Two‐phase self‐thinning in stream‐living juveniles of lake‐migratory brown trout Salmo trutta L. Compatibility between linear and non‐linear patterns across populations?

The self-thinning rule establishes that, in resource limited populations, increased per-capita requirements by individuals with indeterminate growth may limit density. This invariant rule relating log-log linear declines of cohort density with increased body mass has been well established in sessile organisms. However, few studies have assessed the occurrence of self-thinning in mobile organisms and still fewer have demonstrated its operation in natural situations. In Bisballe Baeck, where brown trout individuals experienced the simultaneous operation of density-dependence on growth and mortality, increased body mass explained 89.0–98.4% of the variations in density. However, relative to the invariant linear line, the decline in density with increased body mass described distinctly nonlinear, two-phase trajectories. An initial phase of variable slopes continued until body mass attained ≈10.2 g, at which the direction of the trajectory switched to a second phase of substantially steeper slopes or severe declines of density per mass unit increment. As inferred by the highly significant relationships between these two sets of slopes and annual recruitment, the two phases were consistent with a major assumption of self-thinning: the rate at which density declined with increased body mass increased with the intensity of intraspecific competition. The results of this study provide prima facie evidence for the occurrence of self-thinning in a mobile species based on observational data. A further comparison among density–body mass relationships described for resident, anadromous and lake-migratory brown trout suggests consistent two-phase self-thinning patterns across populations.     

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.     

The difference between above- and below-ground self-thinning lines in forest communities

Quantifying the self-thinning process in various plant communities has been a long-standing issue in both theoretical and empirical studies. Most studies on plant self-thinning have centered only on aboveground parts, and rarely on belowground parts. There is still a general lack of comparison between above- and belowground self-thinning processes, especially for forest communities. The fundamental mechanistic difference and the functional association between above- and belowground competition indicate that the self-thinning process of belowground parts may be different from that of aboveground parts. We investigated the self-thinning lines for above-ground (M _A), below-ground (M _B), and total biomass (M _T), respectively, across forest communities in China. The results showed that neither the classical self-thinning rule (−3/2 exponent) nor the universal scaling rule (−4/3 exponent) can apply to all the self-thinning relationships across these forest communities and that the self-thinning lines for belowground biomass were flatter and lower than those for aboveground biomass across most of these forest communities.     

A method of selecting data points for fitting the maximum biomass-density line for stands undergoing self-thinning

Abstract A problem that has always existed in self-thinning studies is the lack of objectivity in determining which data points to include when fitting the maximum biomass-density line. This paper demonstrates a more objective method of selecting data points using data from 12 even-aged Pinus radiata stands undergoing self-thinning. The method involves the division of a cluster of data points into a specified number of intervals. From each interval, that point having the maximum stand biomass is selected to contribute to the fitting process. This process is repeated a specified number of times using an increasing number of intervals. Each time an estimate of the line is obtained. From these estimates, one can be chosen to represent the maximum biomass-density line. The method may also be useful in other areas involving the definition of a boundary condition.     

In Defence of the -3/2 Boundary Rule: a Re-evaluation of Self-thinning Concepts and Status

The -3/2 power rule, or -3/2 self-thinning rule, was accepted10 years ago as an important generalization, but has recentlybeen questioned by a number of authors. This paper assesseswhat remains of the rule. While it has been empirically establishedthat size-density trajectories followed by self-thinning plantpopulations do not necessarily follow a -3/2 slope, a more generalpower rule describing a density-dependent upper limit to meanshoot biomass per plant (the '-3/2 boundary rule') remains largelyintact. Principal component analysis (PCA) overestimates the steepnessof the thinning slope if y:x variance ratio is greater than1:1. Lonsdale's (Ecology 71: 1373-1388) overall mean PCA slopeof -0·6 for biomass-density suggests a true mean slopeclose to the theoretical value of -0·5. Reduced majoraxis (RMA) regression appears a reasonable approximation forthe -3/2 but not the -1/2 formulation of the rule. Fitting ofa linear functional relationship (LFR) is a more appropriateslope estimation procedure, not previously used for data onthinning. None of these procedures estimates a boundary linethat is not transgressed by any data point except through errorsof measurement. Mortality due to overcrowding ensues when a small, suppressedplant no longer holds its leaves high enough in the canopy tomaintain a positive carbon balance. It follows that LAI shouldremain constant during thinning, and that self-thinning theoryshould be developed in terms of maximum leaf area index andthe biomass required to support it. A derivation is presentedand some of its consequences are examined.Copyright 1995, 1999Academic Press Self-thinning, -3/2 power rule, -3/2 self-thinning rule, boundary line, size-density compensation, regression methods     

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.     

Trade-offs between the metabolic rate and population density of plants

The energetic equivalence rule, which is based on a combination of metabolic theory and the self-thinning rule, is one of the fundamental laws of nature. However, there is a progressively increasing body of evidence that scaling relationships of metabolic rate vs. body mass and population density vs. body mass are variable and deviate from their respective theoretical values of 3/4 and -3/4 or -2/3. These findings questioned the previous hypotheses of energetic equivalence rule in plants. Here we examined the allometric relationships between photosynthetic mass (M(p)) or leaf mass (M(L)) vs. body mass (beta); population density vs. body mass (delta); and leaf mass vs. population density, for desert shrubs, trees, and herbaceous plants, respectively. As expected, the allometric relationships for both photosynthetic mass (i.e. metabolic rate) and population density varied with the environmental conditions. However, the ratio between the two exponents was -1 (i.e. beta/delta = -1) and followed the trade-off principle when local resources were limited. Our results demonstrate for the first time that the energetic equivalence rule of plants is based on trade-offs between the variable metabolic rate and population density rather than their constant allometric exponents.     

Layering, the effective density of mussels and mass-density boundary curves

Overall intraspecific mass-density patterns have seldom been explored in animals. Instead, self-thinning studies have predominated. The analysis of 253 samples in a multilayered mussel showed that the classical approach is biased by layering or crowding effects, suggesting a transition zone between density independence and self-thinning, without a C–D effect. However, when the effective density (density corrected by layer effect) is used, space/allometric constraint expectations are met. Layering and crowding effects on self-thinning and the mass-density boundary should be common in mussels and other taxa showing aggregated distributions.     

Colorimetry and prime colours – a theorem

Human colour vision is the result of a complex process involving topics ranging from physics of light to perception. Whereas the diversity of light entering the eye in principle span an infinite-dimensional vector space in terms of the spectral power distributions, the space of human colour perceptions is three dimensional. One important consequence of this is that a variety of colours can be visually matched by a mixture of only three adequately chosen reference lights. It has been observed that there exists one particular set of monochromatic reference lights that, according to a certain definition, is optimal for producing colour matches. These reference lights are commonly denoted prime colours. In the present paper, we intend to rigorously show that the existence of prime colours is not particular to the human visual system as sometimes stated, but rather an algebraic consequence of the manner in which a kind of colorimetric functions called colour-matching functions are defined and transformed. The solution is based on maximisation of a determinant determining the gamut size of the colour space spanned by the prime colours. Cramer’s rule for solving a set of linear equations is an essential part of the proof. By means of examples, it is shown that mathematically the optimal set of reference lights is not unique in general, and that the existence of a maximum determinant is not a necessary condition for the existence of prime colours.     

Patterns of diversity, depth range and body size among pelagic fishes along a gradient of depth

Studies of geographical patterns of diversity have focused largely on compiling and analysing data to evaluate alternative hypotheses for the near‐universal decrease in species richness from the equator to the poles. Valuable insights into the mechanisms that promote diversity can come from studies of other patterns, such as variation in species distributions with elevation in terrestrial systems or with depth in marine systems. To obtain such insights, we analysed and interpreted data on species diversity, depth of occurrence and body size of pelagic fishes along an oceanic depth gradient. We used a database on pelagic marine fishes native to the north‐east Pacific Ocean between 40°N and 50°N. We used data from the Pacific Rim Fisheries Program that were obtained from commercial, management and scientific surveys between 1999 and 2000. Depth of occurrence and maximum body length were used to assess the distributions of 409 species of pelagic fishes along a depth gradient from 0 to 8000 m. A presence–absence matrix was used to classify the depth range of each species into 100‐m intervals. Atmar & Patterson's (1995 ) software was used to quantify the degree of nestedness of species distributions. Pelagic fish species diversity decreased steeply with increasing depth; diversity peaked at less than 200 m and more than half of the species had mean depths of occurrence between 0 and 300 m. The distribution of species showed a very strong nested subset pattern along the depth gradient. Whereas species with narrow ranges were generally restricted to shallow waters, wide‐ranging species occurred from near the surface to great depths. The relationship between maximum body size and mean depth range differed between teleost and elasmobranch fishes: being positive for teleosts, but negative for elasmobranches. Results support hypotheses that some combination of high productivity and warm temperature promote high species diversity, and reject those that would attribute the pattern of species richness to the mid‐domain effect, habitat area, or environmental constancy. The data provided a clear example of Rapoport's rule, a negative correlation between average depth range and species diversity.     

Allometry of territory size and metabolic rate as predictors of self-thinning in young-of-the-year Atlantic salmon

1. Self-thinning is a progressive decline in population density caused by competitively induced losses in a cohort of growing individuals and can be depicted as: log₁₀ (density) = c − β log₁₀ (body mass).
2. In mobile animals, two mechanisms for self-thinning have been proposed: (i) the space hypothesis predicts that maximum population density for a given body size is the inverse of territory size, and hence, the self-thinning slope is the negative of the slope of the allometric territory-size relationship; (ii) the energetic equivalence hypothesis predicts that the self-thinning slope is the negative of the slope of the allometric metabolic rate relationship, assuming a constant supply of energy for the cohort.
3. Both hypotheses were tested by monitoring body size, population density, food availability and habitat for young-of-the-year Atlantic salmon ( Salmo salar ) in Catamaran Brook, New Brunswick. The results were consistent with the predictions of the space hypothesis. Observed densities did not exceed the maximum densities predicted and the observed self-thinning slope of −1·16 was not significantly different from the slope of −1·12, predicted by the allometry of territory size for the population under study.
4. The observed self-thinning slope was significantly steeper than −0·87, predicted by the allometry of metabolic rate, perhaps because of a gradual decline in food abundance over the study period. The decline in density was more rapid in very shallow sites and may have been partly caused by a seasonal change in water depth and an ontogenetic habitat shift rather than solely by competition for food or space.
5. The allometry of territory size may be a useful predictor of self-thinning in populations of mobile animals competing for food and space.     

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.     

Three Interpretations of the Self-thinning Rule

The self-thinning rule states that w, the mean biomass per plantin a dense monospecific stand, is related to p, the number ofplants in a unit area of that stand, by the power law relationshipw = Kp^–, where is approximately three-halves. The supportfor this rule, theoretical as well as experimental, has thusfar been largely empirical. In an effort to provide a firmertheoretical basis for it, three different and somewhat independenttheoretical models are propounded and shown to lead to powerlaw relationships with characteristic exponents in the vicinityof three-halves. theoretical models, mathematical model, three-halves law, density-effect, self-thinning     

The effect of clipping on self-thinning in Trifolium pratense

The self-thinning rule describes mortality in a crowded even-aged stand as a function only of biomass accumulation. We tested this prediction by stopping biomass accumulation with clipping treatments in Trifolium pratense. Moderate levels of clipping slowed thinning mortality considerably, but did not stop it entirely.     

Comparison of self-thinning models: an exercise in reasoning

Self-thinning of forest stands is one of the clearest and best-documented examples of natural selection. Besides their theoretical interest, understanding of self-thinning is important for forest practice because it produces estimates of stand density and stocking. There is a considerable diversity of views on the processes causing self-thinning, predicting variables, and analytical form of models. The most popular model was proposed by Reineke (J Agric Res 46(7):627–638, 1933) over 70 years ago. This study compares existing models of self-thinning and provides evidence that the virtually unknown model developed by Artur Nilson describes self-thinning more realistically than Reineke’s. While in the Reineke model the rate of mortality (the slope of self-thinning line) is assumed to be constant, it changes from 0 to −2 in Nilson’s model. As a result, Nilson’s model is slightly but consistently more accurate than Reineke’s. Although both models are empirical, their analysis suggests several general conclusions about self-thinning.     

Environmental correlates of body size distributions of European springtails (Hexapoda: Collembola)

Aim Species–body size distributions (SBDs) are plots of species richness across body size classes. They have been linked to energetic constraints, speciation–extinction dynamics and to evolutionary trends. However, little is known about the spatial variation of size distributions. Here we study SBDs of European springtails (Collembola) at a continental scale and test whether minimum, average and maximum body size and the shapes of size distributions change across latitudinal and longitudinal gradients and whether SBDs of islands and mainlands differ. We also test whether the island rule and the positive body size–range size relationship of vertebrates also holds for Collembola. Location Europe. Methods We use a unique data set on the spatial distributions of 2102 species of European springtails across 52 countries and larger islands together with associated data on body size, area, climate variables, longitude and latitude. Differences in the central moments of SBDs are inferred from simultaneous spatial autoregression models. Results The SBD of the European Collembola and its largest suborder Entomobryomorpha is unimodal and symmetrical. Average, minimum and maximum body weight and the skewness of the mainland/island SBDs peaked at intermediate latitudes. We could not find simple latitudinal gradients in minimum and maximum body weight. Average and maximum body size increased with country/island area in accordance with the island rule in vertebrates, while minimum body size did not significantly differ between islands and mainlands. Finally, we found a weak but statistically significant positive correlation of range size and body size. Main conclusions We provide evidence for differences in body size distributions between islands and mainlands that are in part in line with the island rule in invertebrates. We also find evidence for an interspecific body size–range size relationship similar to that of vertebrates although the vertebrate pattern is much stronger than the springtail pattern. Our results on latitudinal gradients of maximum and average body size imply the need to account for species richness and area effects in the study of latitudinal gradients in body size. We recommend implementing sample size and area effects in the study of body size distributions on islands and mainlands.     

数据点选择与参数估计方法对杉木人工林自疏边界线的影响

自疏边界线是指植物种群发生密度依赖死亡时种群最大收获量的上边界线。已有研究由于在拟合自疏边界线的过程中对数据点的选择和参数估计的方法存在诸多的差异,进而导致产生对自疏法则的争议。该研究采用26年生杉木(Cunninghamia lanceolata)人工林的定位观测数据,对视觉法、死亡率法、等距区间法和相对密度法等4种数据点选择方法以及最小二乘法、降维分析法、分位数回归法和随机边界方程等4种参数拟合方法进行对比分析,以探寻客观选择自疏拟合数据和正确拟合方法的途径。比较4种不同的数据选择方法得出:视觉法具有较强的主观性;对于没有发生非密度依赖死亡的林分,死亡率法可以准确地确定林分自疏的起始点;等距区间法可以减少非密度依赖死亡的影响,得到的数据点能充分反映林分的自疏过程;相对密度法可以保证临界密度阈值以上的数据点拟合林分自疏边界线的有效性,并能排除非密度依赖死亡的影响。比较分析4种不同的拟合方法发现:最小二乘法和降维分析法拟合的林分自疏边界线均从实测数据"中心"穿过,与林分自疏边界线为林分收获量上边界线的涵义不相符合,无法真实反映林分的自疏进程;分位数回归和随机边界方程的拟合结果均与实测数据一致,能够较为准确地反映林分自疏的真实过程,但二者的统计推断要求都比较严格。分位数值的正确选取和残差足够小且趋于0,是分位数回归法和随机边界方程能否正确反映林分自疏动态的前提。     

Biological power laws and Darwin's principle

Assuming that the repertoire of responses by living systems to perturbation gives a measure of their Darwinian fitness in a rapidly fluctuating environment, those that fulfill allometries (power laws) are described by means of catastrophes, whose variables and parameters are smooth functions of biological attributes. Using empirical allometries from a given system as input, a method is proposed to construct its associated catastrophe, allowing specific predictions on its susceptibility to perturbation and related properties, based on general results from catastrophe theory. The method is discussed within the macroecological context, and an example is provided by applying it to ecological systems that satisfy the self-thinning rule.     

Decline in exotic tree density facilitates increased plant diversity: the experience from Melaleuca quinquenervia invaded wetlands

The Australian tree Melaleuca quinquenervia (melaleuca) formed dense monocultural forests several decades after invading parts of Florida and the Caribbean islands. These dominant forests have displaced native vegetation in sensitive wetland systems. We hypothesized that native plant diversity would increase following recent reductions in density of mature melaleuca stands in south Florida. We therefore examined data on changes in melaleuca densities and plant species diversity derived from permanent plots that were monitored from 1997 to 2005. These plots were located within mature melaleuca stands in nonflooded and seasonally-flooded habitats. Two host-specific biological control agents of melaleuca, Oxyops vitiosa and Boreioglycaspis melaleucae, were introduced during 1997 and 2002, respectively. Also, an adventive rust fungus Puccinia psidii and lobate-lac scale Paratachardina pesudolobata became abundant during the latter part of the study period. Overall melaleuca density declines in current study coincided with two to four fold increases in plant species diversity. The greatest declines in melaleuca density as well as the greatest increases in family importance values and species diversity indices occurred in nonflooded as compared to seasonally-flooded habitats. Most pioneer plant species in study sites belonged to Asteraceae, Cyperaceae, Poaceae, and Ulmaceae. The rapid reduction in melaleuca density and canopy cover during the study period may be attributed to self-thinning accelerated by the negative impact of natural enemies. Densities of other woody plants, particularly Myrica and Myrsine, which were sparsely represented in the understory by a few suppressed individuals also declined during the same period, possibly due to infestation by the generalist lac-scale. These findings indicate that natural-enemy accelerated self-thinning of melaleuca densities is positively influencing the native plant diversity and facilitating the partial rehabilitation of degraded habitats.