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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.
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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.
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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)
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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
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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
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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).
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Nb/Ne = 0.485 + 0.758 × log(AL/a)
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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
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(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.
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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.
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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 can be (and often is) larger than Ne per generation, and that Ne/N . 1 is quite plausible for species with delayed age at maturity. The key ratio Nb/Ne is explained by a few simple life-history traits across diverse taxa—in particular, the fraction of reproductive cycles that occur after as opposed to before sexual maturation. An important applied consequence of our results is that it will now be much easier to estimate and monitor Ne per generation based on data for Nb per reproductive cycle, which is much easier to estimate for iteroparous species.
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