Models for plant self-thinning

. Plant self-thinning, which is density-dependent mortality, has several observed characteristics, including a certain mathematical relationship between growth and density. The original equation that describes self-thinning is log (cid:1) w ¼ C (cid:2) ð 3 = 2 Þ (cid:3) logdensity, (cid:1) w = mean weight. The basic equation is supported by data from ecology and forestry, but there have been a number of reported slopes that differ from (cid:2) 3/2. This study proposed that change in plant density over time decreases exponentially and that plant growth (weight or volume) increases over time according to one of two models: either exponential growth or sigmoid growth. Exponential growth with a ﬁ nite time limit, in conjunction with exponential decrease in density, led to the equation log w = C + a / c 9 log density, where a / c < 0, w is total weight, but did not imply any particular value for its slope. Sigmoid growth, in conjunction with exponential decrease in density, led to the equation log ( w /( a 1 + a 2 w )) (or v ) = C + a 1 / c 9 log density, where a 1 / c < 0, w (or v ) is a total, but did not imply any particular value for its slope. For the data examined, exponential density decrease was supported by all data sets. For eight data sets, exponential growth was supported and in 7 of 8, log w = C + b log density was a good ﬁ t with a / c a good predictor of b . For 31 data sets, sigmoid growth was supported, and in 29 of 31, log ( w /( a 1 + a 2 w )) (or v ) = C + b log density was a good ﬁ t with a 1 / c a good predictor of b . The numerical value of b can be regarded as an index, and there was some indication that a wide range of values (or lack) is associated with a wide (or narrow) range of environments to which the species is adapted. For the larch data, the initial spatial distribution of trees was aggregated but changed toward a random distribution of individuals over time.


INTRODUCTION
Plant self-thinning describes a process in crowded, even aged populations of one species in which mortality is believed to occur due to competition between members of the population, and thus is density dependent, which results in a definite relationship between plant size and number of individuals (Yoda et al. 1963, Harper 1977, Mohler et al. 1978, Weller 1987, Zeide 1987. It has been observed to occur at higher densities and continue over time, has been observed to be greater at lower light intensities, and has been observed to be greater on sites with richer soil conditions (Harper 1977). Reineke (1933) proposed a formula relating density and average stem diameter at breast height for trees ðlog density ¼ A À 1:605 Â log dÞ. Yoda et al. (1963) proposed a formula relating mean weight and density ðlog w ¼ C À ð3=2Þ Â log densityÞ, usually referred to as the -(3/2) power rule of plant self-thinning; these authors provided a detailed explanation, which they termed a "crude approximation," for linearity on a log-log plot and a slope of -(3/2). The two formulas can be shown to be equivalent (Yoda et al. 1963, Avery andBurkhart 2002) by substituting weight = c 9 d 2.4 into Yoda et al.'s formula.
Data on wheat plants (Puckridge and Donald 1967), forestry data in Britain (Bradley et al. 1966), and data from their own experiments were examined by White and Harper (1970) in an attempt to discover whether the rule is widespread. Often, the slope of the linear regression line, on a log-log plot, was approximately -(3/2). The constant C differs and is a characteristic of the species under the given conditions. These reported studies used the mean weight (or volume in the case of trees) rather than the total weight or volume per unit area. Weller (1987) gives a detailed discussion regarding the use of mean weight compared to the total weight per unit area; he advocates using total weight. He observes that the denominator of the mean is the number of individuals per unit area, which is also the density, and therefore, weight = ∑w/n will be correlated with density = n/area. Also, the mean will increase when below average individuals die, even if the survivors have not grown. Last, he notes that the slope of log w vs. log density can be obtained by adding 1 to the slope of log w vs. log density. Subsequent to Weller's report, there have been articles that challenge and articles that defend the rule. On the basis of numerous reported slopes differing from -(3/2) (or -(1/2)), Lonsdale (1990) concluded there is no evidence to support a thinning rule. Weller (1990) suggested that the dynamic self-thinning line and the species boundary line are different concepts and that the dynamic thinning line may fall below the boundary line for various reasons. Sackville Hamilton et al. (1995) later defended the rule as an upper species limit (boundary) but concluded that the -(3/2) dynamic self-thinning rule is theoretically untenable.
Some recent articles have related plant mass to resource use and metabolism and thence to density (Enquist et al. 1998, Niklas et al. 2003. These authors are using maximum population density in this context and conclude that biomass scales as the -(4/3) power of maximum density, which is equivalent to a slope of -(4/3) on a log-log plot. Data from tree-dominated communities support their model. Pretzsch (2006) examined four tree species in un-thinned plots relative to this slope, Reineke's stand density index, and Yoda et al.'s rule; his results do not appear to support any of the three.
Others (Sterba 1987, Li 2002, Vospernik and Sterba 2015 discuss the relationship between the competition-density (C-D) rule and the self-thinning rule. Sterba (1987) and Vospernik and Sterba (2015) use the C-D rule, with stem diameter in place of volume (which provides a direct link to the Reineke equation), to estimate the maximum self-thinning line (representing potential density) for a given species.
Other recent articles have focused on the best statistical methods to estimate the self-thinning boundary line and to determine whether the slope is invariant across all species or only invariant within a species (Zhang et al. 2005, Zhang et al. 2013, Weiskittel et al. 2009, VanderSchaaf and Burkhart 2010, Cao and Dean 2015. Plant growth is a result of a number of processes. For example, above-ground dry matter production is directly related to transpiration (Arkley 1982, Perry et al. 2009). And growth, measured as weight or volume, is a function of time. Many researchers believe that plant growth, when observed over a long enough period of time, must exhibit a sigmoid model of growth. The sigmoid model of choice has usually been the logistic function (Fresco 1973, McMartin 1979, Causton and Venus 1981, Morris and Silk 1992, Weiner et al. 1998, Yin et al. 2003. Weiner et al. (1998) attempted to fit both logistic and exponential functions to their data. They commented that "because many plants were still growing exponentially by the end of the study, it was not possible to fit the logistic model to all plants. " Yoda et al. (1963) commented that "in July, an unusually long spell of rainless days caused the death of a greater part of standing plants, and the experiment was forcibly discontinued." Experiments with herbaceous plants are often conducted in a greenhouse, which modifies the normal climate. Data in forestry tables are produced with an eye toward management that results in thinning suitable to maximize yield. These normal yield tables are based on temporary sample plots; the self-thinning law should be tested using long-term data of permanent sample plots or yield tables based on such data (Zeide 1987). It therefore seems necessary to consider two possible models for growth: exponential, for which there is a limit to the time that a plant may grow according to this model, and sigmoid, for which no such limit seems necessary.
Like growth, plant density is a result of many factors and is a function of time. Harper (1977) comments that "the time trend of the self-thinning process can also be analyzed by following the change in the log of plant numbers. A linear relationship between log numbers and time implies a constant exponential decrease-a risk of death to the individual which, like the decay of an isotope, remains constant over time." Such a decrease is the type II survivorship curve (Deevey 1947). If such a model is to be used for density decrease due to self-thinning, it seems necessary to impose a limit to the time this occurs because, at some point, distances between survivors will be too great for competitive effects to exist, and mortality will be due to densityindependent causes.
There are four goals of this study. First, the above statements regarding growth and density will be formalized as mathematical hypotheses, and a method of relating these will be introduced, with the objective of obtaining formulas that relate growth and density. Second, a variety of data sets will be examined to determine whether the hypotheses are supported by the data. Third, assuming the data supports the hypotheses, the same data sets will be examined to determine to what extent, if any, the formulas that relate growth and density are supported. Last, the process of self-thinning and its slope indices will be examined relative to ecological attributes of species that are believed to be undergoing self-thinning.
Hypotheses for density and growth (weight or volume) The density of a population and the growth of plants are both functions of time, which is usually recorded as the number of days since emergence in the case of herbaceous plants, or age in years for trees. Density: The rate at which population density per unit area, denoted by x, decreases at any instant in time is proportional to the density at that instant, symbolized by dx/dt = cx, c < 0, t 0 < t < T 1 , which yields This is the type II survivorship curve (Deevey 1947). Growth (weight or volume), where this variable is defined as the total amount per unit area: Either the rate of growth at any instant in time is proportional to total amount per unit area at that instant, symbolized by or dw=dt ¼ a 1 w þ a 2 w 2 , a 1 > 0, and a 2 < 0, (Lotka 1956), which yields the logistic function Lotka's method of solution uses the Taylor series for dw/dt and the associated characteristic equation when the series terminates with the term of degree two. It is also possible to integrate dw=dt ¼ a 1 w þ a 2 w 2 using the method of partial fractions, followed by substituting w 0 =ða 1 þ a 2 w 0 Þ for e a 1 C , where C is the constant of integration, to obtain Eq. 4. As t ! 1; w ! ða 1 =Àa 2 Þ: Eq. 2 will be fitted to points denoted as (time, number of individuals), which will make x(t) a discrete variable. For the purposes of the hypothesis and subsequent mathematics, x(t) must be regarded as a differentiable function. Since weight and volume are measured rather than counted, no such statement needs to be made for w(t) or v(t). The difference between these types of variables in the context of plant self-thinning has been discussed by Zeide (1987).

Introducing the chain rule
The three variables, time, total weight or volume, and density, can be related by means of the chain rule, dw=dt ¼ ðdw=dxÞ Â ðdx=dtÞ. Substituting dw=dt ¼ aw for weight and dx=dt ¼ cx for density in this formula yields aw ¼ ðdw=dxÞ Â cx; which is a separable differential equation whose solution is Because all referenced reports use log 10 , natural logarithms were converted to log 10 , which are written as log x. The equation (compare to Eq. 1) becomes for which the general equation would have b in place of a/c. If dw=dt ¼ a 1 w þ a 2 w 2 ; substitution in the chain rule yields a 1 w þ a 2 w 2 ¼ ðdw=dxÞ Â cx; which is a separable differential equation whose solution is As above, natural logarithms were converted. The equation becomes for which the general equation is If it is established that the variables on opposite sides of Eq. 1 or Eq. 7 exhibit a linear relationship, then b is the true slope whose value can be estimated, and the predicted value, either a/c or a 1 /c, may, or may not, be close to the estimate. Eq. 7 is algebraically equivalent to

Description of data sets
There are 27 cited data sets in this study, which have been grouped into three categories. Initially, these were divided into two groups: eight that exhibited exponential growth and 19 that exhibited sigmoid growth. As explained at the beginning of the Discussion, the group that exhibited sigmoid growth underwent a further reorganization in which all 19 data sets were analyzed as a complete set (subset A) of all data, and as a set of 12 (subset B) for which some age classes were omitted. Data sets that are in subset B are identified by a " †" symbol. The data sets include a diversity of situations as to species, experimental methods, and environmental conditions. The leftmost entry in Table 1 gives the citation for the data, including the senior author and other specifics; the remaining tables have citations numbered identically. The first nine entries are for herbaceous plants, of which six were conducted in the field while the remaining three were conducted under greenhouse conditions. The remaining entries are for trees, of which all but four are from yield tables. These four data sets are from long-term studies of Douglas fir and western larch plus a compilation of many data sets for lodgepole pine. The data for western larch are from stands in the Coram Experimental Forest in northwest Montana that were naturally regenerated in the early 1950s and never thinned nor treated in any way. The data for Douglas fir are from a plantation, established with two-yearold seedlings in 1949 at the Iron Creek site in western Washington and also contains unthinned plots (Curtis and Marshall 2009). For the herbaceous plants, w is g/m 2 and x is number of individuals/m 2 . For the trees, the original data gave v as ft 3 /ac (the data from Bradley et al. 1966 required a conversion to ft 3 ) and x as number of trees/ac.
The choice of data sets to use in this study was subjective, but not capricious. Once chosen, however, statistical calculations were performed and results for all data sets are reported. Two initial adjustments in the cited data were made prior to statistical analysis. Puckridge and Donald (1967) did not report a density for harvests 2 and 5. These values were estimated from Fig. 5 of White and Harper (1970). The original data from Coram 1 and Coram 2 report ages in years as 9, 14, etc., but can be 1-9, 6-14, etc. To reduce exaggeration in a 1 , the first age group was omitted from volume and statistical calculations.

Statistical analysis
In general, ordinary least-squares (OLS) regression is acknowledged to be the best method to estimate the mean of one random variable given a fixed value of another. In this study, OLS is employed to estimate the fit of growth and density vs. time, because time is fixed. Sokal and Rohlf (1995) and Zhang et al. (2005) provide a ❖ www.esajournals.org discussion of several methods for estimating functional relationships such as Eqs. 1, 7 (the self-thinning boundary line) where both variables are truly random, and OLS regression is judged to be inappropriate. The above references provide a comparison among several methods and the associated formulas to estimate the parameters. In this study, both major (or principal) axis (MA) and reduced (or standardized) major axis (RMA or SMA) regressions are used. When the variances of  Table; Tr, Treatment. Results are for x ¼ x 0 e ct from ordinary least-squares regression (one-tailed). The first nine citations are herbaceous species for which x and x 0 = number of individuals/m 2 , and t = days since germination (t = weeks for citation 6). The remainder are tree species for which x and x 0 = number of trees/ac or number of trees/ha, and t = age in years. The first x 0 value is number of trees/ac, and the second is number of trees/ha; x 0 (ha) = 2.471 9 x 0 (ac). The number that precedes some N values is the site index or yield class. Plot number and Tr 5 are the experiment identification numbers used by Yoda et al. and Puckridge and Donald. † Identifies data sets for which x(t) has been recalculated after some early age classes were omitted. Of these data sets, citations 11, 13, 16, 17, and 21 also have some older age classes omitted from the calculated x(t).
*P < 0.05; ÃÃ P < 0.01; ÃÃÃ P < 0.001. the two random variables (growth and density) are very different, RMA may be preferred over MA.

Change of units
As noted, the original data for trees were in ft 3 / ac. The parameters a, c and a 1 are dimensionless rates and are not affected by the choice of units. A conversion to m 3 /ha was employed for tree data (citations 10-27) and affects the exponential decrease function and the logistic growth function. For the exponential decrease function, x 0 (ha) = 2.471 9 x 0 (ac), which is noted in Table 1. For the logistic growth function, the only difference between the two sets of units is in the parameter a 2 . To convert, a 2 ha ð Þ ¼ a 2 ac ð Þ= ð0:0283 Â 2:471Þ ¼ a 2 ac ð Þ=ð0:0699293Þ; which is noted in Table 3.
The situation for the MA and RMA regression equations and associated confidence intervals that are shown in Table 5 follows. Eq. 7 contains logðv=ða 1 þ a 2 vÞÞ on the left side and log x on the right side. In converting from ft 3 /ac to m 3 /ha, the denominator on the left side is unchanged and dimensionless (the units of a 2 and v are reciprocals). Both log v and log x do change with a change in units. Sokal and Rohlf (1995) provide a discussion and formulas for both RMA and MA regression lines and associated confidence intervals. For RMA, standard deviations of both variables are necessary. For MA, the covariance, the variance of log v, and the larger eigenvalue of the covariance matrix are components of the estimated slope, and both eigenvalues are components of the confidence interval. Let A and B be conversion factors, X and Y be random variables, and Var be the symbol for variance. Then, Var (log AX) = Var(log A + log X) = Var(log X). If E denotes expected value and Cov the symbol for covariance, then following the definition of covariance, Covðlog AX; log Therefore, the covariance matrix for logarithmically transformed variables, and its eigenvalues and corresponding eigenvectors, is invariant under a change in units. So, the estimated slopes and associated confidence intervals for MA and RMA are also invariant under a change in units.
The means of log AX and log BY do differ from the means of log X and log Y, so a change in units generally changes the value of the intercept C.
The new value of C can be determined from the requirement that the regression line passes through the bivariate mean.
Eqs. 2, 3, 4 will be fitted to various data sets. Eqs. 2, 3 will be linearized by taking natural logarithms, after which parameters are estimated. Eq. 4 cannot be linearized; its parameters are estimated with a program noted in the Acknowledgments.

RESULTS
In eight of the data sets, it was possible to fit an exponential growth function. An attempt was made to fit a logistic function to these same data sets. In six of these, no estimated values for any of the three parameters of the logistic function were generated. Values of the three parameters were generated from the other two data sets. In one, the probabilities ranged from 0.15 to 0.52; in the other (citation 7), the probability for a 1 < 0.05, but the other two probabilities were 0.09. The conclusion is that these data sets represent exponential growth for the time intervals of the respective experiments. In the remaining data sets, it was possible to fit both an exponential and a logistic growth function. Weiner et al. (1998) were also able to fit both an exponential and a logistic function to some of their data. At this juncture, the data sets were divided into two groups: those that exhibit exponential growth, namely those for which only a satisfactory exponential function could be fit, and those that exhibit sigmoid growth, namely those for which both an exponential and a logistic function could be fitted. Table 1 shows the citation, species, and exponential decrease function x(t) for all data sets. All are significant at 0.05 or less. All but two x(t) are significant at 0.001. Table 2 shows the exponential growth function for the data sets that exhibit exponential growth, and Table 3 shows the logistic growth function for the data sets that exhibit sigmoid growth. No attempt has been made to provide notation that differentiates between population parameters and sample statistics for a, c and a 1 . Table 4 shows results for the self-thinning equation from those data sets that exhibit exponential ❖ www.esajournals.org growth. The first set of results is for a one-tailed linear correlation test between log w and log x, the variables from Eq. 1. All but one of the eight, citation 9, show a significant correlation at 0.05 or less. The next set of results shows the estimated values of the intercept and b (C and b) in Eq. 1 from both MA and RMA regression. The final results in Table 4 show the predicted value of b from Eq. 5, which is a/c, and the 95% MA and RMA confidence intervals for b. Table 5 shows results for the self-thinning equation from those data sets that exhibit sigmoid growth. The correlation tests and the regressions are now for Eq. 7, rather than Eq. 1. Otherwise, the format of the results is the same as in Table 4, with a 1 /c from Eq. 6 now the predicted value of b. Fig. 1 shows data from citation 1 plus a plot of the fitted exponential growth function. This is typical of all eight such data sets and accords with the conclusion that these data sets exhibit exponential growth during the time interval of the respective experiments. Fig. 2 shows data from citation 14, Douglas fir, plus plots of the exponential and logistic growth function that were fitted to the data. This is typical of all remaining data sets. Probabilities for the logistic functions were generally smaller than those for the corresponding exponential functions. Visually, it also seems reasonable to conclude that these data sets represent sigmoid (logistic) rather than exponential growth. That is the conclusion here, and the reason for the separation into the two groups.

Actual size-density values vs. fitted size-density functions
After the initial adjustments to the three data sets (citations 6, 25, 26) mentioned in the Methods section, all data sets were examined in the following two ways. Plots of number of individuals vs. time were graphed along with the fitted exponential decrease function. Second, plots of actual weight or volume vs. time were graphed along with the fitted function, either exponential or logistic. Fig. 3 is typical of almost all data sets for exponential decrease. The actual number of individuals in the first two age classes is much greater than predicted by the (original) exponential decrease function x(t). So, actual initial mortality is much greater than expected compared to the fitted x(t). Mortality as a result of competition from other members of the population is densitydependent mortality, with the descriptor selfthinning. However, x(t) is fitted to the decreasing number of individuals over time, regardless of the cause(s) of mortality to those that died. One might make the initial assumption that total mortality is composed of density-independent and density-dependent factors which act additively to produce c. However, it is just as valid to suppose that there is, at least initially, interaction between the two types of factors so that total mortality is greater than expected. Fig. 4, from the same data set that produced Fig. 3, shows a graph of actual total volume from 20 to 160 yr of age along with the original fitted logistic growth T. subterraneum 7 554.00 0.00797 * 0.603 Notes: N, number of data points. Results are for w ¼ w 0 e at from ordinary least-squares regression (one-tailed). w and w 0 are in g/m 2 and t = days since germination. Citations are numbered identically to those in Table 1.
function. Actual total volume is less than that predicted by the logistic function for the first two age classes; this is typical of nearly all data sets that exhibit sigmoid (logistic) growth. And it is what would be expected if mortality is greater than expected for these two age classes. Accordingly, both x(t) and v(t) were recalculated for most of these data sets. Generally, the difference between actual and fitted x(t) is 20% or more in those that were recalculated.
To follow up from the previous paragraph and Methods, both of the following are presented in Tables 1, 3, 5: (1) All 19 data sets that exhibited sigmoid growth are analyzed with the complete set of data reported in the cited publications, and (2) 12 of the 19 are analyzed with some age classes omitted. For example, citation 11 has two entries in these tables, denoted as 11 and 11 †. For 11, growth and density functions are calculated using all data; for 11 †, growth and density  Table 1. For citation 6, w and w 0 = g/m 2 and t = weeks since germination. The rest are trees for which v and v 0 = total ft 3 /ac or total m 3 /ha, t = age in years, and units of a 2 are ac/ft 3 or ha/m 3 . For ft 3 /ac, use a 2 (ac/ft 3 ), and for m 3 /ha, use a 2 (ha/m 3 ). a 2 (ha/m 3 ) = a 2 (ac/ft 3 )/0.0699293. v max = a 1 /Àa 2 . There are probability values P < p for a 1 , a 2 , and 1 + (a 1 /(a 2 v 0 )). The value shown is the largest of the three. For citations 6 and 25, p values for a 1 and a 2 < 0.01, and for citation 27, p values for a 1 and a 2 < 0.01 and 0.05. If X = the estimated value of 1 + (a 1 /(a 2 v 0 )), then v 0 ¼ ða 1 =a 2 Þ Â ð1=ðX À 1ÞÞ.
† Identifies those data sets for which some early age classes are omitted from the estimated v(t) to match those of the estimated x(t) in Table 1. Of these data sets, citations 11, 13, 16, 17, and 21 also have some older age classes omitted from the calculated v(t).
‡ Standard volume formulas for trees this small are not in use, so volumes were calculated using the formula for the volume of a cone. Ages are reported as 9, 14, etc., but can be 1-9, 6-14, etc. *P < 0.05; ***P < 0.001.
functions are calculated using a reduced data set that has some age classes omitted. N is the number of data points for the given data set. A similar difference between the actual number of individuals and x(t) is also true for the data sets that exhibit exponential growth. Except for citation 7, all of these data sets have 5-7 data points. Because of the relatively small number of data points, it was judged better to retain all data for statistical calculations.
The only exception to the difference shown in the two earliest age classes in Fig. 2 is the data set for Iron Creek (citation 18). In 1949, two-year-old Douglas fir seedlings were planted at the Iron Creek site in western Washington. Actual mortality is definitely less than expected compared to the fitted x(t). As noted, this is a plantation with regular spacing between plants. Also, it is normal to plant only healthy, vigorous seedlings. Fig. 4 also shows one other difference between actual volume and the fitted logistic function. Actual volume is greater than that predicted by the logistic function for the two oldest age classes. A similar difference is also apparent in Fig. 2 for a different data set. This is typical for all data sets that exhibit sigmoid growth. One interpretation of this difference is that these trees are now large mature trees for which mortality due to self-thinning is no longer a major factor. In Figs. 2, 3, 4, there is a point on the graphs at which a density of 81 trees/ac (200 trees/ha) is shown. This density is shown only as a reference point. For equal spacing as in a plantation, whether the trees in every row are aligned in columns or whether trees in alternating rows are aligned in columns, distances between trees will be~23-25 ft (7 + m) for this density. Although shading will occur at certain times, this distance may be too great to create a competitive effect for water or soil nutrients between nearest neighbors for many species. In most cases, the percent difference between actual and fitted is slight for the oldest age classes. Citations 11,13,16,17,and 21 are the only ones for which a few of the oldest age classes were omitted, after which logistic and density functions were recalculated.
A density of 784 trees/ac (1937 trees/ha) corresponds to a distance of 6-7 ft (2 m) between nearest neighbors. Citations 13, 14, and 15 have an estimated x 0 that is definitely smaller than this density. Trees in these data sets may not have experienced a great deal of self-thinning.

Self-thinning slopes as indices
A considerable portion of the results, as well as the equations in the section on Methods, deals with slopes of lines that relate logarithmically transformed variables. Harper (1977) and Zeide (1987) place importance on the relationship between production and population. If it is true that there is a linear relationship between log w (or v) and log x, then the numerical value of the slope of this line is an index number that relates the observed yield and population size over time. If b = À1, adding log n, n = 2, 3, . . ., to both sides of Eq. 1 produces log n þ log w ¼ C À log x þ log n, from which log nw ¼ C À log x n . Notes: CI, confidence interval; MA, major axis; N, number of data points; RMA, reduced major axis. Results are for onetailed linear correlation test between log w and log x; estimated intercept and slope for MA and RMA regression for log w = C + b log x; predicted value of b, which is a/c; 95% MA and RMA confidence intervals for b. Citations are numbered identically to those in Table 1.
If n = 2, then halving the number of individuals through mortality results in a doubling of total biomass. Since there are now half as many individuals as there were previously, the average survivor is now four times as large as previously. If b = À0.5, adding 0.5 log n to both sides of Eq. 1 results in log w ffiffiffi n p ¼ C À 0:5 logðx=nÞ. If n = 4, then reducing the number of individuals to one-fourth of the previous amount results in a doubling of total biomass, which means the average survivor is now eight times as large as previously. Production of the fraction that survives more than compensates for lost production of those that died, and the numerical value of the slope is an index value of the magnitude of increase in size of the survivors. The ratio of the absolute values of two slopes, larger divided by smaller, defines the ratio of relative size increase. To illustrate both examples, Fig. 5 shows two graphs: one from citation 1 of Table 4, with MA slope = À0.5385, and the other is hypothetical with a slope of À1.0. Continuing with the assumption of a linear relationship, and assuming slopes vary among species, and perhaps 15 À0.918, *** within a species in different environments, one may then compare and contrast. One possible goal is to relate this parameter to other parameters or attributes of biological significance. Since Eq. 7 differs from Eq. 1 by the term "log (a 1 + a 2 w)," the situation relating a decrease in number of individuals to an increase in total biomass is more complicated. For longleaf pine (citation 10) which has a slope of À3.488, a doubling of volume equates to a reduction in number of individuals that varies from about a third to a half, with an increase in individual size varying from about three to four times as much as before mortality. For western hemlock (citation 20 †) which has a slope of À2.4609, a doubling of volume equates to a reduction in number of individuals that varies from about a third to an eighth, with an increase in individual size varying from about three to fourteen times as much before mortality.
If the goal is to compare two species, one may consider the ratio of two slopes. For both Eqs. 1, 7, these slopes can be written as Dy 1 =Dx 1 and Dy 2 =Dx 2 for the two different lines. To simplify, assume the denominators represent the difference between identical x values. The ratio becomes Dy 1 =Dy 2 ; the numerator represents a difference between y values that are different than those of the denominator. For citations 2 and 4, the ratio of MA slopes is 1.1, which signifies a 10% greater increase for log w in the data set for citation 2 than for citation 4. This corresponds to a much larger than 10% increase in weight for citation 2 than citation 4. The situation for Eq. 7 is more complicated, because a 1 and a 2 as well as the slope determine the relationship between production and population. Solving for v (or w) yields v ¼ a 1 =ð10 ÀðCþbÂlog xÞ À a 2 Þ, where b is the estimated value of b. For citations 13 † and 15 †, the ratio of MA slopes is 1.29, but the increase in volume for citation 15 † is 4.7% more than for citation 13 †.

Estimated vs. predicted slopes of self-thinning lines
In Table 4, the data sets which exhibited exponential growth, the first set of results is from a Fig. 1. Data are for Erigeron canadensis, citation 1. Solid circles represent actual weights. The fitted function is w = 18.35e 0.01056t with parameter values from Table 2, P < 0.01. This is an example of a data set that exhibits exponential growth for the time interval of the cited experiment.  Table 3, a 1 /Àa 2 = 23,101, P < 0.001. This is an example of a data set that exhibits sigmoid growth. one-tailed linear correlation test between log w and log x. All but one, citation 9, show a significant correlation at 0.05 or less. The next set of results then shows the estimated values of the parameters in Eq. 1 from both MA and RMA. The last results show the predicted value of b from Eq. 5, which is a/c, and the 95% MA and RMA confidence intervals for b. The predicted value, a/c, lies inside the MA and RMA confidence interval in all cases. Fig. 6 shows a graph of predicted vs. estimated slope (from MA); the correlation is significant at 0.001. Equation of the line was estimated from MA.
The format of Table 5, the data sets that exhibited sigmoid growth, follows that of Table 4. For these, all one-tailed linear correlation tests between logðv=ða 1 þ a 2 vÞÞ or w ð Þ and log x are significant at 0.05 or less. There are two data sets, citations 17 and 27, for which the predicted value of the slope, a 1 /c, lies outside the MA confidence interval (CI). However, a 1 /c lies inside the RMA CI in all cases. Fig. 7 shows a graph of predicted vs. estimated slope (from MA); the correlation is significant at 0.001. Equation of the line was estimated from MA.
There is a wide range of estimated slopes shown in Table 5 for Eq. 7. Even if one ignores the two steepest, the range is from À4.5279 to À1.4995 for MA and from À4.3101 to À1.4659 for RMA. There does not appear to be any tendency toward a particular value. In the context of Eq. 1, both Zeide (1987) and Gavrikov (2015) view Fig. 3. Data are for Douglas fir, citations 13 and 13 †. Diamonds represent actual number of trees/ac. Squares and fitted curve represent the fitted density function x = 520e À0.01574t for citation 13. Triangles and fitted curve represent the fitted density function x = 459e À0.01617t for citation 13 †. Parameter values for the fitted density functions are from Table 1, P < 0.001. Citation 13 † has the two youngest, and four oldest, age classes omitted from the data for citation 13. Typically, the fitted density function for the complete data set (citation 13) exhibits a considerable departure (less than) from actual numbers for the early age classes. Removal of the early age classes results in a fitted density function that is much closer to actual numbers. In addition, some data sets, such as citation 13, also exhibit a departure between the fitted function and actual values for the oldest age classes.  Table 3, a 1 /Àa 2 = 22,216, P < 0.001. Inspection shows two common themes for all data sets that exhibit sigmoid growth. With one exception, total actual volume is less than predicted by the fitted function for the youngest age classes and greater than predicted for the oldest age classes. different values of the slope as likely because the numerous factors that contribute to the slope rarely balance.
Although all 31 linear correlations in Table 5 are significant at 0.05 or less (most are significant at 0.001), it is a fact that these same data sets also show significant linear correlations between log v (or w) and log x. Often, data sets from yield tables have a higher r value for Eq. 1 than for Eq. 7. All data sets from experimental stands have a higher r value for Eq. 7 than Eq. 1.
The probable reason for the above observation is the similarity between Eqs. 1, 7. Eq. 7 is log w a 1 þ a 2 w ¼ C þ b log x from which log w À log(a 1 + a 2 w) = C + b log x, which differs from Eq. 1 by the term "log (a 1 + a 2 w)." Because a 1 and a 2 are small numbers, this term is negative. The result is that log w À log (a 1 + a 2 w) is roughly 50% larger than log w (or v in place of w). The effect is to produce considerably steeper slopes for Eq. 7 than for Eq. 1 for all data sets that exhibited sigmoid growth. Figs. 8, 9 illustrate this comparison for two data sets.
Self-thinning relative to ecological attributes Fig. 10 is a subset of Fig. 7. Predicted vs. estimated slopes for three species, Douglas fir, western hemlock, and western larch, are shown. These are the only species with multiple sites that exhibited sigmoid growth. Data points for hemlock and larch are clustered compared to those of Douglas fir which encompass those for hemlock and larch as well as some that are intermediate. If slopes can be regarded as index numbers that relate production and population, one interpretation of Fig. 10 is that a wider range of slopes is associated with adaptation to a wider range of Fig. 5. The two lines are the self-thinning line (Eq. 1), log w = C + b log x. Squares and the fitted line represent a hypothetical data set with slope = À1. Diamonds and the fitted line represent data for citation 1, Erigeron canadensis, with the equation of the latter line from major axis regression, log w = 4.04 À 0.5385 log x, parameter values from Table 4, P < 0.01. Fig. 6. Solid circles represent points (b, a/c) where b is the estimated slope and a/c is the predicted slope of the self-thinning line (Eqs. 1, 5, respectively) for data sets that exhibit exponential growth, with values of b (major axis [MA]) and a/c from Table 4. The blue line was fitted from MA regression; its equation is a/c = À0.03 + 0.948 9 b, r = 0.992, P < 0.001. environments. The handbook by Preston (1961) provides geographical ranges, silvical characters, and habitats for North American trees. Information in the handbook shows that Douglas fir has by far the largest geographical range, a wider range of habitats, and is intermediate in tolerance (Hitchcock et al. 1971 state that Douglas fir is intolerant), compared to either western hemlock or western larch. Western hemlock is described as tolerant and western larch as intolerant. In the context of the Reineke equation, Weiskittel et al. (2009) found that the slope of the self-thinning boundary for Douglas fir is dependent on the site index.
There are 10 permanent plots for Coram 1 and 10 for Coram 2, which were systematically placed. As previously noted, the plots are located in stands that were naturally regenerated, and data were collected every five years from 1961 to 1991. Randomness of plant distribution, determined from the number of plants/plot, is usually judged by comparison to the Poisson distribution, for which the variance equals the mean. A ratio >1 indicates that trees are aggregated (also called a contagious distribution), while a ratio <1 indicates that trees are uniformly spaced. The departure of the ratio from 1 can be tested by a ttest, with standard error = p ð2=ðN À 1ÞÞ where N is the number of plots (Greig-Smith 1964). A value of the ratio in (0, 2.0663) will result in retaining the hypothesis of random distribution of trees at P = 0.05 for df = 9. Fig. 11 shows a plot of the variance: mean ratio for Coram 1 and Coram 2. The ratio was considerably larger than 1 for both sites in 1961 and declined since then. By 1986, the ratio was slightly larger than 1 for Fig. 7. Diamonds and squares represent points (b, a 1 /c) where b is the estimated slope and a 1 /c is the predicted slope of the self-thinning line (Eqs. 7, 6, respectively) for data sets that exhibit sigmoid growth, with values of b (major axis [MA]) and a 1 /c from Table 5. Diamonds represent data sets in subset A (the original complete data set). Squares represent data sets in subset B (those that have some age classes omitted). The line was fitted from MA regression; its equation is a 1 /c = 0.097 + 0.958 9 b, r = 0.981, P < 0.001. Fig. 8. Data are for longleaf pine, citation 10. Squares represent points (log x, log (v/(a 1 + a 2 v))) based on sigmoid growth, where x = actual number of trees/ac and v = actual ft 3 /ac, with values of a 1 and a 2 from Table 3. The corresponding fitted line (major axis [MA]) is the self-thinning line (Eq. 7), logðv=ða 1 þ a 2 vÞÞ ¼ 14:5 À 3:488 Â log x; r = À0.989, P < 0.001, values from Table 5. Diamonds represent points (log x, log v) based on exponential growth, and the corresponding fitted line is the self-thinning line (MA), log v = 7.7 À 1.4827 9 log x, r = À0.975, P < 0.001. In this study, sigmoid growth was thought to be the better choice.

Mathematical connections
Since yield and density are both random variables, the question arises as to which should be regarded as the independent and which the dependent one. Zeide (1987) writes, "since density-dependent mortality is caused by the increase of tree size, it is more natural to use an indicator of the size as the independent variable and the number of trees as the dependent one." A different view of which variable to use as independent can arise from agricultural studies. Willey and Heath (1969) examined a number of possible mathematical functions relating yield and population; population was the independent variable and yield the dependent one. Also, "since yield is an attribute of the crop population, the goal of the agriculturalist is to manage the crop population to produce the optimum (however defined) yield" (Weiner 1990). In this study, the choice was to use density as independent variable and growth as the dependent variable.
Eqs. 3, 4, 5, 6, and all the tables differentiate between a of exponential growth and a 1 of logistic growth. They represent very similar concepts, but they are not mathematically equal. In the case of western hemlock, the average from citations 22, 23, and 24 is 0.03786 for a (not shown in any results) compared to 0.07544 for a 1 . Other comparisons are similar. Fig. 9. Data are for western hemlock, citation 20 †. Squares represent points log x; logðv=ða 1 þ a 2 vÞÞ ð Þ based on sigmoid growth, where x = actual number of trees/ac and v = actual ft 3 /ac, with values of a 1 and a 2 from Table 3. The corresponding fitted line (major axis [MA]) is the self-thinning line (Eq. 7), logðv=ða 1 þ a 2 vÞÞ ¼ 12:72 À 2:4609 Â log x; r = À0.951, P < 0.001, values from Table 5. Diamonds represent points (log x, log v) based on exponential growth, and the corresponding fitted line (MA) is the self-thinning line, log v = 5.6 À 0.5921 9 log x, r = À0.988, P < 0.001. In this study, sigmoid growth was thought to be the better choice.  Table 5, for tree species data from multiple sites. Diamonds represent Douglas fir with complete data sets. Squares represent Douglas fir from the same sites with reduced data sets. Triangles represent western hemlock with complete data sets. Symbols that look like an x represent western hemlock from the same sites with reduced data sets. Asterisks represent western larch.
The chain rule has been used to link the variables growth and density with the variable time. It is an assumption that this connection is reasonable. The hypotheses for dw/dt (or dv/dt) and dx/ dt are hypotheses about their related rates, which are subsequently determined from the data. Just as in a related rates problem that is solvable by calculus, substitution leads to a solution for dw/ dx, which are Eqs. 5, 6. Finally, it must be noted that proof of the chain rule is predicated on the assumption that growth (w or v) is a function of density; it is also a function of time.

CONCLUSIONS
There were four stated goals of this study. The first goal of providing mathematical hypotheses for growth and density and a method of relating these in order to obtain formulas relating growth and density was accomplished. Taken by itself, this could be regarded as a modeling exercise that has questionable relevance. For these hypotheses and formulas to accrue some validity, these statements also need to have some degree of congruence with actual data. Analysis of data does show a reasonable accord between hypotheses and data, and between resulting formulas, both the linear form of the relationship as well as the predicted slope, and the same data. The data that were examined represent a variety of species and environmental conditions. The fourth goal was to examine the process of self-thinning relative to ecological attributes of species that are believed to be undergoing this process. Self-thinning is density-dependent mortality, which leads to questions regarding heritable characteristics of survivors that could be manifested in a variety of ways. Data for one species, western larch, shows a time trend from an aggregated or contagious spatial distribution of individuals toward one of randomness. Data for those tree species that had multiple sites and a corresponding number of data sets indicate an association between a wide numerical range (or lack) of slope indices and existence in a diverse (or narrow) range of environments.

ACKNOWLEDGMENTS
The author used an iterative procedure provided by statpages.info/nonlin.html to fit the logistic function to the data sets that exhibited sigmoid growth. Tim Eichner, one of the natural resource instructors at FVCC, provided the author with the data from Forbes (1961) and made the author aware of the large amount of data available from the Coram Experimental Forest. The data the author reported was generously provided by Drs. D. K. Wright and E. K. Sutherland, Manager and Research Biologist, respectively, Coram Experimental Forest (USFS), Coram, Montana, USA. Two anonymous reviewers pointed out a number of shortcomings during the development of this manuscript, but their comments also provided guidance as to how these shortcomings could be addressed. The author found this very helpful and encouraging.
Pages 209-211 in M. Rechcigl, editor. Handbook of Fig. 11. Broken line plots, blue for Coram 1 and orange for Coram 2, shows the time trend of the variance: mean ratio for number of trees/plot for western larch, citations 25 and 26, during which mortality was approximately 90%. A ratio >1 indicates an aggregated (contagious) spatial distribution, while a ratio of 1 indicates a random spatial distribution. A value of the ratio in (0, 2.0663) will result in retaining the hypothesis of randomness at P = 0.05.