Relationships Between Population Growth and Population Density in Monocultures of Larix leptolepis

One year-old Japanese larch (Larix leptolepis) seedlings were planted in a nursery from 1984 to 1986 at four density levels with four replicates in order to investigate the relationships between plant growth and density of survivors. As the results shown, self-thinning occurred severely in populations growing at high densities. The rate of the self-thinning followed a trajectory defined by the 3/2 power law. The relationship between mean dry weight per tree and population density at a given stage of growth followed the reciprocal equation and power equation at early experimental stages, but changed gradually to follow the 3/2 power law as plants grew further. Shch a change was mainly caused by the constant basal area per ms and decreased growth of tree height along a gradient of the density. The relationship between tree height (h) and density (p) also be discussed and in the 3/2 power law system it could be described by an equation consisting of h=αp（-1/2） where α is a constant.     

The allometric interpretation of the self-thinning rule

Plant populations growing at high densities undergo density-dependent mortality or self-thinning. The density of survivors ({ρ}) is related to their mean biomass (w) by the power equation w = K_ρ^?a, where a is $\text{3}\text{2}$. This is known as the “self-thinning rule”. This relationship is very general for plant populations and represents both an asymptotic time-trajectory for a particular population and a boundary line for juxtaposed joint values of w and p of separate populations. The traditional allometric derivation of the rule is outlined and shown to be unrealistic. An attempt to reformulate the self-thinning rule, based on the traditional allometric derivation, is shown to be unsatisfactory and an alternative allometric derivation is presented. The rule in its traditional statement $\text{w = K}{}_{\mathrm{\rho }}{}^{\mathrm{?3}}\text{2}$ is still its best expression. The nature of the constant K is discussed with particular reference to its dimensionality.     

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     

Density effects on organs in self-thinning <Emphasis Type="Italic">Pinus densiflora</Emphasis> Sieb. et Zucc. stands

Although much research on the density effect in nonself-thinning populations has been conducted, there has been very little research on density effects in self-thinning populations. Furthermore, the density effect of plant organs in self-thinning populations is little reported. The present study analyzed the yield–density (Y–D) effects on organs, such as stem, branch and leaf, together with that on stands of self-thinning Pinus densiflora Sieb. et Zucc.. The stand yield- and organ Y–D effects were well described by reciprocal and parabolic equations, respectively, throughout the experiment. The value of coefficient B in the reciprocal equation decreased monotonically with increasing stand age and became significantly closer to zero at the end of experiment (33-year-old stand), indicating that the constant final stand yield was established regardless of the density realized. The value of the relative growth coefficient h in the allometric equation between mean organ weight and mean aboveground weight was significantly smaller than 1.0 for stem, indicating that stem yield increases monotonically with increasing realized density. The h-value was significantly larger than 1.0 for branch throughout the experiment, and for leaf except at 33 years old, indicating that optimum densities exist. The h-value for leaf was not significantly different from 1.0 at 33 years old, indicating that the leaf yield reached a constant level regardless of realized density. The constant final leaf yield was established at almost the same growth stage as the establishment of constant final stand yield.     

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     

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.     

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。     

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.     

On the Ordered Development of Plants 2. Self-thinning in Plant Communities

It is shown that the hypothesis that a growing point on a vegetativeplant requires a minimum rate of supply of assimilate to continuegrowth can quantitatively describe self-thinning in communitiesof Trifolium subterraneum. The hypothesis can also be used toexplain the different relationships observed between mean plantweight and plant density when plants are grown in full daylight,70 per cent shade, and transferred between the two light environments.The implications of the hypothesis to self-thinning in naturalplant communities are discussed. Self-thinning, assimilate, plant development     

Competition between two similar plant varieties,green shrunk perilla and red shrunk perilla,in mixed cultures

Influences of plant density and time after seeding on the growth of two horticultural forms of perilla (Perilla frutescens var.crispa), green shrunk perilla (f.viridi-crispa) and red shrunk perilla (f.crispa), were examined in a mixed culture experiment. Relationships between mean individual plant weight and plant density in mixed populations were approximated by Ogawa's non-interaction type (NI-type) reciprocal equation. The density conversion factors in the equation for green and red perillas were always, respectively, smaller and larger than unity, suggesting that effects of a green perilla on the other individuals were always stronger than those of a red one in a mixed population. All coefficients in the NI-type reciprocal equation were expressed as functions of time after seeding. As a result, time trends of mean individual plant weights for both species in mixed populations could be reasonably estimated for different plant densities and mixed proportions. The results were also applied to Lotka-Volterra's equation. Time trends of Lotka-Volterra's competition coefficients for both plants could be calculated and were compared with those of density conversion factors.     

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

A new model is proposed to unite the logistic theory of plant growth and the 3/2 power law of self-thinning, which so far have been applied independently to growth analysis. To construct the model the following assumptions are made: a general logistic curve of mean plant weight, a modified form of the formula to show the rule of constancy of the final yield, which is generalized to cover the conditions of different combinantions of density and linear factor supply in a nonself-thinning population and a special population with a specific initial density which follows thew-ρ trajectory of the 3/2 power law type and has an exponential decrease in its density with biological time. Model calculations show that the Sukatschew effect is successfully formulated, that there should be a minimum factor supply below which self-thinning does not occur and that thew-ρ trajectory should be segregated acoording to the level of the linear factor supply.     

A Model for Growth and Self-thinning in Even-aged Monocultures of Plants

A theoretical model is derived from simple postulates to describethe rates of growth and mortality of plants in populations ofdifferent densities. The growth rate is described by a modificationof the logistic growth differential equation in which the increasein weight of an individual plant depends on its area, s_i ratherthan on its weight. The effective area for growth of a plantis reduced by an empirical function, f(s_i) with two terms: oneterm expresses the constraint imposed upon the increasing totalarea of plants by the limited physical area of the plot; theother term allows for a competitive advantage or disadvantagefor plants of varying sizes. Depending on the value of the parametercontrolling the relative competitive advantage term, intrinsicvariability between plants can be amplified or suppressed. Anindividual plant dies if the f(s_i) results in a negative growthrate for that plant. Computer simulations of the growth andsurvival of plants at different population densities were run.The results exhibit characteristics that appear realistic uponcomparison with published data: a survival of the fittest occurringduring thinning; a line of slope close to –3/2 boundingthe graphs of log weight versus log density; and the occurrenceof bimodality, associated with subsequent mortality, on frequencydistribution of log weight. computer logistic model, growth differential equation, density-effect, competition, mortality, self-thinning     

Effects of positive interactions, size symmetry of competition and abiotic stress on self-thinning in simulated plant populations

Background and Aims

Competition drives self-thinning (density-dependent mortality) in crowded plant populations. Facilitative interactions have been shown to affect many processes in plant populations and communities, but their effects on self-thinning trajectories have not been investigated.

Methods

Using an individual-based ‘zone-of-influence’ model, we studied the potential effects of the size symmetry of competition, abiotic stress and facilitation on self-thinning trajectories in plant monocultures. In the model, abiotic stress reduced the growth of all individuals and facilitation ameliorated the effects of stress on interacting individuals.

Key Results

Abiotic stress made the log biomass – log density relationship during self-thinning steeper, but this effect was reduced by positive interactions among individuals. Size-asymmetric competition also influenced the self-thinning slope.

Conclusions

Although competition drives self-thinning, its course can be affected by abiotic stress, facilitation and competitive symmetry.     

Competition between two naturalized dandelions,Taraxacum officinale weber andTaraxacum laevigatum DC., in mixed cultures with different levels of soil moisture

Taraxacum officinala andTaraxacum laevigatum were grown in mixed stands at various plant densities and mixing ratios with various levels of soil moisture to formulate the effect of soil moisture on the competitive relationship between the species. In pure stands, the mean plant weight—plant density relation for each level of soil moisture could be described by the reciprocal equation of the crowding effect. On the other hand, the response of mean plant weight to soil moisture content followed the reciprocal equation for a repulsive growth factor at the respective levels of plant density. By introducing the density conversion factor, the results of mixed stands could be successfully formulated from similar reciprocal equations. The dependence of density conversion factor on soil moisture content was also formulated. From these relations, a comprehensive formula was developed to describe the effects of plant density and soil moisture content on the growth of two species in mixed stands. Changes in the biomass in mixed stands were, examined by means of calculations based on the experimental results.     

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.     

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     

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 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.     

Studies on phosphatidylcholine model membranes. I. Size-heterogeneity effect on permeability measurements

The osmotic shrinking rate of unsonicated egg phosphatidylcholine (PC) liposomes in hypertonic NaCl was studied by determining the initial time rate of change of the reciprocal of the optical density, $\text{d(OD)}{}^{\mathrm{?1}}\text{d}\text{t}$, in a stopped-flow kinetics apparatus, $\text{d(OD)}{}^{\mathrm{?1}}\text{dt}$ was found to be a linear function of reciprocal OD and reciprocal PC concentration, where the linear parameters were quite different depending on the size distribution of liposomes in the dispersion. An approximate theoretical calculation of relative shrinking rates suggests that the larger liposomes mask the osmotic activity of smaller liposomes in the same dispersion. It is concluded that this method should only be used for comparing osmotic permeabilities of liposomes dispersions when both the OD and liposome size distribution of the dispersions are the same.