantations with different initial densities

A stand dynamic model was developed to predict the growth response in even-aged forest plantations of different initial planting densities. The model is based on the integration of three subcomponents: height growth, self-thinning, and diameter increment. The integrated model uses the height of dominant trees to simulate stand response to site quality and internal growth potential. An extended self-thinning submodel is used to simulate mortality in stands due to crowding and inter-tree competition. A diameter increment submodel is used to link the height growth and self-thinning submodels. The height growth submodel is based on an application of the “Pipe Model” theory. The three-parameter self-thinning submodel is developed from an extended self-thinning law that captures self-thinning in stands before they attain full stocking. The diameter increment model is based on the assumption that diameter increment is related to height growth and available growing space described by stand density. The integrated model is applied to data collected from a 45-year-old red pine (Pinus resinosa Ait.) plantation subsectioned with different initial planting densities. For the data used, only two parameters were required to capture 99% of measured variation in height growth. Additional data from sites with different planting intensities are required to formulate a more generalized height growth model. The slope of the linear self-thinning limit for red pine is approximately −1.5. Model predictions are consistent with field measurements.

Clonal building, simple growth rules and phylloclimate as key steps to develop functional–structural seagrass models

The repetitive clonal growth of the seagrasses Zostera noltii Hornem. and Cymodocea no- dosa Aschers at the module level was used to implement a deterministic, individual-based, numerical model using the simplest growth rules. Inter- and intraspecific variability in plant morphology and meadow attributes was simulated, and the results yielded by the model were compared with an exist- ing large data set recorded for both species. The model outputs showed that intra- and interspecific mor- phological variability can be accurately described (r = 0.99, p < 0.0001, n = 19) by a restricted number of parameters (plastochrone interval (PI) and elongation rates for rhizome (RER) and leaf (LER), which are species-specific parameters). Interspecific differences in meadow properties were recorded; however, simulated values were double those observed. This result was mainly attributable to a lack of density- dependence phenomena in the model assumptions, revealing the importance of such phenomena in structuring seagrass populations. In general, species with high PIs displayed longer modules (leaves and internodes) and lower shoot densities, whereas species with lower PI values developed shorter modules and crowded stands. This result corresponds with the relationship indicated by the self-thinning law and by previous studies. The model also showed that plant morphology arises as an emergent property of a simple set of growth rules acting at the module level, and that plant dynamic parameters can be tuned by seagrasses in response to their local environmental conditions. Thus, the whole-plant response to the environment can be determined by the sum of all the modular responses. This model, together with a better knowledge of the regulation of plant dynamic parameters by control variables (light, temperature, nutrients, etc.), provides a conceptual framework that allows the incorporation of module, plant morphol- ogy and meadow properties into functional-structural seagrass models, in which feedbacks among plant morphology, plant development and phylloclimate (i.e. the physical environment actually perceived by each individual organ or plant population) can be included.

Self‐thinning of plant populations from a dynamic viewpoint

Summary 1. In this paper a new formulation of the self-thinning law for plant populations is presented, based on dynamical principles. The basis of the new approach is explicitly to separate the equation of state (total = mean × number) from the change in state ( δ total = mean × δ number + number × δ mean). 2. Using the dynamical approach, an analytical expression for the self-thinning exponent is derived which shows that the self-thinning exponent follows a trajectory, but whenever the total is constant the self-thinning exponent must, by definition, = − 1. Conversely, whenever the total varies the self-thinning exponent must, by definition, differ from − 1. The underlying general principles of self-thinning trajectories are deduced. 3. The new dynamic interpretations are general because they are independent of the details about how the changes in the number and mean occurred. 4. The theory is demonstrated using a typical self-thinning experiment.

Scaling of ramet size and spacing in seagrasses: implications for stand development

The scaling between shoot mass and spacer length was examined across 23 seagrass species by compiling field and literature data on architectural and stand features. The distance between shoots programmed in seagrass architecture was scaled at the 0.24 power of shoot mass. The predicted distance between neighboring shoots resulting from the scaling relationship investigated was compared with the spacing observed in 79 crowded natural seagrass stands. Seagrass architecture predicted 50% of the variability of the distance between neighbors observed in natural stands. Moreover, the growth programme of seagrass species predicted 29% of the variance of maximum aboveground biomass developed by seagrass meadows. The close agreement between the shoot density and biomass developed by closed natural stands and that predicted from seagrass architecture suggested that the upper limit to density and aboveground biomass, such as that defined by the self-thinning law, is already imprinted within the basic architecture of seagrasses (i.e. spacer length in between consecutive shoots).