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Simple life-history traits explain key effective population size
ratios across diverse taxa
Robin S. Waples, Gordon Luikart, James R. Faulkner and David A. Tallmon
Proc. R. Soc. B 2013 280, 20131339, published 7 August 2013
Supplementary data
"Data Supplement"
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Simple life-history traits explain
key effective population size ratios
across diverse taxa
rspb.royalsocietypublishing.org
Robin S. Waples1, Gordon Luikart2, James R. Faulkner1 and David A. Tallmon3
1
Research
Cite this article: Waples RS, Luikart G,
Faulkner JR, Tallmon DA. 2013 Simple lifehistory traits explain key effective population
size ratios across diverse taxa. Proc R Soc B
280: 20131339.
http://dx.doi.org/10.1098/rspb.2013.1339
Received: 27 May 2013
Accepted: 26 June 2013
Subject Areas:
evolution, genetics, ecology
Keywords:
overlapping generations, life history, age at
maturity, adult lifespan, age structure,
iteroparity
Author for correspondence:
Robin S. Waples
e-mail: [email protected]
Electronic supplementary material is available
at http://dx.doi.org/10.1098/rspb.2013.1339 or
via http://rspb.royalsocietypublishing.org.
Northwest Fisheries Science Center, National Marine Fisheries Service, National Oceanic and Atmospheric
Administration, Seattle, WA 98112, USA
2
Flathead Lake Biological Station, Fish and Wildlife Genomics Group, Division of Biological Sciences,
University of Montana, Polson, MT 59860, USA
3
Biology and Marine Biology Program, University of Alaska Southeast, 11120 Glacier Highway, Juneau,
AK 99801, USA
Effective population size (Ne) controls both the rate of random genetic drift
and the effectiveness of selection and migration, but it is difficult to estimate
in nature. In particular, for species with overlapping generations, it is easier
to estimate the effective number of breeders in one reproductive cycle (Nb)
than Ne per generation. We empirically evaluated the relationship between
life history and ratios of Ne, Nb and adult census size (N) using a recently
developed model (AGENE) and published vital rates for 63 iteroparous animals and plants. Nb/Ne varied a surprising sixfold across species and,
contrary to expectations, Nb was larger than Ne in over half the species.
Up to two-thirds of the variance in Nb/Ne and up to half the variance in
Ne/N was explained by just two life-history traits (age at maturity and
adult lifespan) that have long interested both ecologists and evolutionary
biologists. These results provide novel insights into, and demonstrate a
close general linkage between, demographic and evolutionary processes
across diverse taxa. For the first time, our results also make it possible to
interpret rapidly accumulating estimates of Nb in the context of the rich
body of evolutionary theory based on Ne per generation.
1. Introduction
Effective population size (Ne) influences both the rate of random genetic drift
and the effectiveness of natural selection and migration [1], and therefore is crucial for understanding evolutionary processes as well as for conservation
management. Although elegantly simple in concept, Ne is difficult to estimate
in nature, particularly for iteroparous species that can reproduce in more
than one season [2,3]. A number of studies have evaluated life-history factors
that influence the ratio of Ne to census size (N), with the goal of identifying
rules-of-thumb that could be applied to species for which detailed demographic
information is not available (nicely reviewed by Lee et al. [4]). However, these
theoretical analyses have generally made simplifying assumptions (e.g. vital
rates that do not vary with age) and/or evaluated changes in one factor at a
time while holding others constant. But most species do not have constant
vital rates (figure 1), and life-history traits are often correlated and do not
vary independently [5].
Furthermore, no previous studies have evaluated the general relationship
between Ne per generation and the effective number of breeders in 1 year or
breeding cycle (Nb) in iteroparous species. This is a crucial knowledge gap,
as evolutionary processes depend on Ne, but Nb is much easier to estimate in
species with overlapping generations [3,6]. Spurred by recent, rapid advances
in molecular marker development [7], non-lethal DNA sampling in the wild
[8], and sophisticated genetic data analysis software [9], molecular-based estimates of effective size have skyrocketed in the last 5 years [3]. Increases have
primarily come from new single-sample methods [10 –12] that often provide
& 2013 The Author(s) Published by the Royal Society. All rights reserved.
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(a)
2
1.0
annual survival (sx)
0.6
0.2
0
(b)
1.0
relative fecundity
0.8
cod (20)
scrub jay (20)
male elephant seal (15)
wasp (35 days)
0.6
0.4
0.2
0
0.2
0.4
0.6
fraction of adult lifespan (AL)
0.8
1.0
Figure 1. (a) Patterns of age-specific survival and (b) relative fecundity for selected species, starting at age at maturation (0.0 on the x-axis). Within a species,
relative fecundity was standardized by the maximum bx for any age. Maximum age (v, in years unless otherwise noted) is shown in parentheses after species name.
(Online version in colour.)
estimates more relevant to Nb than Ne [3,6]. The relationship
between annual and generational effective size has been
studied in semelparous, age-structured species [13,14], but
this topic is almost completely unexplored for iteroparous
species, which constitute a large fraction of all taxa.
Here, we use an empirical approach to model the relationship between a species’ life-history traits and the key
effective size ratios Nb/Ne, Ne/N and Nb/N. We compiled
vital rates (age-specific survival and fecundity) for 63 iteroparous species from a wide range of animal and plant taxa
(summarized in table 1; for full details, see the electronic
supplementary material, table S1 in appendix S1) and used
a recently developed method (AGENE [15]) to calculate Ne,
Nb and N. Across the species, life-history traits and agespecific patterns of survival and fecundity both varied
widely (table 1 and figure 1).
Results are surprising and informative in several regards.
First, the Nb/Ne ratio varied by a large factor (6x) and
exceeded 1.0 in over half the species—the latter, a possibility
that had not been considered previously. Second, we show
that in some species, Ne/N can be 1—another result at
odds with conventional thinking that this ratio generally
should be 0.5 or less [16]. Remarkably, in spite of the wide
variation in these ratios across species, we find that just
two simple life-history traits (age at maturity, a, and adult
lifespan, AL) explain up to two-thirds of the variance in
Nb/Ne and up to half the variance in Ne/N. These two
traits have long been of considerable interest to both ecologists [17] and evolutionary biologists [18]. Our results thus
demonstrate a close, general linkage between demographic
and evolutionary processes across diverse taxa. For the first
time, it will now be possible to interpret the rapidly accumulating estimates of Nb per time period in the context of the
rich body of evolutionary theory based on Ne per generation;
this in turn will facilitate research in ecology, evolution and
applied conservation biology.
2. Material and methods
(a) Life tables
Our analyses are based on vital rates compiled from published literature for species ranging from coral and aphids to pine trees and
chimpanzees (table 1; electronic supplementary material, S1).
Original sources, life tables for each species and details of their construction are found in the electronic supplementary material,
appendices S1 and S2. The species had to be iteroparous and
have data for both age-specific survival (sx) and fecundity (bx),
where x indicates age; see table 2 for notation. If data were available for only one sex (typically females), we used those for the
other sex as well. Most data are for natural populations, but six
invertebrates were captive. We used some selectivity to broaden
taxonomic coverage across five vertebrate classes.
Proc R Soc B 280: 20131339
urchin (5)
great tit (8)
dolphin (35)
toad (17)
sage grouse (15)
0.4
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Table 1. Ranges of key life-history traits and effective size ratios across 63 species of plants and animals.
maximum
trait
value
species
value
species
age at maturity (a)
1 year
4 days
many
weevil
30 years
hoop pine
maximum age (v)
3 years
18 days
wood frog
weevil
400 years
hoop pine
adult lifespan (AL)
3 years
wood frog
371 years
hoop pine
AL/a
12 days
0.60
mosquito
mosquito
33.8
sachamangua tree
generation length (G)
1.5 years
8 days
gray fox, sea urchin
weevil
169 years
hoop pine
CVf a
adult sex ratiob
0
1.0
dusky salamander, mud turtle
many
1.35
2.68
elephant seal
elephant seal
lifetime Vk†
2.25
aphid
41579
hoop pine
Nb/Ne
Ne/N
0.27
0.19
mosquito
hoop pine
1.69
3.69
sage grouse
mosquito
Nb/N
0.14
wood frog
1.00
razorback sucker
Coefficient of variation of age-specific fecundity.
Number of females : number of males.
b
The core analyses assumed a 1 : 1 primary sex ratio and
random variation in reproductive success among individuals of
the same age and sex, but we explored sensitivity to these
assumptions by manipulating vital rates. For a given species,
the number of offspring produced each time period that survive
to age 1 (N1) was held constant. N1 was chosen to produce Ne per
generation of approximately 500; larger or smaller values of N1
had directly proportional effects on N, Ne and Nb but did not
affect the ratios considered here (Nb/Ne, Ne/N and Nb/N).
Age at maturity was defined as the age with the first nonzero fecundity, and any individuals that reached the maximum
lifespan (v) died before the next time period (see the electronic
supplementary material, appendix S1). Adult lifespan was computed as AL ¼ v 2 a þ 1, and adult census size (N) included all
ages greater than or equal to a [19]. When a, v or AL differed
between sexes, the mean was used.
(b) Effective population size
The underlying model for AGENE is discrete-time, age-structured
and deterministic. Input data are age- and sex-specific vital
rates. At each birthday, an individual of age x produces an
average of bx offspring that survive to age 1; that individual
then survives to age x þ 1 with probability sx. The fraction of
individuals in a cohort surviving to age x is lx, with l1 ¼ 1.
AGENE follows Felsenstein [20] and Hill [21] in assuming that
all reproduction occurs at intervals of exactly one time unit (on
an individual’s birthday), and that survival and fecundity are
independent of events in previous time periods. We only track
individuals that survive to their first birthday, so fecundities
are scaled to result in a stable population that produces N1
age-1 individuals per cohort.
AGENE computes Ne for species with separate sexes and
overlapping generations as [21]
Ne 4N1 G
;
Vk † þ 2
ð2:1Þ
where G is generation length in the relevant time-period units,
Vk† is lifetime variance in reproductive success among the N1
individuals in a cohort, and the population is stable (so the
mean number of offspring produced by a parent over its lifetime
(k† ) is 2). We also used AGENE to calculate Nb using the standard
discrete-generation formula for inbreeding effective size [22]
kN 2
Nb ¼ ;
k 1 þ Vk /k
ð2:2Þ
where N represents adult census size, and Vk and k are computed
across both sexes based on offspring produced in a single reproductive cycle. If survival is random, inbreeding effective size
does not depend on what stage the offspring are sampled [6];
we calculated Nb using the N1 yearling offspring produced in
each cohort.
Nb should not be confused with what Hill [21] termed the
‘annual effective size’ (Ny), which is the variance effective size
of a population that would experience the same rate of genetic
drift in a generation as the focal population does in 1 year.
Because a single reproductive cycle is only part of a generation
for age-structured species, the amount of drift that occurs in
one time period is less than would occur over a generation, so
Ny is always larger than Ne. By contrast, Nb reflects the expected
rate of increase in inbreeding in a single cohort, and it has been
assumed that Nb , Ne when generations overlap [12].
Generation length was computed as the average age of
parents of a newborn cohort [1], using the same units (days/
weeks/months or years) used in that species’ life table. Based
on a simple variation of Wright’s [23] familiar formula, we calculated the expected reduction in Ne (or Nb) owing to unequal adult
sex ratio as E(Ne/N ) ¼ 4fm, where f and m are the fractions of
adults that are females and males, respectively. The range for
4fm extends from 1 (when f ¼ m) and 4/N (when only a single
individual of one sex reproduces). We used 4fm as the value
for the variable Sex.
Proc R Soc B 280: 20131339
a
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minimum
3
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Table 2. Notation used in this study.
effective population size per generation
effective number of breeders in one time period
N
total number of adults alive at any given time
N1
x
total number of offspring produced per time period
age (units can be days, weeks, months, years)
sx
lx
probability of surviving from age x to age x þ 1
cumulative probability of surviving to age
bx
Vk
mean number of offspring produced at age x
variance in reproductive success among adults in one
Vk †
time period
lifetime variance in reproductive success among
k
Vk /k
mean number of offspring per parent per time period
index of overdispersion for same-age, same-sex
individuals in a single cohort
v
a
AL
G
individuals
maximum age
age at maturity (first age with bx . 0)
adult lifespan ¼ v 2 a þ 1
generation length (mean age of parents of a newborn
CVf
cohort)
coefficient of variation of bx for adult lifespan (using
Sex
only ages with bx . 0)
reduction in Nb/N or Ne/N owing to uneven adult sex
Time
Format
time units ¼ years or days/weeks/months
original data in age- or stage-based format
Taxon
Fecundity
vertebrate, invertebrate or plant
separate bx values for males and females—yes or no
Survival
separate sx values for males and females—yes or no
ratio
(c) Data analysis
After removing highly correlated variables (see electronic supplementary material, table S4), we focused on five life-history
traits that are easy to estimate for most natural populations. In
addition to G, a, AL and Sex, we calculated a measure of variation in age-specific fecundity, CVf, which is the coefficient of
variation of bx for all ages with bx . 0. G, a, AL and Ne/N
were log transformed before analysis to reduce the skew towards
high values (see the electronic supplementary material, section
1.A. in appendix S1 for details).
G, a and AL were scaled in the same time units used to construct the life table for each species. This provided a common
currency for evaluating the influence of these life-history traits:
across all 63 species, each time unit represented one season or
episode of reproduction. For example, a indicates the number
of reproductive episodes that occur before an individual reaches
sexual maturity, whereas AL indicates the maximum number of
such episodes that occur over the adult lifetime of a single individual. As our results indicate, for iteroparous species, the key
effective size ratios depend heavily on the number of such reproductive episodes, and which fraction occurs before and after
sexual maturation.
3. Results
(a) Life-history data
The 63 species encompassed a wide diversity of life-history
traits (table 1 and electronic supplementary material, S1),
with generation length ranging from weeks (several invertebrates, including the polychaete Dinophilus gyrociliatus) to
decades or more (several vertebrates and plants, including
the sachamangua tree Grias peruviana) and age at maturity
ranging from four days (the rice weevil Calandra oryzae) to
30 years (hoop pine Araucaria cunninghami). Life-history patterns representing type I survival (e.g. the chimpanzee Pan
troglodytes), type II survival (many birds, including the snow
petrel Pagodroma nivea) and type III survival (e.g. the barnacle
Balanus glandula and the red drum Sciaenops ocellatus) are all
represented (figure 1). Some species had constant fecundity
throughout adult lifespan (as in the mud turtle Kinosternon
subrubrum), whereas in others, fecundity increased continually
with age (many, including Atlantic cod Gadus morhua and the
cushion plant Limonium delicatulum) or was dome-shaped
(highest at intermediate ages, as in cottonsedge Eriophorum
vaginatum and red deer Cervus elaphus).
(b) Theoretical explorations of effective size ratios
In the electronic supplementary material, section II of appendix S1, we analytically evaluated the key effective size ratios,
after translating equations (2.1) and (2.2) into vital rates. Our
goal was to see whether these ratios could be expressed as
functions of simpler variables that are easily measured or
estimated. However, even after making the simplifying
assumptions of constant survival and fecundity with age,
both numerators and denominators of Nb/Ne and Ne/N
remain complex functions of vital rates. Nevertheless, we
did obtain some useful insights
— G ¼ generation length ¼ a þ C(AL21), where C , 1 and
depends on the species’ vital rates. With AL (and hence
lifetime Vk ) held constant, G and hence Ne increase linearly with a (see equation (2.1)), while this change by
itself has no effect on N or Nb/N. Therefore, increasing a
compared to AL (and hence decreasing AL/a) will
increase Ne/N and reduce Nb/N. This provides a theoretical explanation for results shown in the electronic
supplementary material, table S11.
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x (lx ¼ Pxi¼1 si1 , with s0 ¼ l1 ¼ 1)
4
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Ne
Nb
We also evaluated the potential influence of several categorical variables: Taxon ( plants (n ¼ 9), invertebrates (14), vertebrates
(40)), Time units (years (54) or days/weeks/months (9)), Format
(original data in age-based (49) or stage-based (14) format),
Survival (separate estimates for males and females: yes (15), no
(48)) and Fecundity (separate estimates for males and females:
yes (10), no (53)).
For each effective size ratio, we used least-squares regression to
fit models for all possible combinations of life-history traits and
categorical variables, resulting in 1024 models per ratio. Each variable appeared in half of the models. We used a small-sample
corrected version of Akaike’s information criterion (AICc [24,25])
as a model selection criterion [26]. We ranked models according
to AICc (smaller is better) and calculated relative likelihoods for
each model as e2Di/2, where Di is the difference in AICc between
model i and the model with lowest AICc. We used the statistical
package R [27] for all statistical analyses.
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1.0
adjusted R2
0.8
5
Nb / Ne
0.6
0.4
a
AL
a
CVf
AL
a
G
CVf
AL
a
1
2
3
4
0.2
0
1.0
adjusted R2
Nb / N
0.6
0.4
0.2
0
1.0
5
CVf
time
CVf
1
2
format
time
CVf
3
time
sex
G
CVf
time
AL
sex
G
CVf
time
a
AL
sex
G
CVf
5
6
4
format
time
a
AL
sex
G
CVf
7
Ne / N
(c) Empirical patterns across taxa
Electronic supplementary material, figures S1A–C shows the
distribution of the key effective size ratios across taxa, and the
electronic supplementary material, table S4, shows means,
medians and standard deviations of these ratios by taxon.
In some cases, statistically significant differences among taxonomic groups were found, but the variable Taxon was not
included in any of the best-fit models (figure 2). We interpreted this to mean that other variables were more useful
explanatory factors than taxonomy. Therefore, below we
focus on broad patterns and results of the model fitting;
details of statistical tests are presented in the electronic
supplementary material, appendix S1.
(i) Ne/N
As expected based on previous empirical and theoretical
studies [3,4,14,17,28], Ne/N was less than 1 for most species.
However, nine animal species had ratios greater than 1, and
for the mosquito, Ne per generation was 3.7 times as large
as the number of adults alive at any given time (see electronic
supplementary material, table S2 and figure S1B). Ne/N
varied widely among both vertebrates and invertebrates but
was less than 0.8 in all plants. Among classes of vertebrates,
much of the diversity was contributed by amphibians, which
had Ne/N ratios ranging from less than 0.3 (wood frog Rana
sylvatica) to greater than 1.7 (Cascade frog Rana cascadae)
within the same genus.
The best-fit model for Ne/N included six variables (figure 2),
but a subset of three life-history traits provided most of the
explanatory power (adjusted R 2 ¼ 0.64 for AL, a, CVf compared
to 0.74 after adding Sex, Format and Time). By themselves, AL and
a explained half the variance in Ne/N.
(ii) Nb/N
The ratio Nb/N applies to a single reproductive cycle and,
assuming random reproductive success of same-age, samesex individuals (as we did here), is constrained to the range
[0,1]. We found ratios that ranged from less than 0.2 (hoop
pine, wood frog, loggerhead turtle Caretta caretta) to greater
adjusted R2
0.8
0.6
0.4
0.2
0
taxon
AL
a
1
2
CVf
AL
a
format
CVf
AL
a
3
4
no. variables
time
sex
CVf
AL
a
5
time
format
sex
CVf
AL
a
6
Figure 2. Pattern of improvement in adjusted R 2 for fit of explanatory variables to the three effective size ratios. Variables included in the best-fit model
(based on AICc) for each number of variables are shown. In each panel, the
best overall model according to AICc is indicated with an arrow.
than 0.99 (many), with lower values primarily associated
with species with strong age-specific differences in fecundity
(see electronic supplementary material, table S2 and figure
S1C). Among vertebrates, Nb/N was highest for birds
(mean ¼ 0.861; electronic supplementary material, tables S2
and S4).
A single life-history variable (CVf ) explained over 50%
of the variation in Nb/N, and inclusion of additional variables made only small incremental improvements to the fit
(reaching R 2 ¼ 0.74 for the best-fit model that included all
five life-history traits plus the categorical variables Time and
Format; figure 2).
(iii) Nb/Ne
We found more than sixfold variation in Nb/Ne across
species, from 0.27 in the mosquito Culex tritaeniorhynchus to
1.69 in sage grouse Centrocercus urophasianus (see electronic
supplementary material, table S2). Contrary to previous
expectations that Nb , Ne, Nb was more than 1.2 times as
large as Ne in 23 species from all three major taxonomic
groups, whereas only six animal species had Nb/Ne , 0.5
(see electronic supplementary material, figure S1A). Invertebrates had relatively low Nb/Ne ratios (mean 0.84),
vertebrates had intermediated values (1.06) and plants had
Proc R Soc B 280: 20131339
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time
G
CVf
AL
a
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— Nb/Ne has an upper bound at ðVk† þ 2Þ/ð2GkÞ N/Ne ¼
the inverse of the Ne/N ratio (see electronic supplementary material, equation A 13).
— Ne /N 2kG/ðVk† þ 2Þ (see electronic supplementary
material, equation A 18). All else being equal, Ne/N
increases linearly with k, the mean number of offspring
per adult per time period. Following Felsenstein [20],
k ¼ 1/SaþAL1 lx , with the sum taken across the AL years
x¼a
of adult lifetime. Two factors tend to favour larger
values of Slx and hence smaller values of k: (i) high
adult survival rate, which promotes long adult lifespan;
and (ii) early age at maturity, which allows more of the
lifetime lx terms (and in particular, the earlier years
when lx is larger) to be included in Slx. Together, these
factors indicate that a high ratio of AL/a should produce
a large Slx and hence a small k and a low Ne/N ratio,
whereas a low AL/a ratio should have the opposite
effect. We verified this general relationship with artificial
data shown in the electronic supplementary material,
figure S4. Note that k ¼ Slx is independent of patterns
of age-specific fecundity.
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2
observed Nb /Ne
adjusted R2 = 0.674
1
A
B
0
0.5
1.0
log(AL/a)
1.5
observed Nb /Ne
2.0
1.5
Nb/Ne = 0.833 + 0.637 × log(AL)
–0.793 × log(a) – 0.423 × CVf
adjusted R2 = 0.841
1.0
plants
inverts
amphibs + reptiles
mammals
fish
birds
0.5
0
0.5
1.0
predicted Nb /Ne
1.5
2.0
Figure 3. (a) Regression (solid line) of Nb/Ne and the log10 of the ratio of
adult lifespan to age at maturity (AL/a) for 63 species. Letters indicate
semelparous species with variable age at maturity (not used in computing
the regression): A, Chinook salmon, B, annual plant with seed bank. (b) Correlation between observed Nb/Ne and the value predicted using the optimal
combination of three life-history traits identified using AICc. Predicted values
were calculated using the regression equation shown.
relatively high values (1.26) (see electronic supplementary
material, table S4). Birds had even higher mean Nb/Ne
(1.35) than plants, which reflects their relatively high Nb/N
(0.86) and relatively low Ne/N (0.65) ratios.
Despite the sixfold range, a few simple life-history traits
explained most of the variation in Nb/Ne. Three life-history
traits (AL, a and CVf) captured 84.1% of the variance
(figure 3b), and addition of G and Time increased this only
marginally to 86.5% (figure 2). Surprisingly, the ratio of just
two life-history traits (AL and a) alone explained two-thirds
of the variance in Nb/Ne across the 63 species (figure 3a).
We included results for two semelparous species with
variable a in figure 3a. This figure shows that their low
Nb/Ne ratios (less than or equal to 0.25) and short adult lifespan compared with age at maturity place them at one
extreme end of the continuum for iteroparous species (by
definition, semelparous species reproduce in only one time
period, so AL a and log(AL/a) 0).
(d) Sensitivity analyses with artificial life tables
All else being equal, constant fecundity with age maximizes
Nb/N and Nb/Ne, whereas increasing fecundity with age
has the opposite effect (see electronic supplementary
material, table S9). Changes in age-specific survival had relatively smaller effects, but increasing survival with age
reduced Ne/N (and increased Nb/Ne), whereas decreasing
4. Discussion
(a) Life history, demography and evolution
Given the wide range of animal and plant species considered,
it is remarkable that two simple life-history traits (age at
maturity and adult lifespan) by themselves explain up to
half (Ne/N) or up to two-thirds (Nb/Ne) of the variation in
key effective population size ratios. We found some theoretical reasons why these two life-history traits could have a
strong influence on Ne/N and Nb/Ne. Our interpretation of
the analytical results is the following:
— The parameter space that encompasses all possible combinations of vital rates is so vast that its features cannot
easily be captured by simple functions of life-history
traits.
— Real species in nature occupy only a fraction of the possible parameter space that is compatible with viability, and
in this restricted space, the strong influence of a few key
life-history traits can be seen.
Cole [17] and other ecologists have emphasized how
strongly a and AL influence a population’s intrinsic rate of
6
Proc R Soc B 280: 20131339
0
–0.5
survival with age had the opposite effect (see electronic supplementary material, table S10). The dramatic effects of
increasing a while AL and other demographic parameters
were held constant are described above and shown in the
electronic supplementary material, table S11.
We also evaluated sensitivity to some common simplifying assumptions. We assumed an even primary sex ratio; a
skewed sex ratio in newborns has predictable and identical
effects on Ne/N and Nb/N [23], so Nb/Ne is unaffected (see
electronic supplementary material, table S7). In some species,
maximum age (v) was not specified, in which case we used a
simple rule to select v (see detailed methods in the electronic
supplementary material, S1). Truncation of this nominal v
had little effect on Nb/N, but it increased Ne/N and therefore
reduced Nb/Ne. However, even an extreme 40% truncation of
v reduced Nb/Ne by only 7–18% (see electronic supplementary material, table S8). When we redid the analyses using
truncated values for these seven species, the fit between
Nb/Ne and either log(AL/a) or [log(AL) þ log(a) þ CVf ]
was barely affected (electronic supplementary material,
table S13), whereas a slight reduction in fit was observed
for log(Ne/N) under extreme 40% truncation (R 2 reduced
from 0.491 to 0.452 for log(AL/a) and from 0.632 to 0.611
for [log(AL) þ log(a) þ CVf ]).
The above analyses assumed random variance in reproductive success of same-age, same-sex individuals (in which
case Vk k). Allowing the overdispersion factor (Vk /k) to
increase to 2, 4 or 8 reduced Nb/N more than Ne/N in most
species, and hence reduced the Nb/Ne ratio (see electronic
supplementary material, table S12). The magnitude of this
effect varied considerably across species: under the most
extreme scenario (Vk /k ¼ 8), one species showed little (7%)
reduction in Nb/Ne, whereas the others showed reductions of
40–87% (see electronic supplementary material, figure S2).
Assuming an eightfold increase in Vk /k for these six species
reduced the proportion of variance explained for both effective
size ratios by about 9–27 percentage points (see electronic
supplementary material, table S13).
rspb.royalsocietypublishing.org
Nb/Ne = 0.485 + 0.758 × log(AL/a)
Downloaded from rspb.royalsocietypublishing.org on September 19, 2014
For the first time, this paper provides a general understanding
of the relationship between Nb (easier to estimate) and Ne
(more important for evolutionary processes) in species with
overlapping generations. Previous evaluations of Ne and Nb
either were confined to semelparous species [13,14,30], studied
a single iteroparous species [31,32], or theoretically evaluated
only one extreme scenario [33]. We found that Nb/Ne varies
much more than expected (less than 0.3 to greater than 1.6).
All else being equal, increases in a lead to proportional
increases in Ne (see electronic supplementary material, table
S11; see also [4,34]) but have no effect on Nb; conversely,
increasing AL increases Nb more than Ne. CVf, an index of variation in age-specific fecundity, is by far the most powerful
indicator of reduced Nb/N (figure 2), and inclusion of this
third variable substantially improves the ability to predict
Nb/Ne from life history (figure 3b).
Importantly, these results provide a means to translate the
rapidly increasing body of empirical estimates of Nb into
estimates of Ne per generation. Previous studies [13,14] have
shown that, in semelparous species with variable age at
maturity, Ne per generation is larger than Nb by a factor
approximately equal to generation length (Ne GNb), and it
was assumed that the relationship Nb Ne GNb would also
hold for species with overlapping generations [12]. Surprisingly, we show here that this is not the case for iteroparous
species: Nb can exceed Ne by 50% or more. This result has
important practical implications for interpretation of estimates
of Nb in conservation [2,3] and more generally for monitoring
evolutionary changes in natural populations [35,36].
It might seem counterintuitive that Nb for a single time
period could be larger than Ne per generation, but it is easy
to illustrate how this can come about. If a species has constant
fecundity with age (as assumed in many models [4,16,37] and
approximated by many birds and other taxa), and if reproductive success of all adults is random, then Nb ¼ N. In that case,
all that is required for Nb/Ne . 1 is that Ne/N , 1, which commonly occurs for a wide range of species [28] (electronic
supplementary material, table S2 and figure S1B). On the
other hand, delayed age at maturity increases Ne while not
increasing Nb or N, and that is why species with low AL/a
do not have high Nb/Ne ratios (figure 3a).
(c) Effective size to census size ratios
After considering potential influences of a variety of lifehistory traits and mating systems, Nunney ([16] and elsewhere)
concluded that Ne/N should generally be less than or equal
to 0.5. Here, we show that it is quite possible for Ne to be
larger than N, especially if age at maturity is delayed, because
this increases generation length without increasing lifetime
Vk† (see the electronic supplementary material, table S11).
(d) Categorical variables
Although we found some statistical evidence for taxonomic
heterogeneity in key effective size ratios (see electronic supplementary material, table S4 and appendix S1), these
differences were effectively captured by other life-history
traits (figure 2). The only categorical variables included in
the best-fit models were Time (all three ratios) and Format
(Nb/N and Ne/N only), but neither provided the same level
of explanatory power as the key life-history traits AL, a and
CVf (figure 2). Conclusions about categorical variables must
be tempered by small sample sizes and lack of independence
(e.g. all species for which time was measured in units
less than years were invertebrates, and most were captive
populations). Furthermore, the life tables were collected
opportunistically and are not necessarily representative of
the different taxonomic groups; in particular, plants and
invertebrates are underrepresented. Nevertheless, the strong
life-history-effective size correlates that emerged from our
analyses indicate that these potentially confounding factors
did not obscure key underlying patterns that hold across
diverse life histories.
(e) Caveats
Some important caveats should be noted. Like the models it
was based on [20,21], AGENE assumes constant population size
and stable age structure. AGENE results are robust to random
demographic variation in vital rates ([15]; R. Waples &
T. Antao 2013, unpublished data). However, care should be
taken when estimating Ne in species that experience extreme
fluctuations or large pulses in abundance (as in some amphibians and insects, for example). Estimated vital rates are
subject to sampling error, which can affect Ne estimates [39].
This argues for caution in interpreting results for individual
species, but this source of random variation would only be
7
Proc R Soc B 280: 20131339
(b) Relationship between Nb and Ne
This theoretical point has been made before [34,38], but we
are not aware of any demographic estimates of Ne/N nearly
as extreme as what we report for the mosquito (3.7), where
the long generation time (G ¼ 22.5 days; electronic supplementary material, table S2) is owing primarily to delayed age at
maturity (a ¼ 20 days) compared with short adult lifespan
(AL ¼ only 12 days). In Frankham’s [28] seminal review, the
most extreme reported estimate of Ne/N was just over 1.0,
although some recent genetically based estimates are higher
[3]. Our results suggest that Ne/N can be arbitrarily large
for species with prolonged juvenile phases and short adult
lifespans—as applies, for example, to many insects.
Theory (equation (2.1)) suggests that, all else being equal, Ne
should be proportional to generation length in iteroparous
species, but we found that G had no detectable influence on
Ne/N and little on Nb/Ne (figure 2). This can be explained by
the strong positive correlation between log(G) and log(Vk†)
(0.49; electronic supplementary material, table S3): species
with long generation length also tend to have high lifetime
Vk† . Longer adult lifespan increases Vk† , because it exacerbates
the disparity in lifetime reproductive success between individuals that die early and those that reproduce in many time
periods. Vk† appears in the denominator of equation (2.1), so
it largely cancels out the positive effect of longer generation
length. This result illustrates the importance of supplementing
theoretical models with empirical data for multiple, diverse
species to better understand eco-evolutionary processes.
rspb.royalsocietypublishing.org
increase, and the ratio AL/a has been characterized as a ‘lifehistory invariant’ [18]. However, that has been questioned on
statistical grounds [29], and even Charnov [18] noted that the
AL/a ratio differs substantially among taxa. Here, we provide empirical evidence for relationships between AL/a
and key effective population size ratios that remain strong
and consistent when applied across diverse species of
vertebrates, invertebrates and plants. That is, the evolutionary consequences of these two life-history traits are largely
invariant across taxa.
Downloaded from rspb.royalsocietypublishing.org on September 19, 2014
success of same-age, same-sex individuals. Effects appear to
be most pronounced for species with long adult lifespan and
constant fecundity with age, which suggests that a quantitative
adjustment to account for effects of overdispersion is possible
(see the electronic supplementary material, figure S3), but
this topic merits further study.
(f ) Take-home messages
Acknowledgements. This project grew out of analyses initiated as part of
the Genetic Monitoring Working Group (GeM) jointly supported by the
National Evolutionary Synthesis Center (Durham, NC, USA) and the
National Center for Ecological Analysis and Synthesis (Santa Barbara,
CA, USA). We thank Tiago Antao, Dan Doak, Correigh Green, Dave
Gregovich, Eli Holmes, Scott Mills, Friso Palstra, Tom Reed, Eric Ward
and two anonymous reviewers for useful discussions and comments.
Funding statement. G.L. was supported in part by funding from US
National Science Foundation grant nos. DEB-0742181 and DEB-1067613
and Montana Fish Wildlife and Parks.
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Surprisingly, we show that Nb for a single reproductive cycle
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