Society for Conservation Biology

1XPEHURI&HQVXVHV5HTXLUHGIRU'HPRJUDSKLF(VWLPDWLRQRI(IIHFWLYH3RSXODWLRQ6L]H $XWKRU V -RKQ$9XFHWLFKDQG7KRPDV$:DLWH 5HYLHZHGZRUN V  6RXUFH&RQVHUYDWLRQ%LRORJ\9RO1R 2FW SS 3XEOLVKHGE\Blackwell PublishingIRUSociety for Conservation Biology 6WDEOH85/http://www.jstor.org/stable/2387576 . $FFHVVHG

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

Blackwell Publishing and Society for Conservation Biology are collaborating with JSTOR to digitize, preserve and extend access to Conservation Biology.

http://www.jstor.org Number of Censuses Required for Demographic Estimation of Effective

JOHN A. VUCETICH*AND THOMAS A. WAITEt *School of Forestry, Michigan Technological University, Houghton, MI 49931, U.S.A., email [email protected] tDepartment of Zoology, Ohio State University, Columbus, OH 43210-1293, U.S.A., email waite.l@?osu.edu

Abstract: Adequate population viability assessment may require estimation of effective population size (Ne). Butfailure to take into account the effect of temporalfluctuation in population size (FPS) on Ne may routinely lead to unrealistically optimistic viability assessments. We thus evaluate a technique that accounts for the effect of FPS on Ne. Using time series of annual counts of 48free-ranging animal populations, we show that Ne is dependent on timescale: as more census records are incorporated, estimates of FPS tend to increase, and thus estimates of N tend to decrease. Estimates based on, say, 10 annual counts tend to grossly underes- timate the influence of FPS on estimates of long-term Ne, often by more than 100%. Moreover, a newly derived expression for the confidence intervals of Ne/N (in which N is population size) reveals that estimates of Ne based on only a few annual counts are quite unreliable. Our work thus emphasizes the need for long-term population monitoring and provides a framework for interpreting Ne estimates calculated from limited cen- sus data.

Numero de Censos Requeridos para una Estimaci6n Demografica Efectiva de Tamafio Poblacional Resumen: Evaluaciones adecuadas de viabilidad poblacional pueden requerir estimaciones de tamano po- blacional efectivo (N). Sin embargo, omisiones en la consideraci6n del efecto de lafluctuacion temporal del tamanfo poblacional (FPS) en Ne pueden conducir rutinariamente a estimaciones de viabilidad irrealmente optimistas. Es por ello que evaluamos una tecnica que toma en consideraci6n el efecto de FPS en N.. Utiliza- mos series de tiempo de conteos anuales de 48 poblaciones de animales de rango libre, mostramos que Ne es dependiente de la escala temporal. mientras mas registros de censos son incorporados, las estimaciones de FPS tienden a incrementar y por lo tanto las estimaciones de Ne tienden a disminuir. Estimaciones basadas, en por ejemplo, 10 conteos anuales tienden a subestimar la influencia de FPS en las estimaciones de largo plazo de Ne (frecuentemente por mds de un 100%). Mas atn, una nueva expresi6n derivada para intervalos de confianza de Ne/N (en la cual N es el tamanfo poblacional) revela que las estimaciones de N, basadas en solo unos cuantos conteos anuales son poco confiables. Nuestro trabajo hace enfasis en la necesidad de mon- itoreos poblacionales a largo plazo y provee un marco de trabajo para interpretar estimaciones de Ne calcu- ladas a partir de datos de censos limitados.

Introduction Frankham 1995a; Newman & Pilson 1997; but see Brit- ten 1996). Adequate assessment of viability requires, in A principal goal of conservation biology is to maintain part, determining whether the population is large . Requisite for meeting this goal is the con- enough to avoid inbreeding or to maintain adaptive ge- servation of , a key determinant of pop- netic variation. The most appropriate metric for this de- ulation viability (imenez et al. 1994; Keller et al. 1994; termination is effective population size (Ne; Barton & Whitlock 1997), which calibrates the influence of ge- netic drift in a real population to that in an ideal popula- Paper submitted August 11, 1997; revised manuscript accepted De- tion (i.e., one with random mating, even sex ratio, dis- cember 10, 1997. crete generations, and constant size; Wright 1931). An 1023

Conservation Biology, Pages 1023-1030 Volume 12, No. 5, October 1998 1024 EffectivePopulation Size Vucetich& Waite ideal population would lose neutral genetic diversity at a factor, temporal fluctuation in population size (FPS), has rate of 1/2N per generation, where N is population size. perhaps the most important influence on N, (Wright Violations of the ideal assumptions, however, cause real 1938; Frankham 1995b; Vucetich et al. 1997a). Esti- populations, if isolated (Mills & Allendorf 1996), to lose mates based on data collected during a single season genetic diversity at a (typically faster) rate of 1/2N, per (Nunney & Elam 1994) ignore the influence of FPS and generation. Beyond retarding the loss of heterozygosity thus may represent gross overestimates of Ne. We show and allelic diversity (Crow & Kimura 1970), maintaining here that a greater number of periodic counts of a given large Ne increases the likelihood that favorable muta- population typically yields a lower estimate of Ne/N. tions will become widespread (Falconer & Mackay This property arises because longer time series of census 1996) and deleterious will be eliminated data tend to yield higher estimates of FPS(Arino & Pimm (Lynch et al. 1995), and it improves the population's 1995). Using a newly derived expression, we evaluate ability to cope with environmental change (Burger & the confidence intervals for Ne/N estimates as influ- Lynch 1995). MaintainingNe above some threshold thus enced by the number of annual population counts. We is arguably essential for population viability (e.g., Soule also discuss how time series data may be used to adjust 1980; Lynch 1996). estimates of N, for the influence of FPS. An important application of Ne in conservation biol- ogy is the estimation of the minimum viable effective population size. Earlywork suggested that N, must be at Fluctuating Population Size and the least 50 to avoid inbreeding depression and thereby en- Estimation of Ne sure short-term viability, and at least 500 to maintain adaptive and thereby ensure long-term The influence of FPS on long-term Ne is quantified by viability (Franklin 1980; Soule 1980). Recent experimen- Wright (1938), Crow and Kimura(1970), Lande and Bar- ~ tal work, however, suggests that Ne 50 is inadequate rowclough (1987), Grant and Grant (1992), and Vucet- to ensure short-term viability (Latter et al. 1995; New- ich et al. (1997a): man & Pilson 1997). In addition, recent theoretical work suggests that Ne : 500 is inadequate to ensure long-term Ne = Ne,t (1) viability; effective sizes on the order of 1000-5000 may be necessary to maintain adaptive variation and avoid the accumulation and fixation of mildly deleterious mu- where Ne,t is traditionally the effective size for genera- tations (reviewed by Lande 1995a; Lynch 1996). tion t (but see Grant & Grant 1992) and q is the total Effective population size is also useful for defining the number of Net estimates. Equation 1 was originally shown goals of conservation efforts. For example, the mainte- to approximate both the variance and inbreeding effec- nance of 90% of neutral genetic variation over 200 years tive size for cyclical populations with discrete (nonover- has been suggested as a general goal (Franklin 1980; lapping) generations (Wright 1938; Crow & Kimura1970). Soule 1980; Soule et al. 1986; Lande 1995b). Goals of This harmonic-mean estimator has since been shown to this form may be described as Ne = -t/2 ln[Ht]T(Wright approximate the effective size of populations undergo- 1931), where t is the time horizon of the goal, Ht is the ing more general fluctuations (e.g., Motro & Thomson fraction of neutral heterozygosity to be maintained, and 1982) and to yield estimates for populations with over- Tis generation time. Within this framework, Ne has been lapping generations falling within the range of indepen- applied to species recovery plans (e.g., U.S. Fish & Wild- dently derived values (Vucetich et al. 1997a). Moreover, life Service 1987, 1993, 1994). Ne has also been used to we have recently shown that equation 1 is inversely re- specify proposed risk categories of the World Conserva- lated to a standard measure of FPS, the standard devia- tion Union (i.e., critical, endangered, vulnerable; Mace tion of log-transformed population size (Vucetich et al. & Lande 1991) and to prioritize salmon stocks for con- 1997a). Because equation 1 quantifies FPS, and FPS esti- servation (Allendorf et al. 1997). Ne thus is an important mates are independent of sampling frequency (Gaston & tool in conservation biology. McArdle 1994), whether Ne,t is used to represent the ef- The application of Ne, however, has been limited by fective size for generation t or for year t is inconsequen- difficulty in obtaining accurate estimates, whether by ge- tial. Moreover, Net quantifies the rate of drift on a per- netic methods (reviewed by Neigel 1996) or demo- generation basis, whether estimated from data collected graphic methods (reviewed by Caballero 1994; Husband during a single year (Nunney & Elam 1994) or across & Barrett 1995; Nunney 1995). The usual approach for multiple years (see below). obtaining demographic estimates of Ne is to model the Because Ne is strongly dependent on population size, influence of variance in lifetime reproduction, uneven which varies both within and between populations, the- sex ratio, and overlapping generations, while assuming oretical evaluation of factors influencing Ne is facilitated that the population size is constant (Crow & Denniston by standardizing equation 1. The most straightforward 1988; Caballero 1994; Nunney & Elam 1994). Another standardization is the ratio of effective size to census

Conservation Biology Volume 12, No. 5, October 1998 Vucetich& Waite EffectivePopulation Size 1025

size (e.g., Nunney 1991, 1995; Briscoe et al. 1992; longer a population is observed (Lawton 1988), estimates Frankham 1995b; Husband & Barrett 1995; Vucetich et of N, should decrease as q increases. We now evaluate the al. 1997a). Here, we use the ratio of the quantity in number of annual estimates (q) required to account ade- equation 1 divided by the average census size (e.g., quately for the influence of FPS on Ne. Frankham 1995b; Vucetich et al. 1997a): q q Increased Fluctuation over Time and the q/ Ne,tq qI Estimation of Ne (Ne/N)q = ( Nt Ne,t q K , (2) qN/ t=l- - Various mechanisms have been proposed to explain the well-documented observation of increasing FPS with in- creasing numbers of periodic population counts (re- where Net is the effective size calcu- (per-generation) viewed by Ariio & Pimm 1995). For example, the tempo- lated in is the census size in and is the year t, Nt year t, q ral dynamics of many environmental factors are likely to number of annual estimates. This is approach appropri- be characterized by small, short-term trends superim- ate if the is stable or if the population trajectory ap- posed on larger, longer-term trends (i.e., a reddened is used to estimate deterioration proach past genetic (Ta- spectrum; Williamson 1984; Steele 1985). The variance ble But if N is or over time and 1). increasing decreasing in such systems appears to increase over time. This pat- the estimate will be used to forecast deteriora- genetic tern is unlikely to be simply an artifact of measurement then the denominator should the ex- tion, represent error because such error is typically assumed to be a size the of interest. In pected population during period white-noise phenomenon (e.g., Halperin & Gurian 1970), such the denominator could be with the cases, replaced and the observed variance should thus asymptote over of recent values of for or an average N, example, extrap- short time scales (Arifio & Pimm 1995). Perhaps because olation of the trend. already-observed are partly driven by fluctuations in The on the left side of 2 subscript q equation empha- the abiotic environment, population size also tends to ex- sizes our main focus: Ne on the of depends number annual hibit increasing fluctuation over time (Lawton 1988; estimates incorporated. Because FPS tends to increase the Pimm & Redfearn 1988; Arifio & Pimm 1995; Curnutt et al. 1996), suggesting that incorporating more population counts will tend to yield lower estimates of Ne. To the effect of FPS on esti- Table 1. An illustrationof the accuracyof equation2 for explore increasing Ne estimatinggenetic deterioration.* mates, we analyze the effect of the number of popula- tion counts (i.e., q) incorporated on estimates of N. Our Fractional analysis is based on a collection of 48 time series of pop- t Nt Net/Nt Ne,t Ht Ht* error ulation counts. These time series, each at least 18 years 1 26 1.000 1.000 0.000 0.5 13 long, represent a variety of animal taxa, including 24 2 22 0.5 11 0.955 0.963 -0.009 mammalian, six avian, and 18 invertebrate 3 22 0.5 11 0.911 0.927 -0.018 species 4 17 0.5 8.5 0.858 0.893 -0.041 (Packer et al. 1991; Turchin & Taylor 1992; Foley 1994; 5 18 0.5 9 0.810 0.860 -0.062 Vucetich et al. 1997b; C. Bocetti, unpublished data). To 6 20 0.5 10 0.769 0.828 -0.076 explore the influence of q on the ratio Ne/N, we assume 7 23 0.5 11.5 0.736 0.797 -0.083 that the populations are ideal except for FPS. Thus, we 8 24 0.5 12 0.705 0.768 -0.088 assume that Net = This unrealistic is nec- 9 31 0.5 15.5 0.683 0.739 -0.083 Nt. assumption 10 41 0.5 20.5 0.666 0.712 -0.069 essary to isolate the influence of FPS. In practice, the in- 11 44 0.5 22 0.651 0.685 -0.053 fluence of other factors known to depress Ne should also 12 34 0.5 17 0.632 0.660 -0.045 be incorporated. Ignoring these factors here, we calcu- 40 0.5 20 0.616 13 0.636 -0.032 late (Ne/N)q over a range of q for each time series using 14 0.602 0.612 43 0.5 21.5 -0.017 2. Consistent with the behavior of the 15 50 0.5 25 0.589 0.589 0.000 equation FPS, sample median of (Ne/N)q decays asymptotically as q in- *We illustrate the a population with dynamics (Nt) of hypothetical creases (Fig. 1). Overall, our analysis suggests that esti- an increasing trend. Time (t) is measured in years. is as- Nc,t/Nt mates based on a few counts tend to underestimate signed arbitrarily. (In practice, Net/Nt would be estimated from only equation 4.) Ne, is obtained by multiplying columns 2 and 3. The the influence of FPS on the Ne/N, thereby tending to is calculated in the tra- proportion of heterozygosity remaining (Ht) overestimate N,. ditional way (Ht_l)(1-1/2Net). Ht* is calculated as (Ht_l*)(1-1/ 2Ne), where Ne is calculated as (Ne/N)N = (0.445)(13.491) Because only a few annual counts are often available (following equation 2). The hypothetical population is assumed to for populations of conservation concern, we examine have a time 1 The error tends to be generation of year. fractional the possibility that Ne estimates based on short time se- negative (positive) for increasing (decreasing) trends but tends to zero toward the end of the time series. The magnitude of the error ries adequately account for the influence of FPS. Specifi- depends on the size ofNe (Crow & Kimura 1970). cally, we calculate the percent error between (Ne/N)a

Conservation Biology Volume 12, No. 5, October 1998 1026 EffectivePopulation Size Vucetich& Waite

1.0 1.0- 4 annual counts 0.8

0.8 .... 0.6 0.4

0.2 0.6 0.0

1.0 6 annualcounts 0.4 - 0.8 x A 0.6

0.2 0.4 . . 0.2 X 0.0 0.0 :2 1 3 5 7 9 11 13 15 17 a. 1.0 10 annualcounts Number of annual population counts 0.8 Figure 1. The estimated relationship (equation 2) be- 0.6 tween Ne/N and the number of annual population 0.4 counts (q) included in the Ne/N estimate (based on 0.2 48 empirical time series). The solid line represents the 0.0 median, and the dotted lines represent the first and ~'--' cm aita 0 08888 8 8 8 08 the x-axis is 18 be- third quartiles. The upper limit of Error(X%) cause the shortest time series consisted of 18 annual counts. (Ne/N), = I because FPS cannot be observed Figure 2. Theprobability of obtaining >X% error for when q = 1. The median appears to converge on 0.5, various numbers of annual censuses included in the consistent with a previous analysis showing that the estimate. The error is defined as the percent difference median Ne/N was approximately 0.5 when popula- between the long-term Ne/N estimates and those based tions were assumed ideal exceptfor the influence of on 4, 6, or 10 annualpopulation counts. FPS (see Fig. I of Vucetich et al. 1997a).

percent error tends to decrease, as with increases in q, and (Ne/N)qax for each of the 48 time series, where q falling perforce to zero at q = qmax) Then, for each of represents some specified number of annual estimates the time series, we observe the number of counts (i.e., (e.g., 4, 6, or 10) and qma is the total number of counts the value of q) for which (Ne/N)q first falls below the in the time series. The frequency distribution of these er- specified level of error. Based on these observed values rors (Fig. 2) suggests that estimates based on short time of q, we estimate cumulative distributions of the mini- series should be viewed skeptically. In fact, these errors mum number of counts required to account for the in- probably represent underestimates because longer time fluence of FPS on Ne estimates. Figure 3 clearly shows series would tend to increase FPS even further (i.e., that more than just a few counts will typically be re- long-term Ne/N will be lower than ([Ne/N])qma). Over- quired to adjust Ne adequately for the effect of FPS. all, Ne/N estimates based on less than 10 counts are ap- We conclude these analyses by considering two de- parently subject to very large errors. The magnitude of tails. First, populations characterized by small FPS may these errors depends on the value of Ne,t/Nt (unrealisti- require relatively few annual counts for accurate estima- cally assumed to equal unity). In practice, these esti- tion of Ne. Though plausible, this supposition appears to mated errors should be adjusted by multiplying by the be incorrect. For example, the minimum number of average Ne,t/Nt, whenever such an estimate is available. counts required to achieve less than 10%error is not sig- In many cases, it would be appropriate to use Ne t/Nt nificantly correlated with (Ne/N)qmax (Spearman's rank 1/2 as a benchmark (Nunney 1991, 1993, 1995; Nunney correlation: rs = -0.22, p = 0.17). Thus, accurately esti- & Elam 1994). mating long-term Ne may often require as much data for Extending the analysis, we assess the number of a population with high Ne/N (low FPS) as for one with counts required to account for the influence of FPS low Ne/N (high FPS). Second, recall our assumption that within a specified level of error. To do so, we calculate Ne/N would not decrease further even if more than qma the percent error between (Ne/N)q and (Ne/N)qm, for counts were available. This assumption may be routinely each of the time series, where q = 1,2, . . qma.. (The violated because FPS is unlikely to approach an asymp-

Conservation Biology Volume 12, No. 5, October 1998 Vucetich& Waite EffectivePopulation Size 1027

1.0 (N /N)(X2( - )2 2 2' (q [N-eJ-Nq- (lqc,q))) X 0.8 (Ne/N)(X2((2, q))2 c 2 (3)

-' 0.6 where X2(p q-l) denotes the pth percentile of the Chi- square distribution with q - 1 degrees of freedom. o 0.4 These computations show that estimates based on small q may contain little information about Ne/N, as reflected by the wide confidence intervals. In addition, geometric 0E .-- 10%error increases in q only result in linear decreases in confi- 0.2 - 25% error dence interval width. when is minor in- -A- 100% error Thus, q small, creases in q lead to substantial decreases in confidence- interval width. When q is large, however, substantial 0.0 -.... increases in q are required to reduce confidence-interval 0 10 20 30 40 50 60 70 width appreciably. We emphasize that because these Numberof annualcounts confidence intervals are based on the assumption that populations are ideal except for FPS (i.e., Ne,t = Nt), 3. The estimated cumulative distribution Figure fre- they are presented simply to illustrate the large number the minimum number of annual quency for popula- of annual counts required to account adequately for the tion counts to achieve <10%, <25%, or <100% error in influence of FPS. the estimation of Ne/N relative to the long-term Ne/N (based on 48 time series). The long-term Ne/N is esti- the time series. on a matedfrom complete Any point ComprehensiveEstimates of N trajectory is interpretable as the minimum number of counts (x to be 100 annualpopulation required axis) Until now we have assumed that populations are ideal (1 - CF)% certain that an Ne/N estimate will be except for FPS (i.e., Net = Ne). To account hypotheti- within or 100% the estimated 10%, 25%, of long-term cally for the remaining influential factors (i.e., variance Ne/N. (CF is the Three data exceed 70 y-axis.) points in fecundity, uneven sex ratio, and overlapping genera- years. This figure suggests, for example, that the me- tions), as one would do in practice, Ne,t in equation 1 dian number of counts required to estimate Ne/N with <10% error is -12 years.

100 tote within the short time frames over which popula- tions are studied Arifio & Pimm usually (Murdoch 1994; Eo 80- 1995). A positive correlation between the minimum req- L8 uisite number of counts and the of the time series 8o 0 4 length U o 60 (rs = 0.60, p < 0.0001; Fig. 4) suggests that this assump- tion was probably violated for some, if not all, time se- - V oo ries. Therefore, we have probably underestimated the 40 number of counts to account for required adequately ' - *- S the influence of FPS (Figs. 2 & 3). Z 20. . . * Z

ConfidenceIntervals for Ne/N 0 40 80 120 160 Length of time series (years) Besides underestimating the effect of FPS, Ne estimates based on only a few counts may also be quite unreliable. Figure 4. The relationship between the total length of To evaluate this assertion, we computed confidence in- time series (q) for 48populations and the estimated tervals around Ne/N for populations that are ideal ex- minimum number of annual censuses required to cept for FPS (Fig. 5; an Appendix for its derivation is achieve <10% error (relative to the Ne/N estimate available from the authors upon request): based on the complete time series).

Conservation Biology Volume 12, No. 5, October 1998 1028 EffectivePopulation Size Vucetich& Waite

Significance level rioration thus should be limited accordingly. Moreover, ----- 1 0 --.--- 50% ---- 80% 95% conservation goals aimed at preserving some fraction of neutral genetic diversity over long time scales (e.g., 100-200 years) often may be unrealistic because proper 0.8 - _ evaluation of such goals may be infeasible. These con- clusions are based on the recognition that fluctuation in 0.6 - - -- population size is dependent on time scale (Arino & 04 ' ------Pimm 1995) and that FPSis an important predictor of Ne - - 0 .4 .-, ...... 8 (Frankham 1995b; Vucetich et al. 1997a). Because FPSis also an important predictor of the demographic compo- nent of risk (Lande 1993; Foley 1994), fore- 0.2 - casts of demographic extinction risk should similarly be .' / , -?__ limited to realistically short time scales. We do not intend to however, that demo- 0.0 .- ...... imply, 1 3 5 7 9 11 13 15 17 graphic estimators of Ne are completely uninformative or that estimators should used. Number of annual population counts only genetic be Judicious use of genetic estimators (reviewed by Neigel 1996) re- Figure 5. Average Ne/N values (based on 48 empiri- quires similar considerations (Nunney & Elam 1994). For cal time series) for q annualpopulation estimates example, it would be dubious to use Ne estimates based (solid line) and corresponding confidence intervals for on the temporal method (Waples 1989) to calculate the estimated values of Ne/N (equation 3). rates of drift for time scales exceeding the sampling in- terval. Recent advances in molecular techniques (e.g., polymerase chain reaction [PCR] amplification of micro- can be replaced by an estimate of the (variance) effec- satellite markers), however, permit the comparison of tive size (Nunney 1991): DNA samples from museum specimens ("ancient" DNA; Herrmann& Hummel 1993) with those from extant popu- Net= Nt{4r(l - r) T/[rAf(l + /Af) + lations, so it is now feasible in some cases to estimate long- (1 - r)A,( 1 + IA) + (1 - r)Ib, + rI] }, (4) term Ne (Waples 1989). Demographic estimators should where r is the sex ratio, T is the average generation time, remain useful, though, partly because they can reveal the Af and Am are the average adult life spans for both sexes, relative importance of underlying ecological processes in- IAfand IAmare the standardizedvariances in adult life span, fluencing Ne. Thus, rather than discouraging the use of de- and Ibf and Ibm are the standardized variances in reproduc- mographic estimators outright, we emphasize the need for tive success. Practical considerations for estimating these long-term population monitoring and cautious application parameters are described elsewhere (Nunney & Elam when their use relies on few periodic estimates. 1994). In many cases, the availablenumber of Nt estimates Although the difficulty of obtaining accurate estimates may exceed the availablenumber of Net estimates. In such of long-term N, for any individual population is discour- cases, the long-term effective size should be estimated by aging, the successful recovery of a threatened or endan- substituting Net in equation 1 with the quantity zNt, gered population may depend little on accurate estima- where z represents the average Netl/Nt computed from the tion of its effective size. Recent theoretical work limited estimates available. Such estimates should also be suggests that long-term minimal viable Ne is roughly interpreted cautiously because Ne t/Nt estimates may vary 1000-5000 (Lande 1995a; Lynch 1996), and recent substantiallyover time (e.g., Peterson et al. 1998), and be- work suggests that values of Ne/N may average about 0.1 cause the assumption of independence between N,,/Nt (Frankham 1995b; Vucetich et al. 1997a). Taken to- and FPShas not yet been thoroughly evaluated. gether, these considerations suggest that a typical mini- mum viable population size may be roughly 5000- 50,000. Because most populations do not receive seri- Discussion ous conservation attention until their numbers have dwindled below well below 5000, it would seem mis- Our results show that N, is strongly dependent on time guided to be overly concerned with the precision and scale (Fig. 1). As longer time scales are considered, esti- accuracy of Ne estimates; Ne may be several orders of mates of Ne tend to decrease, and thus estimates of the magnitude smaller than the predicted minimal viable Ne. rate of tend to increase. The Ne values de- Nevertheless, additional studies aimed at the accurate es- rived from few annual estimatesunderestimates- will likely timation of long-term Ne/N may prove to be valuable. mate long-term rates of genetic deterioration because For example, large numbers of accurate estimates would they typically will fail to reflect the influence of infre- permit the clarification of any taxonomic patterns in quent population bottlen ec asts of genetic dete- long-term Ne/N (Frankham 1995b; Waite & Parker 1996;

Conservation Biology Volume 12, No. 5, October 1998 Vucetich& Waite EffectivePopulation Size 1029

Vucetich et al. 1997a). Then, for any taxon character- Crow, J. F., and C. Denniston. 1988. Inbreeding and variance effective ized a narrow range of Ne/N, converting to an ap- population numbers. 42:482-495. by S. L. and B. A. Maurer. minimal viable size would be Curnutt,J. L., Pimm, 1996. Population variability proximate population across space and time. Oikos 76:131-144. straightforward. Falconer, D. S., and T. F. C. Mackay. 1996. Introduction to quantitative In summary, because fluctuations in population size genetics. Longman, New York. (FPS) typically have a strong influence on Ne (Frankham Foley, P. 1994. Predicting extinction times from environmental sto- 1995b; Vucetich et al. 1997a), accurate estimation of chasticity and . Conservation Biology 8:124-137. R. 1995a. and extinction: a threshold effect. Ne annual population Frankham, Inbreeding long-term typically requires many Conservation Biology 9:792-799. counts. We thus urge cautious application and interpre- Frankham,R. 1995b. Effective population size/adult population size ra- tation of estimates of Ne based on limited numbers of tios in wildlife: a review. Genetical Research 66:95-107. counts. Unfortunately, no universal prescription can be Franklin, I. R. 1980. Evolutionary changes in small populations. Pages offered as to the number of counts that should be con- 135-149 in M. E. Soukl and B. A. Wilcox, editors. Conservation bi- Sinauer Massachusetts. ducted. Our do a suite of ology. Associates, Sunderland, findings, however, provide Gaston, K. J., and B. H. McArdle. 1994. The temporal variability of ani- conditional prescriptions (Fig. 3). For instance, about 20 mal abundances: measures, methods, and patterns. Philosophical annual counts may often be required to obtain (over)es- Transactions of the Royal Society B (London) 345:335-358. timates of long-term Ne with less than 10% error (and Grant, P. R., and B. R. Grant. 1992. Demography and the genetically ef- fective sizes of two of Darwin's finches. 95% confidence limits differing by less than a factor of populations 73: 766-784. a number of counts two). Although discouragingly large Halperin, M., andJ. Gurian. 1970. A note on estimation in straight line is required to obtain unbiased and reliable estimates, our regression when both variables are subject to error. Journal of the findings provide a basis for meaningful interpretation of American StatisticalAssociation 66:587-589. Ne estimates calculated from limited census data. Herrmann, B., and S. Hummel, editors. 1993. Ancient DNA. Springer, New York. Husband, B. C., and C. H. Barrett. 1995. Estimating effective popula- tion size: a reply to Nunney. Evolution 49:392-394. Acknowledgments Jimenez, J. A., K. A. Hughes, G. Alaks, L. Graham, and R. C. Lacy. 1994. An experimental study of inbreeding depression in a natural habi- tat. Science This work was a McIntire-Stennis to 266:271-273. supported by grant Keller, L. F., P. Arcese, J. N. M. Smith, W. M. Hochachka, and S. C. T.A.W. We owe an intellectual debt to L. Nunney and Stearns. 1994. Selection against inbred Song Sparrows during a nat- thank him for valuable discussion. We thank C. Bocetti ural population bottleneck. Nature 372:356-357. and C. Packer for sharing unpublished time-series data. Lande, R. 1993. Risks of population extinction from demographic and environmental and random American We thank N. Arguedas for several key references; S. L. stochasticity catastrophes. Naturalist 142:911-927. Pimm for and L. Vucetich, thought-provoking dialogue; Lande, R. 1995c. and conservation. Conservation Biology 9: R. Waples, and two anonymous reviewers for helpful 782-791. comments on the manuscript. Lande, R. 1995b. Breeding plans for small populations, based on the dynamics of quantitative genetic variance. Pages 318-340 in J. D. Ballou, M. Gilpin, and T. J. Foose, editors. Population management Literature Cited for survival and recovery. Columbia University Press, New York. Lande, R., and G. F. Barrowclough. 1987. Effective population size, ge- Allendorf, F. W., D. Bayles, D. L. Bottom, K. P. Currens, C. A. Frissell, netic variation and their use in population management. Pages 87- D. Hankin, J. A. Lichatowich, W. Nehlson, P. C. Trotter, and T. H. 123 in M. E. Soule, editor. Viable populations for conservation. Williams. 1997. Prioritizing Pacific salmon stocks for conservation. Cambridge University Press, New York. Conservation Biology 11:140-152. Latter,B. D. H., J. C. Mulley, D. Reid, and L. Pascoe. 1995. Reduced ge- Arinio, A., and S. L. Pimm. 1995. On the nature of population extremes. netic load revealed by slow inbreeding in Drosophila melano- EvolutionaryEcology 9:429-443. gaster. Genetics 139:287-297. Barton, N. H., and M. C. Whitlock. 1997. The evolution of metapopula- Lawton, J. H. 1988. More time means more variation. Nature 334:563. tions. Pages 183-210 in I. A. Hanski and M. E. Gilpin, editors. Lynch, M. 1996. A quantitative-genetic perspective on conservation is- biology: ecology, genetics, and evolution. Aca- sues. Pages 471-501 in J. C. Avise and J. L. Hamrick, editors. Con- demic Press, San Diego, California. servation genetics: case histories from nature. Chapman and Hall, Briscoe, D. A., J. M. Malpica, A. Robertson, G. J. Smith, R. Frankham, New York. R. G. Banks, andJ. S. F. Barker. 1992. Rapid loss of genetic variation Lynch, M., J. Conery, and R. Biirger. 1995. Mutation accumulation and in large captive populations of Drosophila flies: implications for the extinction of small sexual populations. American Naturalist the genetic management of captive populations. Conservation Biol- 146:489-518. ogy 6:416-425. Mace, G., and R. Lande. 1991. Assessing extinction threats: towards a Britten, H. B. 1996. Meta-analyses of the association between multilo- re-evaluation of IUCN threatened species categories. Conservation cus heterozygosity and . Evolution 50:2158-2164. Biology 5:148-157. Burger, R., and M. Lynch. 1995. Evolution and extinction in a changing Mills, S., and F. Alendorf. 1996. The one-migrant-per-generationrule in environment: a quantitative-genetic analysis. Evolution 49:151-163. conservation and management. ConservationBiology 10:1509-1518. Caballero, A. 1994. Developments in the prediction of effective popu- Motro, U., and G. Thomson. 1982. On heterozygosity and the effective lations size. Heredity 73:657-679. size of populations subject to size changes. Evolution 36:1059-1066. Crow, J. F., and M. Kimura. 1970. An introduction to population genet- Murdoch, W. M. 1994. Population regulation in theory and practice. ics theory. Harper and Row, New York. Ecology 75:271-287.

Conservation Biology Volume 12, No. 5, October 1998 1030 EffectivePopulation Size Vucetich& Waite

Neigel, J. E. 1996. Estimation of effective population size and migra- Soule, M. E., M. Gilpin, N. Conway, and T. Foose. 1986. The millen- tion parameters from genetic data. Pages 329-346 in T. B. Smith nium ark: how long a voyage, how many staterooms, how many and R. K. Wayne, editors. Molecular genetic approaches in conser- passengers? Zoo Biology 5:101-114. vation. Oxford University Press, Oxford, United Kingdom. Steele, J. H. 1985. A comparison of terrestrial and marine ecological Newman, D., and D. Pilson. 1997. Increased probability of extinction systems. Nature 313:355-358. due to decreased genetic effective population size: experimental Turchin, P., and A. D. Taylor. 1992. Complex dynamics in ecological populations of Clarkia pulchella. Evolution 51:354-362. time series. Ecology 73:289-305. Nunney, L. 1991. The influence of age structure and fecundity on ef- U.S. Fish and Wildlife Service. 1987. Recovery plan for the Puerto fective population size. Proceedings of the Royal Society of Lon- Rican parrot, Amazona vittata. Atlanta, Georgia. don, Series B 246:71-76. U.S. Fish and Wildlife Service. 1993. Grizzly bear recovery plan. Mis- Nunney, L. 1993. The influence of mating system and overlapping gen- soula, Montana. erations on effective population size. Evolution 47:1329-1341. U.S. Fish and Wildlife Service. 1994. Desert tortoise (Mojave popula- Nunney, L. 1995. Measuring the ratio of effective population size to tion) recovery plan. Portland, Oregon. adult numbers using genetic and ecological data. Evolution 49: Vucetich, J. A., T. A. Waite, and L. Nunney. 1997a. Fluctuating popula- 389-392. tion size and the ratio of effective to census population size (Ne/N). Nunney, L., and D. R. Elam. 1994. Estimating the effective population Evolution 51:2015-2019. size of conserved populations. Conservation Biology 8:175-184. Vucetich, J. A., R. O. Peterson, and T. A. Waite. 1997b. Effects of social Packer, C., A. E. Pusey, H. Rowley, D. A. Gilbert, J. Martenson, and S. J. structure and prey dynamics on extinction risk in gray wolves. O'Brien. 1991. Case study of a population bottleneck: lions of the Conservation Biology 11:966-976. Ngorongoro Crater. Conservation Biology 5:219-230. Waite, T. A., and P. G. Parker. 1996. Dimensionless life histories and ef- Peterson, R. O., N. J. Thomas, J. M. Thurber, J. A. Vucetich, and T. A. fective population size. Conservation Biology 10:1456-1462. Waite. 1998. Population limitation and the wolves of Isle Royale. Waples, R. S. 1989. A generalized approach for estimating effective Journal of Mammalogy 79:829-841. population size from temporal changes in allele frequency. Genet- Pimm, S. L., and A. Redfearn. 1988. The variability of animal popula- ics 109:379-391. tion abundances. Nature 334:613-614. Williamson, M. H. 1984. The measurement of population variability. Soule, M. E. 1980. Thresholds for survival: maintaining fitness and evo- Ecological 9:239-241. lutionary potential. Pages 151-170 in M. E. Soule and B. A. Wilcox, Wright,S. 1931. Evolutionin Mendelianpopulations. Genetics 16:97-159. editors. Conservation biology: the science of scarcity and diversity. Wright, S. 1938. Size of a population and breeding structure in relation Sinauer Associates, Sunderland, Massachusetts. to evolution. Science 87:430-431.

2' i

:-:j?F?- :.?? Ir :???.

aii 'r.R?LL !Y:? 1?:?"5:

I;? ?::'?'?:B '??:? ?:?I- i :

Conservation Biology Volume 12, No. 5, October 1998