Oikos 119: 455–464, 2010
doi: 10.1111/j.1600-0706.2009.18002.x,
© 2009 The Authors. Journal compilation © 2010 Oikos
Subject Editor: Karin Johst. Accepted 31 July 2009
Pitfalls and challenges of estimating population growth rate from
empirical data: consequences for allometric scaling relations
William F. Fagan, Heather J. Lynch and Barry R. Noon
W. F. Fagan ([email protected]) and H. J. Lynch, Dept of Biology, Univ. of Maryland, College Park, MD 20742, USA. – B. R. Noon,
Dept. of Fish, Wildlife and Conservation Biology, Colorado State Univ., Fort Collins, CO 80523, USA.
The intrinsic rate of increase is a fundamental concept in population ecology, and a variety of problems require that estimates of population growth rate be obtained from empirical data. However, depending on the extent and type of data
available (e.g. time series, life tables, life history traits), several alternative empirical estimators of population growth rate
are possible. Because these estimators make different assumptions about the nature of age-dependent mortality and densitydependence of population dynamics, among other factors, these quantities capture fundamentally different aspects of
population growth and are not interchangeable. Nevertheless, they have been routinely commingled in recent ecoinformatic analyses relating to allometry and conservation biology. Here we clarify some of the confusion regarding the empirical estimation of population growth rate and present separate analyses of the frequency distributions and allometric scaling
of three alternative, non-interchangeable measures of population growth. Studies of allometric scaling of population growth
rate with body size are additionally sensitive to the statistical line fitting approach used, and we find that different approaches
yield different allometric scaling slopes. Across the mix of population growth estimators and line fitting techniques, we find
scattered and limited support for the key allometric prediction from the metabolic theory of ecology, namely that
log10(population growth rate) should scale as –0.25 power of log10(body mass). More importantly, we conclude that the
question of allometric scaling of population growth rate with body size is highly sensitive to previously unexamined assumptions regarding both the appropriate population growth parameter to be compared and the line fitting approach used to
examine the data. Finally, we suggest that the ultimate test of allometric scaling of maximum population growth rates with
body size has not been done and, moreover, may require data that are not currently available.
The intrinsic rate of increase is a fundamental concept in
population ecology. This measure, commonly denoted r or
rmax, is key to understanding diverse problems including success of invasive species, recovery of endangered species, and
the stability of populations. For example, estimates of r are
commonly used in allometric studies (Savage et al. 2004,
Duncan et al. 2007) and in multispecies conservation
research (Pereira et al. 2004, Pereira and Daily 2006). Recent
Congressional hearings on the US Endangered Species Act
have focused on the time needed for species recovery, which
is intimately linked to the rate of intrinsic rate of population
increase. Because of the importance of r in such diverse
settings, it is important that field researchers, informatics
researchers, and population modelers have a common frame
of reference when talking about the rate at which a population can grow.
Unfortunately, the symbol r, and its variants rm and rmax,
have several fundamentally different, but entrenched, meanings in different subdisciplines of ecology (Slade and Balph
1974, May 1976, Caughley 1980). Such longstanding differences stem, in part, from inconsistent terminology for
quantifying aspects of population growth rate (Fenchel
1974, Blueweiss 1978, Hennemann 1983, Thompson 1987).
This terminological vagueness, coupled with a variety of
different approaches for estimating different measures of
population growth rate from empirical data (described
below), creates many opportunities for confusion. The
problem has been compounded in recent years as the development of ecoinformatics as a research approach has instigated the development of compiled datasets for hypothesis
testing. Though compiled datasets have proven tremendously useful in testing important, general predictions from
ecological theory and in promoting the development of
new theory, it is critical that the data involved be biologically meaningful, internally consistent, and interpreted
appropriately. We note that all three of these requirements
have been violated in various analyses of allometric scaling
of maximum population growth rate with body size. First,
large databases of maximum per capita growth rates for
mammalian populations regularly include rates so high as
to be biologically meaningless (up to r > 16 year-1 [i.e. λ
(= er ) exceeding 106 year-1 , where λ (per time step; typically
per year) is referred to as the asymptotic population multiplier]; Thompson 1987, Duncan et al. 2007; see also
Caughley 1980 for related discussion). Secondly, serious
difficulties arise when growth rate metrics with different
455
meanings are commingled as in a recent paper (Duncan
et al. 2007) that reports a test of a key prediction of the
metabolic theory of ecology (Savage et al. 2004). Finally,
terminological confusion has led to the inappropriate interpretation and application of particular measures of population growth, as in Calder (2000) where physiological
maximum rates of population growth are analyzed in a conservation application.
Although previous efforts have attempted to clarify the
key issues associated with the empirical estimation of population growth rate (Davis 1973, Slade and Balph 1974, May
1976, Caughley 1980, Hayssen 1984), significant confusion
still exists. In this paper, we address two assumptions underlying much of the recent work on population growth rates
and its scaling with body size, principally context-specific
metrics of maximum population growth rate and appropriate line fitting methods for drawing inference about allometric scaling.
The paper is organized as follow. First we provide a brief
review of alternative ways of quantifying population-level
rates of change. We then discuss several alternative, commonly used frameworks for estimating the intrinsic rate of
population increase from empirical data for vertebrate populations, namely the Euler equation used in conjunction with
full life table data, the Euler equation used in conjunction
with summary life history traits (mean life span, mean fecundity etc.), and time series approaches. Second, we introduce
the various line fitting approaches that have been used by
other authors to examine allometric scaling relationships
between population growth rate and body size. We demonstrate that these different approaches yield fundamentally
different scaling relationships, and results derived from different line fitting approaches cannot be compared. In the
third and final portion of our analysis, we apply each of the
alternative line fitting approaches to datasets representing
different metrics of maximum population growth to investigate fully the question of allometric scaling of maximum
population growth rates. In our re-analysis of the published
data, we find only scattered and limited support for allometric predictions from the metabolic theory of ecology (Savage
et al. 2004). We conclude with a discussion that highlights
the widespread need for empirical estimates of intrinsic rates
of population increase from scant data and suggest ways in
which data compilation efforts may be streamlined to yield
improved estimates of interspecific variation in the potential
for population growth.
Some of these points have been raised before ­(Caughley 1980)
but the important distinctions among ­metrics appear to have
eroded through time, as recent efforts have misapplied or confounded alternative measures of population growth (Calder
2000, Duncan et al. 2007). We summarize notation, definitions, and appropriate usage of different empirical measures of
population growth rate in Table 1.
To start, consider several common usages of the generic
symbol r in studies of population growth. For example, if
one conducts a detailed demographic study of some age-,
size-, or stage-structured population and then analyzes the
data via demographic matrix techniques (Caswell 2001,
Morris and Doak 2002), one typically obtains an estimate of
λ (the asymptotic population multiplier). In this context,
the symbol r (= ln λ) (year-1) is the corresponding rate of
population change in an exponential form, and it represents
the rate of population growth that should be ultimately
expected ‘if the conditions under which the data were
obtained were to persist into the future’. A parallel tradition
exists in wildlife management contexts wherein annual population census data (denoted Nt with t being time in years)
are routinely used to calculate an observed rate of change
using raw counts rather than age-structured data (Caughley
and Birch 1971, Hone 1999). In this context, r, termed the
rate of increase, is typically interpreted as the slope of ln (Nt)
against t (Caughley 1980), and reflects population growth
‘that has already been observed over some period of time’. In
ecological modeling, the symbol r (and its variants rm and
rmax), are taken to be the intrinsic rate of natural increase,
which corresponds to ‘the maximum growth rate that a population can achieve under density-independent conditions
in a particular environment’ (Kot 2001). Note also, that an
entirely different suite of problems involving unreasonably
high values of population growth rate may occur because r
(used generically) did not always refer to growth on the log
scale. This issue may be the source of some of exceptionally
high values of population growth found in very early papers
(predating those discussed here). Rather than attempt to reconcile these entrenched alternative uses of the symbol r, we
simply note that the symbol has fundamentally different
connotations in different contexts. This paper focuses on
the empirical estimation of the intrinsic rate of natural
increase, and for consistency with some previous treatments
(Caughley 1980), we will use a series of specific symbols
throughout our paper when referring to different measures
of population growth rate (Table 1).
Key measures of population-level rates of
change
Alternative approaches to estimating
­population growth rate from empirical data
To develop a common foundation for our analyses, we briefly
review several related measures of population-level rates of
change. These rates range from experimentally measured
instantaneous rates of change (denoted ~r ) to theoretical maximum rates of change derived from life history information
that may, or may not, reflect biologically reasonable parameters for wild populations. This review is an important first step
because a fundamental distinction exists between subdisciplines concerning notation for measures of ­population change.
Several non-interchangeable frameworks exist for estimating
population growth rate, from empirical data. We discuss
three here: the Euler equation with life table data (which
yields an estimate denoted ~r ; note alternative methods for
estimating ~r in Table 1), approximations to the Euler equation employing life history traits (which yield estimates
denoted ~r , rmax , or ρ depending on what one assumes about
survivorship), and time series of population counts (which
yield an estimate denoted rm).
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Table 1. Different measures of population growth rate, their assumptions, and their applications1.
Symbol
Definition
Function of
Estimation approach
Relevant to
~r
Per-capita rate
of population
growth
estimated from
field studies
Per-capita rate of
population
growth
estimated from
life history
traits
1) logarithm of the dominant eigenvalue of
a demographic matrix (Caswell 2001)
2) solving the Euler equation for a specific
lifetable from a wild population2 (Eq. 1, 2)
3) direct estimation from count data, i.e.
regressing log(Nt) on t (Caughley 1980)3
4) mark–recapture analysis (Pradel 1996,
Nichols and Hines 2002)
Approximating a solution to the Euler
equation using a suite of in-the-wild life
history parameters in the absence of a
lifetable and assuming exponentially
distributed mortality (type II survivorship, Pereira and Daily 2006) (Eq. 4)
Any context-specific analysis
r̂
rm
Per-capita
population
growth rate at
very low
density
1) intercept of linear regression of ln(Nt+1/
Nt) vs Nt from a time series of population
counts or densities (Fagan et al. 2001)
2) intercept of regression based on
alternative density dependent forms
(Sabo et al. 2004)
1) Parameterization of
population dynamic
models4
2) Characterizing population
recovery potential of at-risk
species
rmax
Maximum
density-independent
per-capita
population
growth rate
Physiological
maximum
per-capita
population
growth rate
Intrinsic organismspecific limits,
stochastic mortality,
environmental
conditions,
available resources
(food, shelter etc.),
population density
Intrinsic organismspecific limits,
stochastic mortality,
environmental
conditions,
available resources
(food, shelter etc.),
­population density
Intrinsic organismspecific limits,
stochastic mortality,
environmental
conditions,
available resources
(food, shelter etc.)
Intrinsic organismspecific limits,
stochastic mortality
Solving the Euler equation for a specific
lifetable from a laboratory or captive
population maintained under low density,
high-resource conditions (Eq. 1, 2)
Allometric scaling and other
multispecies comparisons
Intrinsic organismspecific limits
Approximating a solution to the Euler
equation using a suite of life history
parameters in the absence of a lifetable
and assuming uniform threshold
mortality (type I survivorship, Cole 1954)
(Eq. 3)
Purely theoretical investigation involving the Euler
equation (Cole 1954) and/
or the sensitivity of r to
components of the Euler
equation
ρ
Any context-specific analysis
in many previous papers, these metrics have been referred to inconsistently as r, rm, or rmax, leading metanalyses and compilation studies to
combine estimates from different sources when in fact those estimates are not comparable (Calder 2000, Duncan et al. 2007). This set of
definitions builds on those introduced by Caughley (1980).
2
solving the Euler equation under the assumption of exponential mortality (type II survivorship) is an alternative approach for estimating r̂ that
does not require a specific lifetable (Pereira and Daily 2006). For many mammal species (over a wide range of mammal body sizes), this
approach yields estimates of r̂ that approximate those from the corresponding full life tables (r~) (Lynch and Fagan 2009).
3
this estimation approach, which is based on count data without regard to age structure and affords a retrospective summary of what the
population growth rate has been, differs from estimation approaches 1 and 2, which are based on demographic data and yield asymptotic
estimates of what the population growth rate will be.
4
because it reflects density-independent population growth rate, this measure most closely mirrors the mathematical parameter r that appears
in standard theoretical models of population growth such as the logistic or Lotka-Volterra equations.
1
Using the Euler equation to estimate population
growth rate from life table data
β
∑l m e
The first framework has its roots in the science of demography (Keyfitz 1985) and in life-table studies in ecology.
When full life tables are available, estimates of r can be
obtained from the Euler equation (Roughgarden 1996,
Kot 2001)
∞
∫ l(x)m(x)e
0
−r x
(1)
dx = 1
where l(x) is the survivorship to age x, and m(x) is the per
capita fecundity of female offspring at age x. Note that
m(x) can be defined in any number of ways. For example,
one example would be to use a delta function formulation
(discussed below). Alternatively, one can use the discrete
analog
− rx
=1
(2)
where α is the age at first reproduction, β is the age at last
reproduction, lx is the probability of surviving to age x, and
mx is the number of female offspring produced per female of
age x (Cole 1954). Note that r from the Euler equation and
log(λ) from an age-structured matrix model should be the
same, if the matrix is constructed correctly. As discussed by
Caughley (1980), it is important to recognize that values of
r obtained from Eq. 1 or 2 are specific to the particular population, habitat, and conditions under which the life table
data were obtained, just as estimates of λ are specific to the
conditions under which data were obtained for their corresponding demographic matrices (Caswell 2001). Consequently, values of r obtained from Eq. 1 or 2 do not
necessarily correspond to the density independent measures
of population growth that theoreticians intend when they
x =α
x
x
457
write ‘r’ in an equation for population dynamics (Caughley
1980). Put another way, while a value of r obtained from Eq.
1 or 2 does characterize a rate of exponential growth, that
rate is not necessarily the same exponential rate that would
be observed if the population were growing under conditions of ample resources and benign conditions (Table 1).
Approximations to the Euler equation employing
life history traits
In many situations, full life table data will not be available,
and approximations to Eq. 1 or 2 will be necessary to estimate r. Reliance on such estimates has a long history in ecology (Cole 1954) and their use continues to the present day
(Pereira and Daily 2006, Duncan et al. 2007). To obtain
such an approximation, one common approach is to assume
that all individuals in a population survive to a common age
(the average longevity, denoted L) at which point all individuals die (i.e. lx =1 for all x < L and lx = 0 for x ≥ L); this
corresponds to an extreme type I survivorship curve (Pearl
1928). In this scenario, Eq. 1 can be rewritten as
e − ρ + me − ρ( α ) − me − ρ(β +1) = 1 (3)
where m is the average number of female offspring produced
per female per year (Cole 1954) and ρ is the estimated maximum population growth rate (Caughley 1980; Table 1).
Many authors have adopted this approximation, hereafter
referred to as the “Cole approximation”, as an estimate of the
maximum population growth rate (Blueweiss et al. 1978,
Henneman 1983, Schmitz and Lavigne 1984, Robinson and
Redford 1986, Ross 1992, Fisher et al. 2001). Duncan et al.
(2007) have compiled estimates of ρ from the aforementioned six studies into one table. To that compilation, we add
data from Thompson (1987), yielding a total of 285 mammalian species for which ρ has been estimated using Eq. 3
(Supplementary material Appendix 1). In general, values of ρ
from the approximation in Eq. 3 are somewhat correlated
with, but generally much larger than, values of r obtained
from Eq. 1 when full life table data are available (Lynch and
Fagan 2009). The quantity ρ has been used to represent the
maximum population growth rate of a species although even
low density, high resource populations experience stochastic
mortality (i.e. losses due to predation, disease, and other
sources that induce mortality before maximum lifespan), thus
violating the primary assumption that leads to the derivation
of Eq. 3.
A different approximation to Eq. 2 arises if one assumes
that mortality is independent of age and occurs at a rate µ
per unit time, thus l ( x ) = e −µx (and Sx=1x+1/1x is constant).
This corresponds precisely to a type II or exponential survivorship curve (Pearl 1928). Starting from Eq. 1, Pereira and
Daily (2006) used this assumption to derive
∞ ∞
 ∫ ∑ δ ( x − y∆ − α ′ ) e −(ˇr + µ )x dx = 1
m
0 y =0
(4)
 is the number of female offspring per litter, ∆ is
where m
the interval between litters, and α ′ is the age at first
­reproduction. Births are viewed as occurring in pulses,
458
leading to a birth pulse function δ(x) that evaluates to 1/T
for 0 ≤ x ≤ T and is 0 elsewhere. Pereira and Daily (2006)
used this approach to obtain an estimate of r, denoted r̂
(Table 1), for 155 species of mammals from Costa Rica, for
both native and countryside habitats. These data are available as an appendix to Pereira and Daily (2006) so we do not
repeat them here. For many mammal species (over a wide
range of mammal body sizes), Eq. 4 yields estimates of r̂
that closely mirror values of r obtained from full life tables,
which is not true for Eq. 3 (Lynch and Fagan 2009). Thus,
unlike the more widely used estimate represented by ρ, the
quantity r̂ could be a realistic characterization of survivorship within a population of animals.
One reason for the popularity of approximations to the
Euler equation, especially Eq. 3, is that estimates of r can be
obtained from scant life history trait data (e.g. estimates of
litter size, age at first reproduction). This ease of estimation
would appear to be a real benefit when dealing with multiple
species or with species that have not been intensively studied. However, as we discuss below, estimates of ρ and r̂ (Eq.
3 and 4, respectively) are fundamentally different, and moreover, these estimates systematically diverge as functions of
average individual mass.
As discussed above, generic applications of Eq. 1 or 2 do
not imply anything about the rate of growth of a population
at low density under benign conditions. On the other hand,
if life table data were available for a laboratory or captive
(e.g. zoo) population wherein population growth conditions did involve low population densities and ample
resources, one could use Eq. 2 to obtain an estimate, denoted
rmax, that reflects a species’ capacity for population growth
under the best of all possible circumstances while also
acknowledging natural variation in longevity among individuals within a population (Table 1). The metric rmax thus
contrasts with ρ from Eq. 3 because the latter estimate
assumes that all individuals live to a common age before
dying and strictly annual reproduction (Table 1). Savage et
al. (2004) present rmax estimates for a couple of mammal
species (e.g. laboratory rodents), but data for rmax have not
yet been systematically compiled for use in mammalian
allometric studies.
Estimates of population growth rate from time series
In contrast to the above approaches based on the Euler equation,
another framework for estimating r derives from wildlife population monitoring studies, wherein one would count or otherwise
census the abundance of populations repeatedly in different years
to obtain several estimates of Nt. Within this framework, one can
estimate a different variant of r, denoted rm, as the intercept of the
regression of ln(Nt+1/Nt) versus Nt (Morisita 1965, Fagan et al.
2001). (Note this is just one approach to fitting models to timeseries, and many other more realistic modeling approaches and
parameter estimation techniques exist, Kendall et al. 1999, Sabo
et al. 2004). However, the simple approach has the advantage of
a long history in ecology that has resulted in its application to a
great diversity of species, which is why we rely upon it here).
Note that rm ≠ rmax; unlike any estimates derived from the Euler
equation, rm captures a population’s instantaneous per-capita
­population growth rate at very low densities (Table 1) and is not
the time-averaged population growth rate over the course of the
timeseries. In many cases, linear regression will suffice to estimate rm on a plot of ln (Nt+1 /Nt) versus Nt (Morisita 1965, Fagan
et al. 2001). Such a linear regression corresponds to an assumption that the density dependence in Eq. 1 follows that of a Ricker
function (e.g. f(Nt) = Nt /K, where K is the carrying capacity of
the population). Fagan et al. (2001) used this approach to estimate rm for 72 mammalian species from the Global Population
Dynamics Database (NERC 1999). Duncan et al. (2007) have
compiled nearly 100 additional estimates of mammalian rm values from the literature. Combining the two datasets we have
compiled estimates of rm for 119 mammalian species.
The impact of estimation assumptions on the
allometric scaling of r
Multi-species analyses have not previously addressed the
relative performance of the different metrics of population growth (Table 1). In particular, systematic multispecies comparisons of the different life-history based
approximations to the Euler equation (Eq. 3–4) and
regression techniques are lacking. We conduct these comparisons here, demonstrating that the estimation
approaches yield fundamentally different estimates of
population growth rate for the same species. These differences have far-reaching implications that extend to recovery planning for at-risk species and allometric theory.
Unfortunately, there is almost no overlap in which species
are represented in the different empirical datasets. This
precludes a direct comparison of different techniques
applied to a common dataset.
As might be expected, estimates of the various population
growth rate metrics (Table 1) depend on which values of life
history traits are used for a species. In some cases, estimates
of life history traits necessary for the calculation of growth
rate differed among sources. For consistency when comparing among estimation techniques, we used estimates
 , ∆ from the YouTHERIA Project (formerly
for α, β, m
­PanTHERIA, Bielby et al. 2007) and the AnAge database
(de Magalhaes 2005). Estimates of µ (necessary for Eq. 4)
were used directly from the ‘native habitat’ compilation of
Pereira and Daily (2006).
Analyzing the relationship between population growth
rate and body mass has a long history (Fenchel 1974).
Because mammals span almost nine orders of magnitude in
mass and because comparatively good data on body mass,
other life history traits, and population change are available
for mammals, many authors have examined allometric scaling of mammalian population growth rate (McLaren 1967,
Hennemann 1983, 1984, Schmitz and Lavigne 1984,
Robinson and Redford 1986, Thompson 1987, Ross 1992,
Slade et al. 1998, Duncan et al. 2007). Recently, Savage et al.
(2004) extended the metabolic theory of ecology (West et al.
1997) to predict that population growth rate should scale as
body mass raised to the –0.25 power.
To test this prediction, Duncan et al. (2007) compiled
estimates of mammalian ρ and rm from diverse sources. The
analysis of Duncan et al. (2007) combined estimates of
ρ and rm in a single ordinary least squares regression analysis
and, for comparison, in a single phylogenetically corrected
generalized least squares regression. As we discussed above,
ρ and rm are fundamentally different quantities that capture
different aspects of a population’s capacity for growth
(Table 1). As such, these measures should not be commingled in a statistical analysis.
To test if different estimation techniques yield estimates
of population growth rate that vary systematically among
mammalian species as a function of body size, we performed a series of allometric regressions. For each of the
three compilations of mammalian population growth rate
estimates (i.e. the datasets for ρ, r̂, and rm), we performed
four different allometric regressions. First, we conducted a
standard ordinary least squares (OLS) regression of the
estimate of population growth rate (generically ‘r’) on body
mass. Second, we conducted a phylogenetically corrected
OLS regression in which phylogenic relatedness among
species, which violates the sample independence assumed
by OLS regression, is accounted for by explicitly including
the phylogenetic correlation structure into the error structure of the regression model (Supplementary material
Appendix 2; Ives et al. 2007). Third, we conducted a standardized major axis (SMA, also called RMA) regression of
the estimate of ‘r’ on body mass; SMA regression accounts
for uncertainty in both the x- and y-axis measures and is
appropriate when seeking to characterize the relationship
between two variables (as opposed to using one variable to
predict another), such as in tests of predictions from allometric theory (Warton et al. 2006). Fourth, we conducted
a phylogenetically corrected SMA regression. Both the
OLS and SMA phylogenetically corrected regressions
adopted Grafen’s method (power = 1) for characterizing
phylogenetic divergence (Grafen 1989). An alternative
approach, assuming all branch lengths are of equal length,
produced qualitatively similar results. For all allometric
analyses, we adopted the standardized, species-specific estimates of female mass (denoted M) given in the YouTheria
Project (Bielby et al. 2007). All variables were log10 transformed before each of the four analyses. For more detailed
descriptions of our regression methods and their statistical
assumptions, see Supplementary material Appendix 2.
Given the phylogenetic structure of the error, and recognizing that the purpose of allometric line fitting is to summarize a relationship between two variables (as opposed to
using one variable to predict another), we consider results
from phylogenetically corrected SMA regression to yield the
best estimates of allometric slope. However, as discussed in a
thorough analysis by O’Connor et al. (2007), phylogenetically corrected SMA regression may yield allometric slope
estimates that are too steep. Likewise, slope estimates from
phylogenetically corrected OLS will be systematically too
shallow (O’Connor et al. 2007). Another approach, least
squares variance-oriented residual (LSVOR) regression, proposed by O’Connor et al. (2007), seeks to minimize this bias
by estimating the extent of standard error in the x and y
dimensions for a single point relative to the sample variance
in x and y. In our case, however, it was impossible to estimate
the standard error in the y values (the population growth
rates) because those errors are not generally reported along
with the point estimates in the literature. Therefore, for completeness, we report results for all four analyses described
above, but we deem all results from regressions lacking
­phylogenetic correction to be inappropriate given the phylogenetic structure evident in the datasets.
459
Figure 1. Histograms of empirical estimates of the different measures of the rate of population increase from approximations to the Euler
equation (ρ and r̂) and from time-series analyses (rm). See Table 1 for definitions of the different measures of population growth rate.
Results
Figure 1 presents frequency histograms of empirical estimates of mammalian ρ, r̂ and rm compiled from published
datasets. For the datasets reporting estimates of population
growth rate via approximations to the Euler equation (Eq. 3,
4), approximately 8% of species had ρ ≥ 3, whereas 21% of
species had r̂ ≥ 3. These rates correspond to the capacity to
increase >20 fold over the course of one year (Fig. 1). Approximately 1% of the estimates of ρ and r̂ exceed 10, corresponding to annual population increases in excess of 22 000
fold. For the dataset derived from time series, 7% of rm estimates exceed 3, and the largest value of rm is 6.5. Note however that all of these estimates assume no density dependence
kicks in after the first litter, and as such can be misleading.
Supplementary material Appendix 1 gives the names, standardized biomasses, and estimates of population growth rate
for those species that do not already appear in the appendices
to Duncan et al. (2007) or Pereira and Daily (2006).
When we plot the data from Fig. 1 as a function of body
size, we find significant evidence for allometric scaling in
each of the three datasets but different allometric slopes
among datasets (Table 2). Figure 2 summarizes the allometric relationships for each of the three metrics of population growth rate, and best fit curves from phylogenetically
corrected SMA regressions are shown for comparison. In
Fig. 2C, where rm is plotted against body mass, the best fit
curve appears strikingly skewed from the central tendency
of the data cloud. This reflects the high degree of phylogenetic structure in this particular dataset. This dependence
on phylogeny is made clear in Supplementary material
Appendix 3 where we replot the regression including the
relative weighting of each datapoint in determining the
model fit.
As judged by the 95% confidence intervals around each
fitted regression slope, only six of the 12 possible regressions
(three datasets × four regression techniques) yield allometric
slopes that are consistent with the predicted –0.25 slope
from metabolic theory (Table 2). When the empirical estimates of mammalian ρ, r̂ and rm are treated separately in
allometric regressions, only those OLS regressions involving
rm and ρ are consistent with the predicted –0.25 slope.
However, all three slopes from the phylogenetically corrected
OLS regressions are consistent with the theoretical prediction. For SMA regression, the allometric slope from the
count based estimates (rm ) is consistent with metabolic theory, but neither of the slopes derived from the Euler equation (ρ and r̂) matches the theoretical prediction. For the
phylogenetically corrected SMA regressions, none of the
allometric slopes are consistent with the prediction from
metabolic theory, with all three slope estimates being considerably steeper than metabolic theory predicts (Table 2).
Overall, allometric relationships based on rm are consistent
with predictions from metabolic theory for three of the four
regression approaches, but not for the phylogenetically corrected SMA regression.
Table 2. Allometric slopes (± 95% CI) of three measures of population growth rate as estimated by four regression techniques.
Regression technique
Population growth
estimator
r̂
ρ
rm
OLS
Phylogenetically
corrected OLS*
SMA
–0.36 (–0.41,–0.31)
–0.28 (–0.31,–0.253)
–0.20 (–0.23,–0.17)
–0.19 (–0.29,–0.10)
–0.26 (–0.30,–0.22)
–0.25 (–0.34,–0.15)
–0.49 (–0.55,–0.44)
–0.38 (–0.41,–0.35)
–0.27 (–0.30,–0.24)
Phylogenetically
corrected SMA
–0.64 (–1.00,–.41)
–0.43 (–0.49,–0.38)
–0.57 (–0.79,–0.41)
*Note: from a statistical perspective, the OLS model with phylogenetic correction is no longer an OLS model but rather a generalized least
squares (GLS) model, but we have retained the OLS terminology in the table and elsewhere in the manuscript for simplicity and ­comparison
with the other models.
460
Discussion
Significant confusion exists in the literature both with regard to
the existence of alternative, non-interchangeable estimators of
population growth rate and with regard to what statistical analyses are appropriate to gauge interspecific variation in those
estimators. Our main goals here have been to bring some terminological clarity to the first issue and, via parallel statistical
analyses of compiled data on mammalian population growth
rates, some quantitative illustrations of the second issue. We
discuss each of these two points in turn before providing some
summary suggestions for productive ways forward.
Alternative, non-interchangeable estimators of population growth rate
Figure 2. Allometric scaling of different measures of population
increase in mammals derived from empirical data. Panel (A)
gives the best fit allometric scalings for ρ derived from Cole’s
(1954) approximation to the Euler equation that assumes type
I survivorship using data compiled across multiple sources
(Thompson 1987, Duncan et al. 2007). Panel (B) gives the
allometric scaling for r̂ from Pereira and Daily’s (2006) dataset
for Costa Rican mammals. Panel (C) gives the allometric scaling for rm from time series (Fagan et al. 2001, Duncan et al.
2007). Note that the data for this panel are highly structured
phylogenetically, and this structuring drives the location of the
regression slope. In all panels, we plot allometric relationships
obtained from phylogenetically corrected SMA ­regression
(O’Connor et al. 2007).
To check the generality of our findings, we reran all the
allometric analyses excluding all species from the datasets for
Fig. 1A and 1C that were larger than the largest species in
Fig. 1B (tapir). With the modified datasets, our estimates of
the allometric slopes were largely comparable to those in
Table 2 (Fagan et al. unpubl.). The major exceptions being
that two slope estimates (for ρ using ­phylogeneticallycorrected OLS and for rm using SMA) shift slightly such that
their confidence intervals no longer overlap –0.25.
The Cole (1954) approximation, arguably introduced as an
algebraic convenience to the (then) computationally difficult
Euler equation, has been used and cited so broadly that its
application has been effectively disconnected from the unrealistically harsh assumptions it makes about inter-individual differences in survivorship. By reducing the estimation of a
population’s growth rate from a life table to the far simpler
problem of obtaining a few life history traits, the Cole (1954)
approximation greatly broadens the range of species for which
such growth rates may be calculated. However, the estimates
obtained via this approximation are unreasonably large, especially for small bodied species (Lynch and Fagan 2009). Given
these problems, we argue that ρ is not a useful metric for characterizing interspecific differences in population growth rate
and strongly discourage further use of this estimator. In place
of the Cole (1954) approximation, we urge wider usage of the
Pereira and Daily (2006) estimator when researchers seek to
estimate population growth rates from life history data. The
Pereira and Daily (2006) estimator, which hinges on exponential (type II) survivorship, provides a far better match to the
survivorship patterns exhibited by a wide range of wild mammal populations (Lynch and Fagan 2009), and consequently,
yields far more reasonable estimates of population growth rate.
Building on Cole (1957), we emphasize that a predictable
relationship exists among the different metrics of population
growth rate (Table 1). Specifically, if one holds life history
traits constant, then for a given species the relationship is
−∞ ≤ r, r^ ≤ rm ≤ rmax ≤ ρ ≤ ∞ (5)
The relationship in Eq. 5 assumes that there are no Allee
effects in operation that reduce rm at small population sizes.
This sequencing of inequalities occurs because, as one moves
from left to right, the metrics systematically exclude more
and more ecological complexities found in real-world populations. When present, these complexities (e.g. stochastic
mortality, resource constraints, population density) act to
reduce a population’s capacity for growth. However, we caution that estimates of population growth rates from ­compiled
datasets may deviate from this relationship for particular
­species. This is because the estimates of population growth
rate calculated via Eq. 2–4 hinge on parameters that reflect
life history traits, and differences among authors in the
values used for these parameters are widespread in the
461
literature. Although we have used a standardized set of
biomass estimates for the allometric regressions in Fig. 2, differences in other traits may easily affect values of ρ, r̂ and rm,
and in many cases, these trait values are not specified in the
original literature sources. Results in Lynch and Fagan (2009)
demonstrate that key elements of Eq. 5 hold true for each of
58 mammal species when life history traits are standardized
within species across metrics.
The methods used to obtain these different measures of
population growth rate make strikingly different assumptions about the factors influencing population growth
(Table 1).
Although these measures often share a common symbol (r),
they are fundamentally different quantities, and are not interchangeable. With the exception of ρ, which we argue should
simply be abandoned in the context of mammalian studies, all
of these measures provide insight on some aspect of a population’s capacity for growth, and will be more or less useful in
different contexts (Table 1). For example, applications to conservation planning should employ estimates of ~r ,r̂ or rm
depending on the species, timeframe, and availability of data
because these estimators are based upon field data. Likewise,
analyses of the impacts of environmental variability, density
dependence and population cyclicity or chaos should carefully
consider their choice of empirical estimator because certain
estimators yield high values for r more frequently than others.
For example, Lande et al. (2003) discuss a variety of techniques
for estimating population growth rate from time series given
environmental and demographic stochasticity, whereas Holmes et al. (2007) focuses on the issue of estimating population
growth rate given observation error. The Ricker model, which
is the foundation for the regression approach used by Fagan et
al. (2001), assumes scramble competition. Consequently, the
Ricker model is not a universally good representation of mammalian density dependence, and other models, such as the
theta-logistic, Gompertz, Beverton-Holt or even delayed Ricker
(for cycling populations) may often be more biologically realistic given the nature of density dependence in wild populations.
Unfortunately, if the time series does not include episodes of
low density, it can be impossible to distinguish these models
statistically, despite the very different implications for densityindependent growth (Sabo et al. 2004, Doncaster 2008).
Statistical issues involved in analyzing the allometric
scaling of population growth rate
Because the different estimators of population growth rate
make different assumptions, they are not interchangeable
and should not be commingled in ecoinformatic analyses.
This need to avoid commingling data from the different estimators is especially important in the context of allometric
regressions. Even if the different estimators had comparable
allometric slopes (which remains a possibility given the wide
confidence intervals associated with the slope estimates,
Table 2), the large differences in the magnitudes of population growth rates generated may strongly influence estimated
allometric slopes when data types are commingled. To see
this consider that different estimators of population growth
rate may be more or less available for animals of different
sizes. For example, the Pereira and Daily (2006) dataset
includes only eight species with mass ≥ 20 kg, and thus only
462
5.1% of the available estimates of r̂ are for such larger species. In contrast, in the compiled dataset of rm estimates,
46.7% of species exceed 20 kg. Thus, disproportionate use of
rm for large mammals together with disproportionate use of
r̂ for small mammals has the potential to drive the estimated
allometric slope artificially shallow because rm tends to be
larger than r̂ for a given species (Eq. 5). Moreover, the consequences of such commingling are made even worse in situations where estimates of population growth rate for particular
species may have extraordinarily large ‘phylogenetic leverage’
resulting from a combination of a high growth rate and their
close phylogenetic relatedness to other species in the sample
(Supplementary material Appendix 3).
In addition to the confusion concerning the existence and
commingling of alternative, non-interchangeable estimators
of variants of r, the important issue of allometric scaling of
population growth rate suffers from significant discord over
what statistical analyses are appropriate. Both OLS and SMA
regression have been widely used to explore such allometric
scaling (Fenchel 1974, Thompson 1987, Savage et al. 2004),
and the importance of correcting for phylogenetic structure
in compiled databases is increasingly recognized (Duncan
et al. 2007, O’Connor et al. 2007). Different statistical
approaches to allometric regression yield fundamentally different results. For example, for the four different approaches
to allometric regression summarized in Table 2, the variance
among regression techniques is more than five times as large
as the variance among the different estimators of population
growth rate. If one adheres, as we do, to the argument that
phylogenetic corrections are essential in analyses of allometric scaling (O’Connor et al. 2007), then the discrepancy
between regression techniques becomes especially clear. All
three of the phylogenetically corrected OLS regressions
(based on different estimators of population growth rate)
show support for the –0.25 scaling prediction from the metabolic theory of ecology (Savage et al. 2004), whereas none
of the phylogenetically corrected SMA regressions show such
support. These results suggest that the issue of whether mammalian population growth rates support the –0.25 slope prediction remains an unsettled question. Our findings that
SMA slopes systematically exceed their respective OLS slopes
matches the ordering found in O’Connor et al. (2007).
What would it take to quantify rigorously the allometric
scaling of population growth rate among mammals?
Given the caveats we have raised concerning the non-­
interchangeable nature of the various metrics of population
growth rate (Table 1), the fundamental problems with using
estimates of growth rate based on the Cole (1954) approximation, and the important statistical issues associated with
testing allometric relationships, what realistic approaches
exist for testing for the –0.25 allometric scaling prediction
across mammals? This is a critical question to answer because
confirming or refuting predictions of the metabolic theory of
ecology (Savage et al. 2004) hinge on getting right both the
underlying dataset of population growth rates and the statistical analysis of allometric scaling. From the data compilation
standpoint, the ideal dataset would be restricted to estimates
of rmax obtained from solving the Euler equation using empirical survivorship data from captive populations reared under
conditions of ample resources. Because many animals do not
(or are not allowed to) breed well in captivity, one would need
to combine survivorship data from zoos with measurements
of maximum potential litter size to obtain estimates of population growth rates (Lynch et al. unpubl.). Barring such a
dataset, it should be possible to obtain reasonable approximations of rmax by parameterizing (Eq. 4) using longevity and
other life history data from captive populations. We denote
these estimates r̂captive to distinguish them from the conventional output of Eq. 3 based on field data. Such an approach
comes close to the ‘laboratory populations’ initially investigated by Fenchel (1974) and matches several of the critical
assumptions made by Savage et al. (2004) in their theoretical
derivation of the expected dependence of intrinsic growth
rate on body mass according to the metabolic theory of ecology. Unlike the Cole approximation (Eq. 3), such an approach
would retain the inter-individual differences in survivorship
that can have such a profound impact on population growth
rate (Lynch and Fagan 2009). Although such datasets would
necessarily exclude certain large bodied species that cannot be
maintained well in captivity (e.g. whales), the resulting dataset should still span six or seven orders of magnitude, which
should be sufficient for quantifying allometric dependencies.
From a data analysis standpoint, the ideal analysis would
employ both OLS and SMA regressions, both with phylogenetic correction, with the expectation that the true allometric
slope would fall in between the resulting estimates (O’Connor
et al. 2007). Until such a dataset is compiled and analyzed,
we suggest that whether or not empirical data on mammalian
population growth rates match the specific predictions of the
metabolic theory of ecology remains an open and unresolved
question.
Acknowledgements – Support for this project came, in part, from the
Dept of Defense SERDP Award SI 1475 to WFF and M. Neel. A.
McGuirk and J. Balachowski made helpful contributions to the
project through data collection.
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