Maintenance cost, toppling risk and size of trees in a self-thinning stand

Wind routinely topples trees during storms, and the likelihood that a tree is toppled depends critically on its allometry. Yet none of the existing theories to explain tree allometry consider wind drag on tree canopies. Since leaf area index in crowded, self-thinning stands is independent of stand density, the drag force per unit land can also be assumed to be independent of stand density, with only canopy height influencing the total toppling moment. Tree stem dimensions and the self-thinning biomass can then be computed by further assuming that the risk of toppling over and stem maintenance per unit land area are independent of stand density, and that stem maintenance cost is a linear function of stem surface area and sapwood volume. These assumptions provide a novel way to understand tree allometry and lead to a self-thinning line relating tree biomass and stand density with a power between −3/2 and −2/3 depending on the ratio of maintenance of sapwood and stem surface.     

Density effects on the growth of self-thinning Eucalyptus urophylla stands

Density effects on the growth of self-thinning Eucalyptus urophylla stands were examined for 7 years. Tree height and stem diameter at breast height were measured during the experimental period. Stems, branches, leaves, bark and roots of 45 E. urophylla trees were sampled in three different density stands in order to establish their biomass equations. Change trends of the biological time τ and density ρ were described used corresponding equations. The stem weight ratio increased and leaf weight ratio decreased, whereas those of branch, bark and root were relatively steady from 2 years after the planting. The competition-density (C-D) effect equation of mean organ weight w _o was derived by combining the allometric power relationship between mean tree weight w and w _o with the C-D effect equation of self-thinning stands. The equations of the C-D effect for w and ρ and for w _o and ρ were used to describe the C-D effects in tree and organs during course of self-thinning, respectively, and showed a good fit to the data. Leaf biomass of different density stands reached a more or less constant level with time elapse. High density produced the greatest biomass and stem biomass, so that it is the best choice in silvicultural practice.     

Ecological and mathematical considerations on self-thinning in even-aged pure stands

It is emphasized in growth analysis of self-thinning populations that relative mortality rate pertains to the difference between relative growth rates and net assimilation rates, each of which are definable on a mean plant size basis or on a biomass basis. The time trends of the ratio of relative mortality rate to relative growth rates to be expected according to Tadaki's, Shinozaki's and Hozumi's models are compared with that of the eastern white pine population, and a good agreement is exhibited. As an alternative to Hozumi's model, a new model is constructed to unite the logistic theory of plant growth and the 3/2 power law concerning self-thinning, which so far have usually been applied independently to growth analysis. To construct the model the following assumptions are made: the fundamental equation to relate mean plant weight with density in self-thinning population proposed by Shinozaki, and a special population with a specific initial density which follows thew-p trajectory of the 3/2 power law type and has an exponential decrease in its density with biological time. Properties of the model are examined from ecological and mathematical viewpoints.     

Theoretical considerations on the C-D effect in self-thinning plant populations

A model for describing the competition–density (C-D) effect in self-thinning populations was developed on the basis of the following three basic assumptions: (1) the growth of mean phytomass follows a general logistic equation; (2) final yield is independent of initial population density; and (3) there exists a functional relationship between actual and initial population densities at any given time. The resultant equation takes the same reciprocal form as the reciprocal equation of the C-D effect derived from Shinozaki–Kira's theory (i.e., the logistic theory of the C-D effect), which deals with the density effect in nonself-thinning populations. However, one of the two time-dependent coefficients is quite different in mathematical interpretation between the two reciprocal equations. The reciprocal equation for self-thinning populations is essentially the same as the reciprocal equation assumed in the derivation of the functional relationship between actual and initial population densities. The establishment of the reciprocal equation is supported by the empirical facts that the reciprocal relationship between mean phytomass and population density is discernible in not only nonself-thinning populations but also in self-thinning populations. The present model is expected to systematically interpret underlying mechanisms between the C-D effect, which is observed at a time constant among populations with various initial densities, and self-thinning, which is observed along a time continuum in a given population. Received: August 5, 1998 / Accepted: January 7, 1999     

An empirical approach to the analysis of forest stratification

Natural forest communities consist of different overlapping elementary subpopulations. Using the results of forest stratification in the preceding study, the properties of mean tree height for subpopulation in a stratified forest stand were examined. Mean tree height decreased as tree density per subpopulation increased. This relation was described by a simple mathematical model consisting of a power equation of tree density and two coefficients. The first coefficient or exponent of tree density was close to −1/2 in its expectation, while the other coefficient depended on life forms, especially in tropical forests. For tropical deciduous forests which suffered from seasonal forest fires, the latter coefficient was smaller than those for tropical evergreen and Japanese forests. This difference of the coefficient was not clear between tropical evergreen and Japanese forests and between deciduous and evergreen forests in Japan. In conclusion, the proposed model is similar to the 1/2 power law of tree height in man-made forests with simple architecture, and is designated the quasi-1/2 power law of tree height.     

The self-thinning process in mangrove <Emphasis Type="Italic">Kandelia obovata</Emphasis> stands

The self-thinning process was monitored in crowded Kandelia obovata Sheue, Liu & Yong stands over four years. The frequency distribution of tree phytomass was an L-shape, which was kept over the experimental period. Spearman’s rank correlation coefficient for phytomass decreased as the time span of the comparison became longer, a result which indicates that the rank of phytomass changes as stands grow. Death of trees resulted from one-sided competition, i.e., death occurred in lower-rank trees. Surviving trees continued to grow. Whatever the current spatial distribution of the trees, death occurred randomly and the spatial distribution gradually became close to random as stands grew. The self-thinning exponent was 1.46, which can be regarded as evidence in favor of the 3/2 power law of self-thinning. Relative growth rate, RGR, decreased in proportion to decreasing relative mortality rate, RMR, with a proportionality constant of 1.57, which was not significantly different from the slope of the self-thinning exponent. This experimental result probably justifies the assumption that the ratio of RGR to RMR in the mean phytomass-density trajectory for any self-thinning population with different densities becomes constant as the growth stage progresses.     

Competitive regulation of plant allometry and a generalized model for the plant self-thinning process

Taking into account the individual growth form (allometry) in a plant population and the effects of intraspecific competition on allometry under the population self-thinning condition, and adopting Ogawa's allometric equation 1/y = 1/axb + 1/c as the expression of complex allometry, the generalized model describing the change mode of r (the self-thinning exponential in the self-thinning equation, log M = K + log N, where M is mean plant mass, K is constant, and N is population density) was constructed. Meanwhile, with reference to the changing process of population density to survival curve type B, the exponential, r, was calculated using the software MATHEMATICA 4.0. The results of the numerical simulation show that (1) the value of the self-thinning exponential, r, is mainly determined by allometric parameters; it is most sensitive to change of b of the three allometric parameters, and a and c take second place; (2) the exponential, r, changes continuously from about -3 to the asymptote -1; the slope of -3/2 is a transient value in the population self-thinning process; (3) it is not a 'law' that the slope of the self-thinning trajectory equals or approaches -3/2, and the long-running dispute in ecological research over whether or not the exponential, r, equals -3/2 is meaningless. So future studies on the plant self-thinning process should focus on investigating how plant neighbor competition affects the phenotypic plasticity of plant individuals, what the relationship between the allometry mode and the self-thinning trajectory of plant population is and, in the light of evolution, how plants have adapted to competition pressure by plastic individual growth.     

Theoretical Relationships Between Mean Plant Size, Size Distribution and Self Thinning under One-sided Competition

As yet there is no comprehensive theory in plant populationecology to explain relationships between mean plant size, sizedistribution and self-thinning. In this paper, a new synthesisof plant monocultures is proposed. If the reciprocal relationshipbetween plant biomass and plant population density among variousstands of even-aged plant populations holds, the same reciprocalrelationship must exist between cumulative mass and cumulativenumber of plants from the largest individual within a population,assuming strict one-sided competition (which is an extreme conditionfor competition for light among plants). The two parametersof the relationship between cumulative mass and cumulative numberwithin a stand both correlate with maximum plant height in thestand. One parameter equals the reciprocal of the potentialmaximum plant mass per area, which is expressed by the productof maximum plant height and dry-matter density. The other parametercorrelates with the potential maximum individual plant mass,which is allometrically related to maximum plant height. Asa stand develops, the growth rate of the smallest individualswill become zero due to suppression from larger individuals,and they will die; i.e. self-thinning will occur. The slopeof the self-thinning line is expressed through the coefficientsof allometry between height and mass and between dry matterdensity and height. When the former coefficient is 3 and thelatter is 0, the gradient exactly corresponds to the value expectedfrom the 3/2 power rule, but it can take various values dependingon the values of the two coefficients. Competition among individualsdetermines size-density relationships among stands, which inturn determine the size structure of the stand. The size structureconstrains the growth of individuals and results in self-thinningwithin the stand.Copyright 1999 Annals of Botany Company. Monoculture, plant population, self-thinning, competition, hierarchy, size-structure.     

Community dynamics of evergreen broadleaf forests in southwestern Japan

The process and rate of revegetation in gaps in an evergreen oak forest were studied by comparing the species composition, tree density, frequency distribution of tree height, and relation between diameter at breast height and tree height among different aged stands. For estimating stand ages, the ages of gap indicators, such as,Symplocos prunifolia andAcer rufinerve, were very useful. It took about 70 years for gaps to be filled by large fully-grown trees. Since the mean residence time of the forest canopy was 180 years, the trees that attain the forest canopy were expected to be canopy trees for 110 years on the average. Tree densities of all broadleaved evergreens exceptS. prunifolia, were independent of stand age. On the other hand, densities of gap indicators,S. prunifolia andA. rufinerve, decreased as stand age increased. Other deciduous broadleaf and coniferous species were scarce as a whole. According to the frequency distributions of height of live and dead trees in different aged stands, it was suggested that shorter trees were more susceptible to death than taller trees. The self-thinning in revegetation process in gaps approximately followed the 3/2 power law, though the power was larger (−1.32) than expected from the law.     

Ecological and mathematical considerations on self-thinning in even-aged pure stands

Ecological and mathematical considerations were made on Shinozaki's, Tadaki's and 3/2 power law models for the mean plant weight-density trajectory under self-thinning in even-aged pure stands, and interrelationships among these models were discussed. To overcome the discrepancy between the observed trajectory of the eastern white pine population and the one predicted from Tadaki's model, a new model was proposed. To construct the model the assumptions were made so as to incorporate the good properties of Tadaki's and Shinozaki's models in early stages of growth into those of the 3/2 power law model observed in later stages. Applicability of the model was tested for the pine population, which showed a good fit to the data. The growth analysis on the basis of the model revealed the growth curve of the pine followed a λw-type logistic cruve and suggested the existence of a lag time, a hyperbolic relationship between biological and physical time and a clear dependence of survivorship curves on initial density.     

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(M)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(γabove-ground和γindividual)分别为-1.293和-1.253,与-4/3无显著性差异(P>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γroot(-1.128)与-1无显著性差异(P>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γstem(-1.263)接近-4/3(P>0.05),平均叶生物量-密度异速指数γleaf(-1.524)接近-3/2(P>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γreproductive(-2.005)显著小于-3/2、-4/3或-1(P<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。     

Density effect, self-thinning and size distribution in Pinus densiflora Sieb. et Zucc. stands

The competition-density (C-D) effect for given times and self-thinning over time in even-aged, natural, pure stands of Pinus densiflora Sieb. et Zucc. were analyzed with the reciprocal equation of the C-D effect in self-thinning stands, and the equation describing the time-trajectory of mean stem volume and stand density. The C-D effect and self-thinning were consistently well explained by the two equations. Differences in mean stem volume and in stand density among the stands tended to merge with increasing stand age. The self-thinning line with a slope of approximately –3/2 was reached by the higher density stand prior to the medium and lower density stands. The skewness of tree height distribution showed positive values, which means that the distribution is more or less L-shaped, and in addition the skewness decreased with increasing mean tree height, which indicates that smaller trees died as the stands grew. This trend is consistent with the asymmetric (one-sided) competition hypothesis that self-thinning is driven by competition for light. The tree height distribution was analyzed using the Weibull distribution. The location parameter h _min of the Weibull distribution increased with increasing stand age, and the scale parameter a tended to increase slightly with increasing stand age. The range of the shape parameter b of the Weibull distribution corresponded to that of the skewness.     

Density-dependence in single-species populations of plants

The general form of yield-density relationships in plant populations is discussed with reference to reciprocal equations and the $\text{3}\text{2}$ power law, which describes the concomitant changes in plant weight and density during self-thinning. A model to describe the pattern of mortality in high density populations is also discussed with particular reference to the nature of intraspecific competition within plant populations.A reparameterized version of a reciprocal equation proposed by Bleasdale & Nelder is used to describe the relationship between individual plant weight and surviving plant density. The biological interpretation of the parameters is discussed in relation to the dry matter production of isolated plants, the density at which mutual interference between neighbours becomes appreciable, and the efficiency of resource utilization at high densities.The reparameterized equation is then used together with an equation which describes mortality during self-thinning as the basis for a new model to describe the relation between total plant yield and sowing density. The law of allometry is used in conjunction with the model to describe the relationship between the weight of a plant part and density, and this then forms the basis for a model of the population dynamics of annual plants with effectively discrete generations. Finally the dynamical behaviour of plant populations is discussed. It is concluded that most plant populations will show neighbourhood stability with exponential or perhaps oscillatory damping towards an equilibrium.     

Stand and self-thinning dynamics in natural <Emphasis Type="Italic">Abies</Emphasis> stands in northern Hokkaido,Japan

Stand dynamics and self-thinning were analyzed in relation to the dynamics of above-ground biomass in natural Abies sachalinensis stands growing on sand dunes in northern Hokkaido, Japan. This was done in order to examine wave-type regeneration in the stands. Fifty-two plots were established in almost pure Abies stands that ranged from saplings to the mature and collapsing growth stages. Above-ground biomass and tree height reached asymptotic levels prior to the collapsing phase, unlike wave-regeneration Abies stands in central Japan and North America. Stand density was high in the young growth stages, but the self-thinning rate, that is, the density decrease per biomass growth in the study stands was greater than in wave-regeneration stands in central Japan, as indicated by a large self-thinning exponent (–1.26 by reduced major axis regression). The range of tree height distribution was very narrow, and the stands vertical structure was typically single-layered. The slenderness ratio of trees was large, except in young stands. In mature and collapsing stands, advanced seedling density increased markedly. These stand and tree characteristics were considered to be correlated with the wave-type regeneration in the study stands, and it is assumed that prevailing winds affect tree mortality.     

Leaf biomass changes with stand development in hinoki cypress (<Emphasis Type="Italic">Chamaecyparis obtusa</Emphasis> [Sieb. et Zucc.] Endl.)

We monitored a permanent plot of 3-year-old Chamaecyparis obtusa seedlings for 11 years after planting. As the stem cross-sectional area at the crown base can be regarded as a good predictor of leaf mass according to the pipe model theory, we measured this parameter to determine temporal trends in leaf biomass. The mean values showed asymptotic growth, maintaining a near-constant level after a stand age of 9 years. Peak values were found at 9 years, followed by a slight decrease because of a continuous reduction in stand density. This temporal trend suggests that the leaf biomass per unit land area attains a peak at an age of 9 years. As the stand density changes with stand age, the relationship between stand stem cross-sectional area at the crown base and stand density showed an optimum curve in which the optimum density was around 9200 ha⁻¹. We propose hypothetical trends in primary productivity and biomass density with stand age, based on the results of measurements of stem cross-sectional area at the crown base and stand density under the assumption of the 3/2 power law of self-thinning.     

Modelling the Time Course of Self-thinning in Crowded Plant Populations

A logarithmic model for the self-thinning of plants is proposed.This model describes the time course of self-thinning very welland fits data from forest stands and yield tables, which followthe 3/2 power law. An approximated expression of this modelshows that plant density decreases with age along a Gompertzcurve. This appears to be a basic property of the time courseof self-thinning in plants. Pinus strobus L., Pinus densiflora Sieb, et Zucc., stand development, self-thinning, 3/2 power law, logarithmic model, mortality     

Species-specific allometric scaling under self-thinning: evidence from long-term plots in forest stands

Experimental plots covering a 120 years' observation period in unthinned, even-aged pure stands of common beech (Fagus sylvatica), Norway spruce (Picea abies), Scots pine (Pinus sylvestris), and common oak (Quercus Petraea) are used to scrutinize Reineke's (1933) empirically derived stand density rule [see text], N=tree number per unit area, [see text]=mean stem diameter), Yoda's (1963) self-thinning law based on Euclidian geometry ([see text] [see text]=mean biomass per tree), and basic assumptions of West, Brown and Enquist's (1997, 1999) fractal scaling rules ([see text] [see text] w=biomass per tree, d=stem diameter). RMA and OLS regression provides observed allometric exponents, which are tested against the exponents, expected by the considered rules. Hope for a consistent scaling law fades away, as observed exponents significantly correspond with the considered rules only in a minority of cases: (1) exponent r of [see text] varies around Reineke's constant -1.605, but is significantly different from r=-2, supposed by Euclidian or fractal scaling, (2) Exponent c of the self-thinning line [see text] roams roughly about the Euclidian scaling constant -3/2, (3) Exponent a of [see text] tends to follow fractal scaling 8/3. The unique dataset's evaluation displays that (4) scaling exponents and their oscillation are species-specific, (5) Euclidian scaling of one relation and fractal scaling of another are coupled, depending on species. Ecological implications of the results in respect to self-tolerance (common oak>Norway spruce>Scots pine>common beech) and efficiency of space occupation (common beech>Scots pine>Norway spruce>common oak) are stressed and severe consequences for assessing, regulating and scheduling stand density are discussed.     

海三棱蔗草种群的密度与生物量动态

本文研究了上海市南汇县东海农场海堤外侧滩涂上海三棱藨草种群的密度动态,高度生长动态、生物量动态以及它们之间及其与环境之间的相互关系。研究结果表明：在环境条件相对稳定的地带A和B内,海三棱藨草种群的高度、高度生长和生物量在生长期内符合Logisfic增长。种群生物量动态与密度动态可分为3个阶段,其中阶段Ⅱ符合Yoda等提出的-3/2自疏定律。地带B为海三棱藨草种群生长的最适地带。地带C内生境条件极不稳定,种群的数量动态变化亦相当剧烈。在不同环境条件下,密度制约因素和非密度制约因素对种群数量动态的相对作用是不同的。在环境条件较稳定的生境中(地带A和B),密度制约因素是决定种群数量动态的主要因素;在环境条件变化剧烈的生境中(地带C),非密度制约因素是决定种群数量动态的主要因素。     

A further analysis of the quasi-1/2 power law of tree height in stratified forest communities

To clarify a statistical basis of the empirical fact designated as the quasi-1/2 power law of tree height in stratified natural forest communities, a derivative of Pearson's type VII distribution was adopted and used for describing the frequency distribution of individual tree height in a subpopulation extracted from a forest stand by the symmetric type difference diagram already proposed for the stratification of samples. Limited by the quasi-1/2 power law of tree height, all the coefficients of Pearson's type VII distribution were expressed as empirical equations of tree density in a subpopulation obtained from the stratification of samples. These empirical equations led to the normalized density function of tree height and gave a statistical basis for the quasi-1/2 power law of tree height. In addition to tree height data, the stem diameter at breast height and tree weight data for a forest stand were also stratified into subpopulations by using symmetric type difference diagrams. In conclusion, a new system was proposed for describing the dependence of mean tree height, mean stem diameter at breast height, and mean tree weight on tree density in a subpopulation.     

A Canopy Photosynthesis Model for the Dynamics of Size Structure and Self-thinning in Plant Populations

A dynamic model for growth and mortality of individual plantsin a stand was developed, based on the process of canopy photosynthesis,and assuming an allometric relationship between plant heightand weight, i.e. allocation growth pattern of plant height andstem diameter. Functions G(t, x), for the mean growth rate ofindividuals of size x at time t, and M(t,x), for the mortalityrate of individuals of size x at time t, were developed fromthis model and used in simulations. The dynamics of size structurewere simulated, combining the continuity equation model, a simpleversion of the diffusion model, with these functions. Simulationsreproduced several well-documented phenomena: (1) size variabilityin terms of coefficient of variation and skewness of plant weightincreases at first with stand development and then stabilisesor decreases with an onset of intensive self-thinning; (2) duringthe course of self-thinning, there is a power relationship betweendensity and biomass per unit ground area, irrespective of theinitial density and of the allocation-growth pattern in termsof the allometric parameter relating plant height and weight.The following were further shown by simulation: (a) competitionbetween individuals in a crowded stand is never completely one-sidedbut always asymmetrically two-sided, even though competitionis only for light; (b) plants of ‘height-growth’type exhibit a greater asymmetry in competition than plantsof ‘diameter-growth’ type, (c) the effect of competitionon the growth of individuals in a crowded stand converges toa stationary state, even when the stand structure still changesgreatly. All of these theoretical results can explain recentempirical results obtained from several natural plant communities.Finally, a new, general functional form for G(t, x) in a crowdedstand is proposed based on these theoretical results, insteadof a priori or empirical growth and competition functions. Canopy photosynthesis, competition mode, continuity equation, self-thinning, simulation, size distribution