Evaluation of modeling strategies for assessing self-thinning behavior and carrying capacity

Abstract

Self‐thinning and site maximum carrying capacity are key concepts for understanding and predicting ecosystem dynamics as they represent the outcome of several fundamental ecological processes (e.g., mortality and growth). Relationships are often derived using alternative modeling strategies, depending on the statistical approach, model formulation, and underlying data with unclear implications of these various assumptions. In this analysis, the influence of contrasting modeling strategies for estimating the self‐thinning relationship and maximum carrying capacity in long‐term, permanent plot data (n = 130) from the mixed Nothofagus forests in southern Chile was assessed and compared. Seven contrasting modeling strategies were used including ordinary least squares, quantile, and nonlinear regression that were formulated based on static (no remeasurements) or dynamic data (with remeasurements). Statistically distinct differences among these seven approaches were identified with mean maximum carrying capacity ranging from 1,050 to 1,912 stems/ha depending on the approach. The population‐level static approach based on quantile regression produced an estimate closest to the overall mean with site‐level carrying capacity depending on tree species diversity and climate. Synthesis and applications. Overall, the findings highlight strong variability within and between contrasting methods of determining self‐thinning and site maximum carry capacity, which may influence ecological inferences.

Full text

1076 8
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Ecology and Evolution. 2 018;8:10768–10779.
www.ecolevol.org
1 | INTRODUCTION
Size–density relationships are essential for understanding and pre-
dicting core ecological processes. These relationships highlight how
the number of individuals in a population decreases with the pro-
gression of time, or more specifically as the individuals increase their
average size. Therefore, a size–density relationship is a fundamental
result of highly dynamic competition and mort ality processes. This
concept is an essential aspect of many ecological disciplines includ-
ing forest (Jack & Long, 1996), wildlife (Jonsson, 2017), and fisheries
ecology (Elliott, 1993). Self- thinning results from a frontier relation-
ship between stand density and tree size (Bi et al., 2000). That is,
Received: 5 March 2018
|
Revised: 30 June 2018
|
Accepted: 10 Jul y 2018
DOI: 10.1002/ece3. 4525
ORIGINAL RESEARCH
Evaluation of modeling strategies for assessing self- thinning
behavior and carrying capacity
Christian Salas-Eljatib1,2 | Aaron R. Weiskittel3
This is an op en access article un der the ter ms of the Creative Commons Attribution License, which permits use, distribution a nd reproduction in a ny medium,
provide d the original work is properly cited.
© 2018 The Aut hors. Ecol ogy and Evolution published by John Wiley & Sons Ltd.
1Centro de Modelación y Monitoreo
de Ecosistemas, Facultad de
Ciencias, Universidad Mayor, Santiago, Chile
2Laboratorio de Biometría, Departamento
de Ciencias Forest ales, Uni versidad de La
Frontera, Temuco, Chile
3School of Forest Resources, Universit y of
Maine, Orono, ME
Correspondence
Christian Salas-Eljatib, Centro de
Modelación y Monitoreo de Ecosistemas,
Faculta d de Ciencia s, Universidad Mayor,
Santiago, Chile.
Email: cseljatib@gmail.com
Funding information
Fondo Nacional de De sarrollo Científico y
Tecnológico (FONDECY T), Grant/Award
Number: 1151495
Abstract
Self-thinning and site maximum carrying capacity are key concepts for under-
standing and predicting ecosystem dynamics as they represent the outcome of
several fundamental ecological processes (e.g., mortality and growth).
Relationships are often derived using alternative modeling strategies, depending
on the statistical approach, model formulation, and underlying data with unclear
implications of these various assumptions. In this analysis, the influence of con-
trasting modeling strategies for estimating the self-thinning relationship and max-
imum carr ying capacity in long-term, permanent plot data (n = 130) from the
mixed Nothofagus forests in southern Chile was assessed and compared. Seven
contrasting modeling strategies were used including ordinary least squares, quan-
tile, and nonlinear regression that were formulated based on static (no remeasure-
ments) or dynamic data (with remeasurements). Statistically distinct differences
among these seven approaches were identified with mean maximum carrying ca-
pacity ranging from 1,050 to 1,912 stems/ha depending on the approach. The
population-level static approach based on quantile regression produced an esti-
mate closest to the overall mean with site-level carr ying capacity depending on
tree species diversity and climate. Synthesis and applications. Overall, the findings
highlight strong variability within and between contrasting methods of determin-
ing self-thinning and site maximum carry capacity, which may influence ecological
inferences.
KEYWORDS
competition, density trajectories, density-dependent, mixed-effects models, mortality,
negative binomial model, quantile regression

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SALA S- ELJATIB And W EISKITT EL
self- thinning occurs when the population density reaches the maxi-
mum possible for a given average individual size, and so any increase
in average size causes a decline in stand density. The self- thinning
phenomenon is one of the few fundamental rules throughout ecol-
ogy and has led to several important concepts such as the - 3/2 law
for plants (Reineke, 1933; Yoda et al., 1963), - 4/3 power rule for ani-
mals (Begon et al., 1986), and maximum carrying capacity (Enquist et
al., 1998). Although the - 3/2 power rule has received some criticism
(Lonsdale, 1990; Weller, 1987), its use for ecological interpretation
of population dynamic s is widely utilized and accepted by ecologists.
The carrying capacity of a population can be expressed in the
same way as the asymptote of a logistic equation in an ecological
model (Gore & Paranjpe, 2001; Gotelli, 2001; Pielou, 1977), as the
maximum density for a given average individual size. This value is
generally termed the maximum stand density index (SDImax), and it
is used for a range of purposes (Avery & Burkhart, 2002; van Laar &
Akça, 2007). The max im um carrying cap ac it y of a fo re st has been re-
cently shown to vary depending upon species composition (Binkley,
1984; Puettmann et al., 1992; Stout & Nyland, 1986), species func-
tional traits (Ducey et al., 2017), and climate factors (Weiskittel et al.,
2009). In addition, it s potential uses for multicohort and structurally
complex forests has been assessed, and general findings suggest a
similar usefulness as for more simpler forests (Ducey & Knapp, 2010;
Sterba & Monserud, 1993). Construction of maximum size–density
relationships provides a basis for quantifying the ecologic al concepts
of self- thinning and carr ying capacity. However, defining this frontier
relationship is difficult and, consequently, it has been derived using
a variety of different approaches that depend upon the statistical
approach, model formulation being used, and the actual data source.
Several statistical models have been proposed for fitting self-
thinning relationships. The most common approach is to fit a base
model, for example, the logarithmic of Reineke (1933), using ordinary
least squares (OLS) and then shif t the intercept upwards by comput-
ing the upper level of the 95% confidence inter val for that estimated
parameter. Nevertheless, this approach does not fully account for
the structure of the data, and more suitable statistical models for
building self- thinning lines have been devised. Zhang et al., (2005)
compared different alternatives for estimating the self- thinning
boundary line including the following: OLS, corrected OLS, deter-
ministic frontier function (DFF), stochastic frontier function (SFF)
or stochastic frontier regression (SFR), and quantile regression (QR).
Their results favored SFR, although QR performed nearly as well.
VanderSchaaf and Burkhart (20 07) elaborated further on the bio-
logical implications of the maximum size—density relationships, and
compared OLS, a first- difference model, and linear mixed- effects
(LME) models for fitting this relationship. They found that the LME
was a better alternative for estimating the slope of the Reineke’s
model in specific situations without accounting for self- thinning pat-
terns of individual stands. Remeasured or dynamic data can be used
to approximate an instantaneous growth rate. Vanclay and Sands
(2009) applied this concept, for analyzing the self- thinning frontier.
In contrast, Weiskittel et al., (2009) used static data and SFR for esti-
mating the self- thinning boundary line in different forest types in the
Pacific Northwest, USA. They also found that site productivity and
the proportion of basal area of the primary species being modeled
were important predictor variables for the size–density relationship.
Recently, Andrews et al., (2018) used QR with mixed- ef fects to de-
termine carrying capacity for several common species in the Acadian
Region of North America and found the method to provide robust
site- level estimates that were influenced by a variety of factors in-
cluding species functional traits and climate.
Similar to alternative statistical approaches, different data
sources have been used for developing self- thinning relationships.
The most common type of data for fitting size–density models is
from static measurements that lack repeat observations. Static
data have been primarily used as they are easy to collect and can
be quickly assessed over a wide range of conditions, whereas their
primary disadvantage for constructing a self- thinning relationship is
that either a full range of conditions must be measured, particularly
high- density sites, or an appropriate statistical model for estimating
it be used. Dynamic data contain site- level attributes that have been
remeasured through time. This source of data has been proposed
by García (20 09) for developing self- thinning lines, based upon dif-
ferential equations. Recently, Trouve et al., (2017) followed García’s
(2009) approach in single- species forests in Australia and found
that the dynamic approach performed similar to a static one, which
was similar to the findings of Kweon and Comeau (2017). Smith and
Hann (1984) also suggested the use of dynamic data for developing
maximum size–density models, which was later modified by Hann
et al., (2003) for using the first site- level measurement along with
future observations to determine a site’s trajectory in self- thinning
space. The third type of data proposed for self- thinning studies are
individual tree- level observations. For example, Ducey and Knapp
(2010) proposed this approach as a way for estimating the maximum
size–density relationships in mixed species and structurally complex
forests. The method was later generalized further by Ducey et al.,
(2017) for incorporating climate and species functional traits.
A variety of core questions remain on the various estimation
strategies for modeling self- thinning relations and site- level maxi-
mum carr ying capacity. The vast majority of research on this topic
had been conducted for single- species stands, but relatively limited
research has been taken for mixed- species forests, particularly spe-
cies rich and productive temperate rainforests. In addition, the de-
pendency of site- level carrying capacity on climate conditions and
other environmental factors has rarely been taken into account. As
indicated by Weiskittel et al., (2009), most studies have used sub-
jective or significantly limited statistical techniques for fit ting the
self- thinning line. Furthermore, most studies have used static dat a,
but rather few have explored and compared the use of dynamic data
for estimating maximum size–density models. Finally, most studies
have ignored the hierarchical structure in both static and dynamic
data, which may have biased findings and limit general inferences to
population- rather than site- level trends. Therefore, we aimed to: (a)
develop alternative strategies for constructing the maximum size–
density relationship that explicitly account for hierarchical data; (b)
compare implied estimates of site- level maximum carrying capacity;

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SALAS- EL JATIB And WEISKIT TEL
and (c) relate observed carrying capacity to various site- level and
environmental variables.
2 | METHODS
2.1 | Data
Our study area covers the secondar y Nothofagus forests in the
southcentral part of Chile (37°–41°S.) Specifically, we focus on the
N. obliqua, N. alpina, and N. dombeyi forest type (Donoso, 1995),
which covers around 600,000 ha. These forests are part of the
temperate rainforests of Chile, which represents the second larg-
est remaining area of this type in the world (Donoso, 1995; Wilcox,
1995) and are internationally recognized for their ecological im-
portance (Davis, Heywood et al., 1994; Olson & Dinerstein, 1998;
Stattersfield, 1998). As highlighted by Salas et al., (2016), these
three species are the most impor tant for commercial and cultural
purposes, which are usually located on the most productive sites in
the Central Depression and foothills of the Andes.
We use data from permanent sample plots established through-
out southcentral Chile where the roble- raulí- coigue forest type
grows (Figure 1). The plots areas ranged from 500 to 10,000 m2 and
were based on conventional tree- level measurements of trees larger
than 1.3 m in height with a diameter at breast- height (d) greater or
equal to 5 cm. We computed stand variables at the plot- level (e.g.,
density N and diameter of the mean basal area tree dg). Plots remea-
sured at least once were considered as “dynamic data,” while others
provided “static data.”
The dynamic plots are clearly shown as a time series, meanwhile
the static plots are only shown as single dots. Figure 2 shows the
relationship between density and quadratic mean diameter (i.e., a
FIGURE1 Permanent sample
plots distribution (dots) in secondary
Nothofagus forests (green) in south- central
Chile. The plots with remeasurements
on time are mark as “dynamic” or “static”
otherwise

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SALA S- ELJATIB And W EISKITT EL
plot-level averaged tree diameter), highlighting the progressive de-
crease in individuals in the population as they increase their size.
The descriptive statistics for the primary site- level variables by type
of data are summarized in Table 1. We have a total of 130 plots, and
because some of them have been remeasured, the total number of
observations is 178. Note that from the available plots, 26 of them
have more than one measurement.
Besides variables representing forest features, we also ob-
tained bioclimatic variables for each plot location from WorldClim
(http://www.worldclim.org) surfaces (Hijmans et al., 2005). These
19 bioclimatic variables represent annual trends (e.g., mean annual
temperature, annual precipitation), seasonality (e.g., annual range
in temperature and precipitation), and extreme or limiting environ-
mental factors (e.g., temperature of the coldest and warmest month,
and precipitation of the wet and dry quarters). Note that these
bioclimatic variables were long- term averages (Hijmans et al., 2005).
In addition, topographic features such as elevation, aspect, and slope
were also evaluated.
2.2 | Modeling strategies
We evaluated different alternatives for constructing and interpret-
ing self- thinning relationships, which were a combination of data
source, model form, and statistical method. The data sources are
defined by the type of measurement (i.e., static or dynamic) and the
level of information (i.e., site- or individual- level). The different mod-
eling strategies are summarized in Table 2 and explained further in
the following paragraphs.
1. Static, site-level data, and linear mixed-effects fit (S-P-LME): We
used the Reineke’s formulation (Reineke, 1933), as the base model
for representing the relationship between stem density—average
tree size, as follows
where:
Nij
and
d
g
ij
the tree density and the quadratic mean di-
ameter for the
j
-th measurement at the
i
-th plot, respectively;
while
eij
is a random error following a Gaussian distribution having
an expected value of 0 and variance
𝜎2
e
ij
. VanderSchaaf and Burkhart
(2007) proposed a method for estimating maximum size—density
lines based on a mixed-effects model by adding random effects
into both parameters of the Reineke’s model, as follows,
where
u0i
and
u1i
are plot-specific random effects to be predicted
and assumed to follow a Gaussian distribution having an expected
value of 0 and variance
𝜎2
0
and
𝜎2
1
, respectively. We fit this model
(Equation 2) using the nlme package (Pinheiro & Bates, 2000) of R
Core Team (2017). In order to estimate the self-thinning line, we
computed the upper level of the 95% confidence interval for the es-
timated intercept-coefficient of the model, by adding the estimated
(1)
ln N
ij
=𝛽
0
+𝛽
1
ln d
g
ij +e
ij
,
(2)
ln N
ij
=(𝛽
0
+u
0i
)+(𝛽
1
+u
1i
) ln d
g
ij +e
ij
,
FIGURE2 Population density versus average individual size
(dg is quadratic mean diameter) for permanent sample plots of
Nothofagus forests. Dots joined by lines correspond to remeasured
plots (i.e., dynamic), and single dots represents plots without
remeasurements (i.e., static)
010203040 50
0
1,000
2,000
3,000
4,000
5,000
dg (cm)
Density (trees/ha)
TABLE1 Descriptive statistics for the primary forest variables segregated by measurement type of plots: static (i.e., no remeasurements)
and dynamic (i.e., at least one remeasurement)
Statistics
Type of measurement
Static (n = 10 4) Dynamic (n = 74)
N (trees/ha) dg (cm) G (m2/ha) PBA (%) N (trees/ha) dg (cm) G (m2/ha) PBA (%)
Minimum 330 13.7 16. 4 48.0 670 11. 5 26.2 3 9.7
Average 1,270 23.7 4 9.5 7 7. 3 1,770 20.2 50.7 82.5
Median 1,160 21.8 47. 5 82.5 1,530 20 .1 50.3 86.6
Maximum 3,580 48.1 106.4 100.0 4,680 34.1 85.5 100.0
CV (%) 50 28.6 32.8 25.9 44 23.8 23.0 17. 5
Note. The variables are population densit y (N; trees/ha), quadratic mean diameter (
dg
; cm), basal area (G; m2/ha), and percentage of basal area of
Nothofagus species (PBA; %).

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SALAS- EL JATIB And WEISKIT TEL
parameter to its respective st andard error multiplied by 1.96 (i.e., the
quantile from the t-distribution for
𝛼=0.05
).
2. Static, site-level data, and quantile regression fit (S-P-LQMM): Both
OLS and LME aim at estimating population-averaged parameters and
therefore do not directly model the frontier relationship that are of
interest in self-thinning estimation. A regression equation that mod-
els the median instead of the expected value, such as a conditional-
mean regression would be a more suitable alternative for a frontier
relationship. Koenker and Bassett (1978) proposed a more general
model, the quantile regression model (QR). The QR corresponding to
a mean conditional model can be expressed as the pth conditional
quantile given
xi
as:
where
Q(p)
[y
ij|
x
ij]
is the p-th quantile for
yij
being determined by the
quantile-specific parameters
𝛽(p)
0
and
𝛽(p)
1
, and the value of
xi
, where
yij
=
ln Nij
and
x
ij =
ln d
g
ij
as in the Reineke’s model. We fit the quantile
regression model (3) in a mixed-effec ts framework by adding random
effects (plot-specific) to the intercept of the model, therefore fitting
a linear quantile regression mixed-effects model (LQMM). We tested
different quantile values as recommended by Ducey and Knapp
(2010). We choose the quantile that gave us the lowest variance es-
timates for the estimated parameters. According to our analyses, the
0.95 quantile was selected. This quantile mixed-effect model was
fitted by assuming a log-likelihood expression based on an asym-
metric Laplace density function, as suggested by Geraci and Bottai
(2007), and using the lqmm package (Geraci, 2014) implemented in
R. Because the quantile regression model was fitted in a mixed-ef-
fect framework, we should not simply use the fixed-effect estimated
quantile-intercept parameter as the corresponding parameter for
constructing the self-thinning line. This would be misleading in the
sense that it does not fully represent the quantile-parameter itself,
but an average of it. Therefore, as in the above strategy, we com-
puted the upper level of the 95% confidence interval of the intercept
parameter for the LQMM model.
3. Static, site-level data, and stochastic frontier regression fit (S-P-
SFMM): Stochastic frontier regression (SFR, Aigner et al., 1997) is an
econometrics model that is often used to determine the technical
efficiency of business firms but has been used in the past for self-
thinning analyses (Bi et al., 200 0; Weiskittel et al., 20 09; Zhang et al.,
2005). We express a SFR model as the following matrix model
where: the vector
Y
contains all the
ln Nij
;
X
is the design matrix
having the observations of
ln d
g
ij
;
𝜷
is a vector of coefficients; the
vector
V
is a random variable representing a portion of the model
error, where
V
ij ∼N(0,𝜎
2
V)
; the vector
U
is a positive random variable
representing the other portion of the model error following a half-
Gaussian with
U
ij ∼
|||
N(0,𝜎2
U
)
|||
. In this analysis, a true random effects
SFR model of Greene (2005) was used to account for the data hier-
archy. The SFR model (4) in a mixed-effects framework (S-P-SFMM)
was fit using PROC QLIM in SAS v9.4, by adding random effects to
the model intercept (4).
4. Static, individual-level data, and linear mixed-effects fit (S-T-LME):
Ducey and Knapp (2010) developed a stand density index based on
tree-level variables, which was determined to be more suitable for
mixed-species and structurally complex forests. Their approach in-
volves fit ting the following system
where
EFiz
is the expansion factor for the z-th tree within the i-th
plot;
dzij
and
SGzij
are diameter and specific gravity, respectively for
the j-th measurement of the z-th tree at the i-th plot. Ducey and
Knapp (2010) did not provide an approach for estimating a self-thin-
ning line, but for computing the maximum carrying capacity, which
can be obtained by
where
Nmaxi
is the maximum density for the i-plot and
SGi
is the av-
erage-specific gravity for that plot. Model (5) was fitted as a quan-
tile regression model with mixed-ef fects using the 95th percentile.
5. Dynamic, site-level data, and density model with linear mixed-effects
(D-P-LME): All the above-explained strategies used static data, which
(3)
Q(p)
[yij
|
xij]=𝛽(p)
0
+𝛽(p)
1
xij +Q
(p)
(𝜖ij
),
(4)
Y
=X𝜷+
(
V−U
),
(5)
𝛼
0
x
0+𝛼1
x
1+𝜖=
1, where
x0=
j
EFiz(dzij )1.6
x1=

j
EFizSGzij dijz
25 1.6
,
(6)
N
maxi=
100
𝛼
0+
𝛼
1
(SG
i
)
,
Modeling approach
acronym
Data source
Statistical model Equation number
Type of
measurement Level
S- P- LME Static Plot LME (2)
S- P- LQMM Plot QR (3)
S- P- SFMM Plot SF (4)
S- T- LME Tre e Ducey (5–6)
D- P- LME Dynamic Plot Density (10)
D- P- NBME Plot Mortality (13)
D- P- NLME Plot 1st measu. (14)
TABLE2 Modeling strategies
evaluated in this analysis. They were a
combination between the data source, model
form, and the statistical model being used

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SALA S- ELJATIB And W EISKITT EL
do not require plots that have been remeasured through time. An alter-
native is using dynamic dat a, which are plots with one or more remeas-
urements, for deriving self-thinning lines. Although there is imbalance
between static and dynamic observations in this analysis, we do not
believe this would greatly influence our general findings since we are
interested in plot-level trends and highlighting the differences across
methods. Therefore, for the following strategies, we used 26 sample
plots with dynamic data of stand variables through time.
Traditionally, in ecology, the rate of change in densit y has been
studied as a density- dependent phenomenon (Dennis & Taper, 1994;
Gotelli, 2001), using the following general differential equation form:
where
f()
is a function relating the rate of change to a vector of param-
eters
𝜃
and N is population densit y. García (2009) proposed to relate
the rate of density change in terms of forest height (H) instead of time,
and proposed the use of the following differential equation form:
where
𝛽0,…,𝛽2
are parameters. Zeide (2010) instead used an aver-
aged stand diameter known as the diameter of the mean basal area
tree (
dg
) as:
A solution of equation (9) is given by Trouve et al., (2017) as the
following density model:
where
N1i
and
N0i
is tree density at the end and at the beginning of the
period for the i- th plot, respectively;
d
g
0i
and
d
g
0i
is the stand quadratic
mean diameter at the end and at the beginning of the period for the i-
th plot, respectively.
𝛽0,𝛽1
, and
𝛽2
are parameters to be estimated and
𝜖i
is a random error for the i- th observation from a Gaussian distribution
with mean 0 and standard deviation
𝜎
.
We fit several variants of model (10) by allowing random
effects on each or all of the parameters of that model. As ex-
plained above, the plot was used as the random factor in order
to take into account the hierarchical structure of the data. Given
that our data have a hierarchical structure, the random effects
capture variation from unmeasured variables at the plot- level
and the individual- level in plots with repeated measures. We
compared models based on the Bayesian information criterion
(BIC; Schwarz, 1978).
For computing the self- thinning estimates from the dynamic
density model, we follow Trouve et al., (2017) by using the following
formulas:
6. Dynamic, site-level data, and mortality model with negative binomial
mixed-effects (D-P-NBME): An alternative approach is to model how
density changes based on the difference in population densities on
time. We fit a negative binomial generalized linear model (NBGLM)
as given by Trouve et al., (2017) in their equation 8, but allowing ran-
dom effects into the intercept of the model
where:
ΔNi=N0i−N1i
;
Δdgi=dg1i−dg0i
; where
u0i
are plot-specific
random effects to be predicted and assumed to follow a Gaussian
distribution having an expected value of 0 and variance
𝜎2
0
; and
𝜃
is
an overdispersion parameter that allows the variance to be scaled
as the square of fitted
̂
Δ
N
i
values; and the other model terms have
been explained above. Note that the last term in equation (13)
does not have a coefficient, which was achieved by using the op-
tion offset implemented in R as suggested by Trouve et al., (2017).
In this way, the model is forced to use the measured value of
Δdgi
as it is.
The corresponding self- thinning line is obtained as in the previous
modeling strategy, by substituting their respective parameter esti-
mates in equations (11) and (12).
7. Dynamic, site-level first-measurement data and average individual
size model with nonlinear mixed-effects (D-P-NLME): Hann et al.,
(2003) proposed a method for estimating the maximum size—
density trajectory. In general, this approach differs from the others
explained above in the sense that it (a) predicts stand diameter
instead of tree density (like in Yoda et al., 1963), and (b) uses
the first available measurement for each plot to evaluate the
trajectory over time.
where
Ni
and
d
g
i
are the tree density and stand diameter for the i-th
plot, while that
N
1
sti
and
d
g
1sti
are the same variables measured for
the first time at the i-th plot. The self-thinning line is obtained by
using the reversed-Reineke’s equation part of (14), and solving for N.
2.3 | Comparisons
We examined t wo important features of the assessed modeling
strategies: (1) model behavior and (2) prediction of carrying capacity.
1. Model behavior. In order to assess how well the self-thinning
lines depicted by each modeling strategy represent the frontier
relationship of population density, we plotted them in both log
(7)
dN
dt
=f(𝜃,N
),
(8)
dN
dH
=𝛽0N𝛽1H𝛽2
,
(9)
dN
d
d
g
=𝛽0N𝛽1dg
𝛽
2
(10)
ln
N1i=ln
(
N1−𝛽2
0i+exp[𝛽0]⋅
1
−
𝛽
2
𝛽1+1
⋅
[
d𝛽1+1
g0i−d𝛽1+1
g1i
])
1
1−𝛽2
+𝜖i
,
(11)
intercept
=
̂
𝛽0
1
−
̂
𝛽
2
+
ln
(
̂
𝛽2−1
)
−ln
(
̂
𝛽1+1
)
1
−
̂
𝛽
2
(12)
slope
=1+
̂
β
1
1
−
̂
β2
(13)
ΔNi∼NB


ΔNi,

ln 
ΔNi=0+u0i+1ln (dg0i)+2ln (N0i)+ln Δdgi
,
var

ΔNi

=
ΔNi+

ΔN2
i
,
(14)
ln
dgi=𝛼0+𝛼1ln (Ni)−




(𝛼0𝛼2)2
𝛼0+𝛼1ln (N1sti)−ln (dg1
sti
)




e−𝛼3

ln (N1sti
)−ln (Ni)


10774
|
SALAS- EL JATIB And WEISKIT TEL
and untransformed space with the observed dispersion of our
data.
2. Prediction of carrying capacity. As a proxy for the carrying
capacity, we computed the maximum st and densit y index
(SDImax), that is, the maximum number of trees at a given refer-
ence average individual size (in our case a diameter) that can
exist in a self-thinning population (Husch, Miller, & Beers, 1972).
As the self-thinning model provides the maximum density for a
given average plot diameter (equation 3), the SDImax is pre-
dicted by
where
̂
Nmaxi
is the predicted maximum density at a base-average tree
size
dbase
at the i-th plot. We predicted SDImax using 25.4 cm as
dbase
,
for each of the modeling strategies. We compared the predicted val-
ues of SDImax among each of the strategies, by computing the multi-
ple comparison test of Scheffé (1953).
2.4 | Modeling carrying capacity
For the best modeling strategy, we further explored the relationship
between the plot- level
̂
SDImax
with various forest, topographic, and
bioclimatic variables. We fit several models of the form
where:
SDI
max
i
is the predicted maximum stand density index at
the i- plot;
Xi
is the predictor variables matrix (with a first column
with ones) at the i- plot;
f()
is a linear or non- lineal function;
𝜽
is a
parameter vector (i.e., coefficients) of the model;
ei
is the random
error term that follows a Gaussian distribution with zero mean and
variance
𝜎2
e
.
We assessed different predictor variables for X in model (16),
therefore having several candidate models. Some of these vari-
ables were based on the results of Weiskittel et al., (2009). We
used the following response variables for representing forest
features: proportion of the primary species (in our case, the ones
belonging to the Nothofagus genus) in density, basal area, and vol-
ume; minimum, maximum, median, standard deviation, coefficient
of variation, and skewness of the diameter distribution; species
richness; Shannon index; and index of species relative importance.
The following topographic features were also used as predictor
variables: elevation, aspect, and slope as in Stage and Salas (2007).
Finally, we also tested to use the bioclimatic variables as predictor
variables in model (16).
The final carrying capacity model was selected after com-
paring the goodness- of- fit of the different model formulations,
prediction capabilities, and the biological behavior of the result-
ing model. Given that we had 19 bioclimatic variables, we first
found the best single predictors of carrying capacity. We then
tested each of the selected predictors with the other potential
predictors.
3 | RESULTS
Based upon all of the modeling strategies explained above, we pre-
dicted the self- thinning lines both in logarithm scale (Figure 3a) and
in untransformed density unit s (Figure 3b). The dynamic - mortal-
ity (D- P- NBME) model strategy followed by the dynamic - density
model (D- P- LME) gave rather high self- thinning lines. Not only is
this behavior unsupported by our observed data, but also is incon-
sistent with the current knowledge of Nothofagus forest dynamics
(Pollmann, 2003; Veblen, 2007; Veblen & Ashton, 1978; Veblen et
al., 1979; Veblen et al., 1996; Veblen et al., 1981). The first-meas-
urement strategy (D- P- NLME) did not provide an appropriate self-
thinning line for our observed data (Figure 3b). On the contrary, both
static plot- level strategies (S- P- LME and S- P- LQMM) of fered us the
best behaviorof the observed dat a, by capturing the limiting rela-
tionship of population density as individual average size increased.
Not only did the estimated intercept and slope parameters for the
self- thinning line differ among modeling strategies, but also their
variances (Table 3).
Al tho ug h bo th S- P- L ME and S- P- LQ MM mod eli ng stra te gi es pr o-
vided the closest approximation to our data, we believe S- P- LQMM
is more suitable for self- thinning estimation because it was able to
better capture the frontier relationship of density, even if it was a
bit higher than the observed maximum densities. This method could
serve as a way for accounting for the sampling error scheme used for
collecting the data, in the sense that we might not be able to sam-
ple some locations with higher densities. Furthermore, a quantile
regression model, as our S- P- LQMM, makes no assumptions about
the distribution of the residual error, which allows correct inference
about other quantiles.
There wer e also import ant and statist ically signifi cant differe nces
in determined carr ying capacity from all of the modeling strategies.
The overall mean value was 1429 ± 121 (mean ± SE) yet ranged from
929 to 3,900 depending on the method. Mean values by method
differed by over 68%, highlighting the large differences across the
modeling strategies. In general, the predicted carrying capacity was
higher for the individual- level and the dynamic strategies (Table 4).
The only exception to this trend was the D- P- NLME strategy where
the predicted carrying capacity was the lowest.
The Schef fe’s test delineated three distinct groups: The first
one was formed by the individual- level and the dynamic mortalit y
model; the second one was formed by the dynamic density model,
followed by static, site- level quantile regression, stochastic regres-
sion, and linear mixed- effects approaches; and the third one was
for D- P- NLME. Although no statistical differences were detected,
the S- P- LME strategy was much closer to the predictions given by
D- P- NLME than quantile regression (S- P- LQMM) and stochastic re-
gression. In addition, the average predicted SDImax for the D- P- LME
strateg y was much higher than any observed forests at a given index
diameter. Both S- P- LQMM and S- P- SFMM provided maximum pop-
ulation densities differentiating by only 78 trees/ha, but S- P- LQMM
overall predicted value was closer to the overall mean value for a
(15)

SDImaxi
=

N
max |dbasei
(16)
SDImaxi
=
f(𝜽,Xi)
+
ei,

|
10775
SALA S- ELJATIB And W EISKITT EL
secondary Nothofagus forests with an average size in diameter of
25 cm. In addition, S- P- SFMM provided several plot- level values that
were generally too low, while S- P- LQMM was more consistent and
had an narrower range of plot- level values. Therefore, we favored
the predictions of carrying capacity by the static, site- level quantile
regression mixed- effects model strategy (S- P- LQMM).
Based on a variety of alternative models, carrying capacity was
found to be ef fectively modeled as a function of climatic, species
diversit y, and abundance of pioneer species composition. The best
model used variables representing: species diversit y (i.e., Shannon
index) and climatic conditions (i.e., precipitation in the driest month).
The model had an error of about 13.5% with respect to the mean
observed value of SDImax. Furthermore, the overall fit of the model
was significantly better than all of the other models examined. The
behavior of the carrying capacity model is shown in Figure 4 by using
fixed (low, mean, and high) values of the predictor variables of pre-
cipitation in the driest month and Shannon’s index. The results sug-
gest that carrying capacit y had a slight unimodal relationship with
pioneer species composition, but was much more sensitive to pre-
cipitation in the driest month and Shannon’s index, which showed
positive relationships with carrying capacity.
4 | DISCUSSION
In this analysis, the difficulty of determining site- level carrying cap ac-
ity was demonstrated given the range of contrasting values obtained
from the various alternative strategies examined. To our knowledge,
relatively few studies have examined the effects of contrasting
modeling strategies using the same dataset, while simultaneously
accounting for hierarchies in the data. Overall, we indicated that the
FIGURE3 Self- thinning lines from different modelling strategies
in logarithmic scale (a) and untransformed units (b). Static site- level
mixed- effects model (S- P- LME), Static site- level quantile regression
mixed- effects model (S- P- LQMM), Static site- level stochastic
frontier regression mixed- ef fects model (S- P- SFMM), Dynamic
site- level stem density mixed- effect model (D- P- LME), Dynamic
site- level mortality negative binomial (D- P- NBME), and Dynamic
site- level based on first measurement (D- P- NLME) represents to
model (3), (10), (13), and (14), respectively
TABLE3 Estimated intercept and slope for the self- thinning line
for each modeling strategy
Modeling strategy Estimated
̂
𝛽0
Parameters
̂
𝛽1
S- P- LME 11.4787 −1. 2741
S- P- LQMM 12.257 −1 . 4742
S- P- SFMM 11.71 −1 . 33
D- P- LME 14.535 −2 .2 396
D- P- NBME 16. 352 −2. 7194
D- P- NLME – –
Note. We do not report the parameters for the D- P- NLME strateg y, be-
cause the parameters are not directly comparable with the other
strategies.
TABLE4 Mean predicted carrying capacity for each modeling
strategy
Modeling strategy
Carr ying capacity
(trees/ha) Scheffe test
S- T- LME 1780 .6 a
D- P- NBME 176 4. 2 a
D- P- LME 1444.2 b
S- P- LQMM 13 49. 8 b
S- P- SFMM 1272.2 bc
S- P- LME 1194. 3 bc
D- P- NLME 10 4 7.0 c
Note. Scheffe’s multiple comparison test results at 5% significance level
(different letters represent statistical differences).

10776
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SALAS- EL JATIB And WEISKIT TEL
relatively simplistic static data covering a range of conditions and
a suitable statistical model, which addressed the hierarchical struc-
ture of the data, produced the most reliable results for estimating
self- thinning and carrying capacity. This is important as it allows the
larger dat aset to be used and suggests that remeasurements may not
provide more robust estimates of self- thinning behavior. It is impor-
tant to highlight that the S- P- LQMM strategy provided both a suit-
able approach for frontier estimation and a method for assessing the
site- level factors that may influence the c arrying capacit y.
The high variation of the predicted carrying capacit y among
the various strategies indicates that the assumptions and results
from any alternative must be reviewed carefully. For example, some
methods had relatively high variation in estimated site- level car-
rying c apacity, while others showed much more limited variation.
This would strongly influence ecological inference and potential ad-
ditional relationships examined as was conducted in this particular
analysis.
Although the individual- level- based strategy (S- T- LME) has been
devised for mixed- species forests like the ones examined in this
analysis, we believe the method gave too wide of a range for carry-
ing capacity and the overall mean that was too high. In comparison
with the site- level LQMM strategy (S- P- LQMM), the individual-
level- based strategy (S- T- LME) provided an estimate of site- level
carrying capacity that was 32% higher. Likewise, the dynamic- based
strategies tend to provide estimates too high for the self- thinning
line. This may indicate that dynamic methods may be overly sensitive
to mortality dynamics and too limited to represent a broad range
of conditions given the strong reduction in available data for this
method. Nonetheless, the density model (equation 10), which was
based on dynamic data, may be better suited for estimating density
trajectories as in traditional growth models, such as in García (2009).
However, the observed trends from the dynamic data in this analysis
may be influenced by the smaller dataset used and assessment with
much larger datasets should be conducted.
As found in this analysis, carrying capacity was not independent
of site- specific conditions. Prior research on self- thinning has sug-
gested that the limiting relationships between population density
and average individual size are independent of site factors (Pretzsch
& Biber, 2005; Reineke, 1933; Yoda et al., 1963). This simplification,
up to some extent, broadens the use of this concept in applied forest
ecology. As previously found by other studies (Andrews et al., 2018;
Bi et al., 2000; Weiskittel et al., 2009; Zhang et al., 2013), we have
also shown, but for the first time in Nothofagus forests, that carrying
FIGURE4 Behavior of carrying capacity depending on:
dominance of pioneer species (i.e.,proportion of basal area in
Nothofagus species); a climate variable (i.e., precipitation in the driest
month); and tree species diversity (i.e., Shannon index). C arrying
capacity is represented by SDImax. Species diversity levels were set to
low (a), medium (b), and high (c) by using the values of 0.05, 1, and 2
for the Shannon index, respectively
20 40 60 80 100
800
1,000
1,200
1,400
1,600
1,800
Prop. of basal area in Nothofagus (%)
Maximum stand density index (trees/ha)
Pp. driest month
20
50
90
20 40 60 80 100
800
1,000
1,200
1,400
1,600
1,800
Prop. of basal area in Nothofagus (%)
Maximum stand density index (trees/ha)
Pp. driest month
20
50
90
20 40 60 80 100
800
1,000
1,200
1,400
1,600
1,800
Prop. of basal area in Nothofagus (%)
Maximum stand density inde
x (trees/ha)
Pp. driest month
20
50
90
(a)
(b)
(c)

|
10777
SALA S- ELJATIB And W EISKITT EL
capacity depends on an array of factors including site productivity,
species diversity, and successional stage of development. Although
several formulations could be used for representing site produc-
tivity (Stage & Salas, 2007), we focused on identifying bioclimatic
variables that had a biologically consistent behavior. For this anal-
ysis, we found that the precipitation in the driest month negatively
affected carrying capacity and ef fectively represented site produc-
tivity. Higher precipitation in the driest month resulted in a greater
carrying capacity, which is consistent with biological expectations.
Therefore, under the current climate change global scenario, we may
expect a decline in maximum carrying capacity for the majorit y of
this forest type, which were similar to the general recent findings
for Andrews et al., (2018) for the Acadian Region of North America.
Interestingly enough, species diversity (represented by the
Shannon index) may help to overcome the ef fect of adverse cli-
matic conditions as greater species diversity increased site- level
carrying capacity. We think that is because of the resource- use
differentiation among different functional groups. In addition, the
propor tion of basal area in the pioneer species (a proxy for stage
of development) has a quadratic effect into carrying capacity. Our
results indicated that there was an optimum proportion of basal
area occupied by pioneer species, which was approximately 50%
in this analysis. Therefore, pure Nothofagus forests will not likely
achieve the maximum potential carr ying capacity. This finding is
in line with the additive basal area concept (Lawes et al., 2006;
Lusk, 20 02), which has also been suggested in Nothofagus fores ts
as well (Donoso & Lusk, 20 07; Donoso & Soto, 2016). This idea
suggested that both shade- tolerant species contributed in adding
more basal area to a forest because of resource- use differenti-
ation among various functional groups. Our findings are also in
agreement with current research on assessing the mixing of spe-
cies in tree growth (Piot to, 2008) as well as forest productivity
worldwide (Liang et al., 2016). However, our findings differ from
those of Weiskittel et al., (2009) who found that site- level carry-
ing capacity increased with primary species composition purit y for
thre e species in the Pacific Nor thwe st, USA. This may highlight an
important distinction between temperate plantations and natural
rainforests as were examined in this particular analysis.
5 | CONCLUDING REMARKS
The modeling strategy involving the use of static, population- level
data and linear quantile regression mixed- effects provided a relia-
ble ecological behavior for both self- thinning estimation and mod-
eling carrying capacity. The t ype of data (i.e., static and dynamic)
heavily influenced the findings for self- thinning and carr ying ca-
pacity with dynamic methods tending to provide much higher es-
timates. In particular, the density model based on dynamic data
tended to overestimate the self- thinning line, but could likely be
a suitable tool for growth modeling. By fitting the equations in a
mixed- effects framework, the evaluation of various external fac-
tors that may influence carrying capacity could be assessed. In
this analysis, climatic, stage of development, and species diversity
were found to be influ enti al. A lthough t he analysis hig hlighted the
strong influence of modeling strategy on self- thinning and maxi-
mum carr ying capacity, the data, particularly the dynamic dat a,
were relatively small despite covering a wide range of conditions.
Additional analyses using more extensive datasets across a vari-
ety of species are likely necessar y to verify the findings presented
for this analysis. Overall, the findings highlight the challenge in
identifying and defining self- thinning relationships and maximum
carrying capacity despite being fundamental concepts in applied
ecology and management.
ACKNOWLEDGMENTS
This study was suppor ted by the Chilean r esearch grant Fond ecyt No.
1151495 and the USDA National Institute of Food and Agriculture,
McIntire- Stennis Project Number ME0- 41516 through the Maine
Agricultural and Forest Experiment Station. This is Maine Agricultural
and Forest Experiment Station Publication Number 3597.
AUTHOR CONTRIBUTION
The study was designed by CS and ARW, and CS collected the data
with help from different sources. CS analyzed the data and wrote
the manuscript, with input from ARW.
ORCI D
Christian Salas-Eljatib http://orcid.org/0000-0002-8468-0829
Aaron R. Weiskittel http://orcid.org/0000-0003-2534-4478
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How to cite this article: Salas-Eljatib C, Weiskittel AR .
Evaluation of modeling strategies for assessing self- thinning
behavior and carrying capacity. Ecol Evol. 2018;8:10768–
10779. ht tps://doi.or g/10.10 02/ece3.4525