Freshwater Biology (1987) 18, 361-372

Cladoceran birth and death rates estimates:experimental comparisons of egg-ratio methods

WILFRIED GABRIEL. BARBARA E. TAYLOR* andSUSANNE KIRSCH-PROKOSCH Department of Physiological Ecology,Max Planck Institute for Limnology, Plon, West Germany

SUMMARY. I. Birth and death rates of natural cladoceran populationscannot be measured directly. Estimates of these population parametersmust be calculated using methods that make assumptions about the formof population growth. These methods generally assume that thepopulation has a stable age distribution.

2. To assess the effect of variable age distributions, we tested six eggratio methods for estimating birth and death rates with data fromthirty-seven laboratory populations of Daphnia pulicaria. The popula-tions were grown under constant conditions, but the initial agedistributions and egg ratios of the populations varied. Actual death rateswere virtually zero, so the difference between the estimated and actualdeath rates measured the error in both birth and death rate estimates.

3. The results demonstrate that unstable population structures mayproduce large errors in the birth and death rates estimated by any ofthese methods. Among the methods tested, Taylor and Slatkin'sformula and Paloheimo's formula were most reliable for the ex-perimental data.

4. Further analyses of three of the methods were made usingcomputer simulations of growth of age-structured populations withinitially unstable age distributions. These analyses show that the timeinterval betvC'een sampling strongly influences the reliability of birth anddeath rate estimates. At a sampling interval of 2.5 days (equal to theduration of the egg stage), Paloheimo's formula was most accurate. Atlonger intervals (7.5-10 days), Taylor and Slatkin's formula whichincludes information on population structure was most accurate.

Introduction information about demographic processes suchas birth, recruitment and death. For example.

To study the factors controlling zooplankton correlations between estimated death rates ofpopulation dynamics, it is helpful to obtain

Correspondence: Dr W, Gabriel. Department of* Present address: Savannah River Ecology Physiological Ecology, Max Planck Institute for

Laboratory, Drawer E, Aiken. South Carolina Limnology. Postfach 165. D-2320 Plon. West Ger-29801. U.S.A. many.

361

362 W. Gabriel, B. E. Taylor and S. Kirsch-Prokosch

the prey population and abundances of preda-tory species have been used to infer whichpredators are most important to the preypopulation (Hall, 1964; Wright. 1%5). Birthand death rates of natural zooplankton popula-tions cannot be measured directly. Instead,they are estimated frotn mathematical modelsof population growth. During the last 30 years,various methods for estimating birth and deathrates have been developed. In this paper weexamine estimators based on the egg ratio (theratio of eggs to animals in the population;Edmondson, 1960), and compare their sensiti-vities to departure from the stable age distribu-tion assumed by these methods.

These estimators are derived from modelsfor exponential population growth. The birthrate b at sampling time /, is estimated from theegg ratio by making assumptions about the agedistribution of the eggs. The populationgrowth rate r is estimated from the change inpopulation size from sampling time r,, tosampling time r̂ . which measures the averagegrowth rate during the sampling interval t2-t\.The death rate (/ is estimated by the differencebetween the birth rate and the populationgrowth rate; this calculation assumes that theestimated birth rate applies for the entireinterval between samples.

Natural processes cause the rates and formsof population growth to vary, so that agedistributions of natural zooplankton popula-tions are unlikely to remain stable, especiallywhen environmental conditions are changing.To test whether departures from a stable agedistribution affect the accuracy of birth anddeath rate estimates based on egg ratios, wecompared estimated and actual death ratescalculated with data from a set of thirty-sevenlaboratory populations of Daphnia pulicariaForbes- We compared death rates because thisestimate includes two important sources oferror: error in the instantaneous estimate ofthe hirth rate; and error in the extrapolation ofthe birth rate over the sampling interval. Thepopulations were grown under constant condi-tions, but the initial age distributions and eggratios were varied. Different initial distribu-tions simulated effects of sudden environmen-tal change on population structure. Six varia-tions of the egg ratio method were applied toexperimental data. The results show that allmethods are affected by departures from the

stable age distribution. A detailed insight intodynamics is difficult to gain solely by labora-tory experiments. Therefore, we performedcomputer simulation as additional "experi-ments" to study if and how the estimatesdepend on the sampling interval.

The birth and death rate estimators

Birth and death rate estimates based on theegg ratio were developed for short-lived anim-als with rapid, continuous reproduction andapproximately exponential population growth.They are derived from quantities that caneasily be measured for freshwater zooplanktonsuch as cladocerans and rotifers, which usuallycarry their developing eggs. These methodshave developed mostly independently ofmethods to estimate demographic parametersfor longer-lived animals, such as fish (e.g.Ricker, 1975) and humans {e.g. Pollard. 1973;Keyfitz, 1977) and for populations that showstrong cohort structure (applications to zoo-plankton populations include Rigler & Cooley.1974, Gehrs & Robertson, 1975, and Hairston& Twombly. 1985). Although methods forestimating secondary production of zooplank-ton rely on similar assumptions, they are alsoseparate from work on population processes(see Winberg, 1971, and Edmondson & Win-berg. 1971).

The egg ratio was first used to estimateparameters of exponential population growthby Edmondson (I960), although Elster^(iy54)calculated a renewal coefficient ('Erneuerungskoeffizient") for the population from the eggratio and egg development period. Edmond-son's method was elaborated and refined byEdmondson tl%8). Caswell (1972), Paloheimo(1974) and Argentesi, Bernardi & di Cola(1974); and a similar method was developed inSmith's appendix to Cooper (1965). Theaccuracy and precision of these methods havesince received considerable attention anddevelopment (Prepas & Rigler, 1978; Seitz,1979; Threlkeld. 1979; Polishchuk. 1980, 1982;DeMott, 1980; Keen & Nassar. 1981; Polish-chuk & Ghilarov, 1981; Taylor & Slatkin,1981; Lynch, 1982, 1983; Dorazio & Lehman,1983).

The six ttiethods that we compared are listedin Table 1. All assume population growth is

Cladoceran birth and death rates estimates 363

TABLE 1. Formulae for estimating birth rates. Variables are: 6=instantaneous per capita birth rate in days '. withsubscript to indicate estimation method; r=insiantaneous per capita population growth rale; £=feniale eggs;Af=fcmale animals; y=juvcnile female animals; /l=adult female animals; Djr^duration of egg stage in days;Z>j=duration of juvenile stage in days; B=finite daily per capita egg hatching rate which is equivalent to thenumber of female eggs hatching per female over a 1-day interval.

Method Equation Source

1. Edmondson: bfjj

2. Caswell: ^CAS

3. Paloheimo: bf^

4. Seiu: ^SEI

5. Taylor & Slatkin: b^^

6. Taylor & Slatkin, modified:

'^^^ e'-1

B=EIND^In (£/N-H)

itpratidn nf . ^' ^-^^ r r^^+b-^-r0̂— ^r—

iteration of E/A=e^^^ {^^^'^—l}

Edmondson, 1960

Caswett. 1972

r aJunClinu, 17 ff

Seitz, 1979

Taylor & Slatkiniy81, equation 7

Taylor & Slatkin1981, equation 22

exponential, so that

where

r=b-d.

(1)

(2)and "̂(̂ 2) are populations at times f] and

fi. and r. b and d are the instantaneousgrowth, birth and death rates of the popula-tion. A stable age distribution is required forexponential growth if birth or death rates varywith the animal's age. Both certainly vary withage for cladocerans. As discussed in detail bySeitz (1979). these methods also assume thatthe death rates among all age classes, includingeggs, are equal. The methods differ in themeasure of the egg ratio (compare methods 1,2 and 3 with 5 and 6) and in the mechanismused to convert the egg ratio to a birth rale(compare methods 1. 2 and 3). Method 6includes a factor intended to compensate par-tially for effects of departures from the stableage distribution, as measured by the propor-tion of adults, on the inferred egg age distribu-tion. In addition to these six methods we alsoanalysed our experiments with the formulas ofArgenlesi et al. Their model, however, impli-citly assumes not only stable but also uniformage distribution in the compartments of eggs,

juveniles and adults. Therefore, under ourexperimental conditions it was not surprisingthat the model of Argentesi et al. gave resultsby one order of magnitude worse than the sixother models.

All of these methods estimate the hatchingrate of eggs from the egg development timeand population structure. Population structureis measured very simply by the egg ratio. Forexample. Edmondson (1960) "finite" birth rateassumes a uniform age distribution of eggs,while Paloheimo (1974) and Taylor & Slatkin(1981) assume that the distributions are thestable age distributions associated with themeasured egg ratios. Threlkeld's (1979). Dora-zio & Lehman's (1983) and Rigler & Down-ing's (1984) methods are based on directmeasures of the egg age distribution. From theegg age distribution the proportion of egghatching in the next instance can be calculated;this number, converted to a per capita mea-sure, is the instantaneous birth rate. Estimat-ing the birth rate thus assumes that the currentegg distribution has been correctly inferredfrom the current population structure. Apotentially serious problem with methods 5and 6 is the determination of the juveniledevelopmental time which is quite sensitive tofood concentrations at low food levels (Roma-

364 W. Gabriel B. E. Taylor and S. Kirsch-Prokosch

novski. 1984). Estimating the death rate as thedifference between the birth rate and thepopulation growth rate assumes that the in-stantaneous birth rate estimated for this instantremains constant during the interval overwhich population growth was measured. Whenenvironmental eonditions or the populationstructure are unstable, birth rates vary, andthis extrapolation can produce large errors.

The error in the estimated death rate in-cludes the sum of errors in the birth andpopulation growth rates. Errors in the deathrate estimate may occur if: (1) the populationdid not satisfy the assumptions of the methodfor estimating the birth rate at that point intime; (2) the birth rate varied during thesampling interval, so that the instantaneousestimate of the birth rate differed from theaverage birth rate over the sampling interval;(3) population densities or development timesneeded to calculate the population growth orbirth rates were measured inaccurately; (4) netmigration to or from the population occurred;(5) sampling errors can also be serious prob-lems (DeMott, 1980; Keen & Nassar, 1981).

Experimental methods

Stock cultures of an obligately parthenogeneticclone of Daphnia pulicaria Forbes were grownin 1.5 litre beakers under dim light. To obtainneonates, several large gravid females wereisolated in the evening, and young (aged0-15 h) were collected the next morning andtransferred to another beaker. This procedurewas repeated on successive days until juvenilesof a range of ages were available to begin theexperiments.

Experimental Daphnia populations weregrown in a flow-through culture system similarto that described by Muck & Lampert (1980).Ten 25(1 ml chambers were placed in atemperature-controlled water bath held at18°C. Culture medium dripped into each cul-ture chamber at the top at 1.5 litre d"̂ andflowed out through a Nitex screen at thebottom into an overflow system. The mediumwas membrane-filtered lake water enrichedwith 1.5 mg C litre"' of Scenedesmus acutusMeyen, which was grown in a chemostatculture, as described by Lampert (1975). Freshculture medium was prepared daily. The food

concentrations were high so that food was nota limiting factor in the experiments.

A total of thirty-seven experimental cultureswere run. Initial populations were ten, elevenor twelve Daphnia of varied ages and fecundi-ties. Initial proportions of juveniles JIN(y=juvenile female animals, A'=female anim-als) were 0.0, U.50, 0.75 and 0.91. The averageadult egg ratio EIA (£=female eggs. A=adultfemale animals) ranged from 2.8 to 15.3.Juvenile populations contained an even dis-tribution of animals aged 1-5 days.

Each experiment lasted 7 days. The experi-ments were terminated by fixing the popula-tion of each chamber in sugar-formalin(Haney & Hall, 1973). All of the animals werecounted and measured, and the egg stage(according to Threlkeld, 1979) was determinedfor ovigerous animals. The chambers werechecked every day for dead animals.

The egg development time D^ was esti-mated to be 2.95 days; the juvenile develop-ment time Dj, 6.5 days. The adult stage wasdefined by body length or the presence of eggsin the brood pouch. The smallest length atwhich 75% of the population carried eggs wasused as the criterion for minimum adult size.Additional details on the experimental setupare given by Kirsch (1982).

Computer simulations

To get additional information on the role ofthe sampling interval for the shortcoming ofdifferent egg ratio methods, we simulated thegrowth of a population by an age-structured,discrete time model. The model contains theessential features of a cladoceran life history:distinct egg, juvenile and adult stages, incuba-tion of the developing eggs by the adult, anditeroparous reproduction. (The term egg isused here to include egg and embryonicstages.) The population was divided into sixty-seven age classes of 0.5 day duration. Thedurations of the egg, juvenile and adult stageswere 5. 12 and 50 time steps, or 2.5, 6 and 25days. The lengths of the stages were adjustedto ensure that the corresponding Leslie matrixwas positively regular (Pollard, 1973), so thatthe age distribution of the population wouldbecome stable over time. An adult produced atotal of ten broods, carrying each brood for the

Cladoceran birth and death rates estimates 365

duration of the egg stage. Strictly parth-enogenetic reproduction was assumed. Thedeath rate was set to 0 for all age classesexcept the last.

The models of age-specific fecundity wereused in the simulations: constant (brood sizeC=5 for all ten broods); and increasing (broodsize Cj=5+5((-l) for brood number (=1, 10).Tliree initial age distributions were simulated:uniform (A/,= I for the juvenile through adultage classes /=5,66); negatively skewed(Ni=62-{i-5) for i=5,66); and positivelyskewed (A',= I+((-5) for (=5,66). Initialpopulations in the egg age classes (=0.4 werecalculated from the numbers and fecundities ofthe adult animals. Proportions of juveniles J/Nwere: 0.19 for the uniform age distribution:0.35 for the negatively skewed distibution; and0.04 for the positively skewed distribution. Atstable age distribution the proportions were0.80 for constant fecundity and 0.86 for in-creasing fecundity.

Birth, death and population growth rateswere calculated for each 0,5 day time step inthe simulations. Birth rates were estimatedaccording to Paloheimo's. Taylor and Slat-kin's. and Taylor and Slatkin's modified for-mulae (methods 3, 5 and 6 in Table 1).Population growth rates were calculated usingequation 1 with time intervals of 0.5, 2.5, 5.7.5 and 10 days. Death rates were calculatedusing the three birth rate estimates and the fivepopulation growth rates in equation 2.

Results

Actual birth and death rates of the experimentalpopulations

In thirty-seven experiments with initial andfinal populations from 920 to over 10.000animals, only eight dead animals were found.This number of deaths corresponds to aninstantaneous death rate £/=0.0(X)3 per day,which is negligibly small. In the followinganalyses, we assume that the actual death ratefor the experimental populations were zero.The death rate of eggs was estimated from theproportion of eggs with arrested development(stage I) in broods at stages Ill-V. Among 350broods at stages III-V, 234 out of 6298 eggswere in stage I, giving a death rate of 3.7%'during the egg stage, or an instantaneous death

04 -

02-

All 0-91 0-75Inilia! ^

C>5

FIG. 1. Population growth rates r for the ex-perimental populations. Rales are shown for allexperimental populations combined and tor popula-tions grouped hy the initial proportion of juvenilesJ/N (y=juvenite femate animals; .V=femalc anim-als). Vertical bars show ranges of values; horizontalbars show medians. The box shows quartiles for thecombined data,

rate of 0.013 per day. This estimate is conser-vative, because eggs which disintegrated orstopped developing after stage I were notdetectable.

Population growth rates r were calculatedusing equation 1 from the total numbers ofanimals at the beginning and end of theexperiment. Because death rates were virtuallyzero, birth rates were equal to growth rates ofthe experimental populations. These birth andpopulation growth rates ranged from 0.17 to0.43 per day (Fig. 1). The rates were affectedby the initial population structure. The treat-ments with low proportions of juvenile (JfN=0.0 and 7^=0.50) tended to have higherbirth rates.

Changes in structure of the experimentalpopulations

Changes in the proportion of juvenile J/Nand the egg ratios E/N and E/A measurechanges in the structure of these populations.(They may also measure response to changingenvironmental conditions, but that effect isnegligible in these experiments, because theanimals were preconditioned to the cultureconditions.) For comparison with the ex-perimental data, we estimated what the eggratios and proportions of juveniles should be atthe stable age distribution for this set ofexperimental populations according to theserelations:

^ 1 (3)

'"^-[) (4)

366 W. Gabriel, B. E. Taylor and S. Kirsch-Prokosch

lO-

o 0-5-

0-

f

(0)

r0-5

Initial

On

tbl

15-

o-

( c )

7-5Initial EM

FIG. 2. Changes in structure of the experimental populations. In all ihree panels shaded bars show theesiimated value of that measure at stable age distribution. Symbols indicate the initial proportion of juveniles( 0 . 0.0; O. 0.50; ^ , 0.75; O^ 0.91). (a) Initial and final proportions of juveniles yw. (b) Initial and final eggratios E/N. (c) Initial and final adult egg ratios E/A. {y = juvenile female animals; A'=female animals;£=female eggs; .4 = adutt fetnale animals.)

and

(5)

Equation 3 is obtained by rearrangingPiiloheimo's formula (method 3, Table 1);equation 4 is Tayior & Slatkin's formula for b(method 5, Table 1) and equation 5 can bederived from equations 3 and 4. The animalsused to initiate the experiments were takenfrom an exponentially growing population, butpicked out according to size and egg number.Therefore they were not perfectly randomlydistributed. We assumed, however, that theadult egg ratios E/A at the end of the experi-ments represent the fecundity of an exponen-tially growing population and can be used for a

rough estimation of proportions at stable agedistribution. The adult egg ratio for this groupof 1137 animals was £M = 10.55. The meanvalue from the thirty-seven experiments was10.65 with a standard deviation of 2.86. Wetook these values as our estimate of EIA atstable age distribution and used it in equation4 to calculate />=0.305 (+0.021. -0.027).Then. EIN=\M (+0.16. -0.19) was calcu-lated from equation 3, and J/N=0.S63(+0.017, -0.027) was calculated from equa-tion 5.

The experimental populations were initiatedwith a wide range of proportions of juveniles.At the end of the 7-day experiments, all of thepopulations were close to the stable proportion

Cladoceran birth and death rates estimates 367

(Fig. 2a). The populations which initially con-tained only adults ended with the highest rangeand median of the proportion of juveniles.Juveniles produced in this set of populationsdid not have time to mature during the experi-ment.

The wide initial range in the egg ratios E/N(from 0,41 to 14.75. median=3.81. mean=3,85, <7=3.13) reflected the wide range in theproportion of juveniles. As the proportion ofjuveniles approached a stable value, so did theegg ratio (Fig. 2b). (Values of the final eggratios ranged from 0.72 to 2.76 withmedian =1.51, mean = 1.58, and a^O.52.)

The adult egg ratio E/A changed drasticallywithin single experiments but the range ofvalues was the most stable among the mea-sures of population structure (Fig. 2c). Thismeasure of population structure is, of course,not affected by the proportion of juveniles,although it may vary with age composition ofthe adult population. The initial median adultegg ratio was 8.2; the final median was 10.4.As we noted above, the adults used to beginthe experiments represent a sample from anexponentially growing population, and theadult egg ratio should consequently be close tothe stable value. Adult egg ratios varied sub-stantially among experimental populations be-cause they contained small numbers of adults,whose individual fecundities were quite vari-able.

Error in the estimated birth and death rates

Methods for the estimating birth and deathrates appeared to be biased (Fig. 3). Five ofthe six methods tended to overestimate deathrate, with errors as large as 3.55 (Caswell) perday (Taylor & Slatkin modified 2.46, Edmond-son 1.35, Seitz 0.69. Paloheimo 0.51). Taylor& Slatkin's method generally underestimatedthe death rate and showed a maximum error of-0.12 per day. Based on the median error.Taylor & Slatkin's method was best for ourdata. It also showed the narrowest range oferrors. Paloheimo's method was next, also itsmedian error and range were much greater.Seitz' method gave results similar toPaloheimo's method. Caswell's method,Taylor & Siatkin's modified method, andEdmondson's method showed substantiallygreater errors.

The magnitude of error depended on thestarting conditions as well as on the estimationmethod. All of the methods performed fairiywell for the initial population structure closestto the stable age structure (see Fig. 2a),J/N=0.9l. As the proportion of adults in-creased, errors increased markedly for allestimates, although Taylor & Slatkin's methodshowed a smaller increase in difference fromexpected than the others. Although Taylor &Slatkin's modified formula was derived tocompensate for departures from the stableproportion of adults in the population, itperformed worst than other methods whendepartures from the stable proportion of adultswere greatest.

.Simulation results

The estimated death rate was used to mea-sure performance of the birth rate formulae.The actual death rates in the simulations werevery close to zero. The number of age classeswas great enough so that the effect of mortalityin the last age class was negligible. (The actualrange of d was O.(K)O-().O11.) Death rates werecalculated using three birth rate estimates andpopulation growth rates measured over fivetime intervals (1, 5, 10, 15 and 20 time steps,or 0.5, 2.5. 5, 7.5 and 10 days). These rateswere calculated at each time step over the firsttwenty time steps of the simulations. Duringthe first twenty time steps (10 days), the initialage distributions altered considerably. For ex-ample, in the simulations with increasing age-specific fecundity, at time step 0 the deviationsfrom the stable birth rate /)=0.326 ranged from+2.033 to +3.727. At time step 19 the devia-tions ranged from +0.025 to +0.059. For eachtime interval and estimation method, the firsttwenty estimates were combined from thesimulations with each of three initial agestructures (uniform, positively skewed, andnegatively skewed) to calculate the resultsshown in Fig. 4.

As the death rates used in the simulationsare zero, the ranges of the estimated deathrates are the most useful indicators of theperformance of these estimators (Fig. 4). be-cause we have no measure of a typical depar-ture from stable age distribution. The be-haviour of each estimator depended stronglyon the time interval over which r was calcu-

368 W. Gabriel. B. E. Taylor and S. Kirsch-Prokosch

04-

0-2-

0-

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06-

06-

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I 2.0-&

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06

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All 0-91 0-75 05 0

d'-b^-r

- 0 2

All 0-91 075 05 0

All 0-91 o r 5 0 5

Initiol proportion of juveni

FIG. 3. Estimated death rates d for the experimental populations. Separate panels show d calculated usingdifferent equations for estimating the birth rates b (see Table 1 for list of equations). Within each panel dataarc shown for <ill experimental populations combined and for populations grouped by the initial proportion otjuveniles JIN (y=juveniie female animals; N=female animals). The deviation of d from 0 measures the error inthe estimate of fr.Vertical bars show ranges of values; horizontal bars show medians; circles show means.Boxes show quartiles for the combined data.

Cladoceran birth and death rates estimates 369

0-5-

-0-5-

-1-0-

r 0-5-1Vo

e 0-BQJ

•D

•O-0-5-t)o

E

lij -l-O

1-0

0-5-

0-

t

-oe-

f i - II II0-5 2-5 5 7-5

Time (ntervol x (doys)10

FIG. 4. Estimated dcaih rates for the simulated popultions. Palohcimo's formula (upper panel). Taylor &Slatkin's formula (middle panel), and Taylor & Slatkin's modified fortnula (lower panel) were used to estimateihc birth rates. Rates were calculated over five time intervals shown on the .r-axis. Vertical bars show theranges of sixty dealh rate estimates. Short horizontal bars indicate where ranges extend beyond the axes of theplot. Longer horizontal bars show medians, and boxes show quartiles. Shaded and unshaded boxes indicatedata from simulations with constant age-specific fecundity and with increasing age-specific fecutidity,respectively.

lated. At a time interval of 2.5 days (equal tothe duration of the egg stage), Paloheitno'sformula looks pretty good most of the time. Itwas substantially less reliable at shorter inter-vals, when it tetided to underestimate rates, orat longer intervals, when it tended to overesti-mate them. Taylor & Slatkin's formula pro-duced fairly accurate estimates at longer timeinteirals. As judged by deviations. Taylor &Slatkin's modified estimate was unreliable atall time intervals. Although the medians errorswere generally slightly below zero, large posi-tive deviations occurred.

The life history of the simulated population

also affected the results. When fecundity in-creased with the age of the adult, rather thanremaining constant, errors in the death rateestimates were greater. This was because theadult egg ratios varied with age structurewithin the adult population, magnifying theeffect of the instability in age structure.

Discussion

The experiments and simulations show thatunstable age structures may produce errors inestimates of birth and death rates using formu-

370 W. Gabriel, B. E. Taylor and S. Kirsch-Prokosch

lae based on egg ratios. The magnitude oferror is influenced by the method chosen andthe population structure. The simulations indi-cate that the amount of error is also influencedby the time interval over which r is calculatedand the animal's life history.

Over a time interval equal to the duration ofthe egg stage O^. Paloheimo's formula esti-mated the population birth rate very accuratelywhen age-specific death rates were equal forall age classes. (The small deviations from 0 inthe death rate estimates made withPaloheimo's formula and r calculated over a2,5 day interval in Fig. 4 reflect the mortalityof the last age class of animals in the simula-tions.) All of the eggs present in the popula-tion at the time f, must hatch (or die) in theinterval [f|. r2=(i + />El- Essentially, the formu-la converts the proportional increment EtN tothe population over the interval D^_ to theequivalent constant instantaneous birth rate.When population structure is changing, errorsin estimates over intervals shorter than Dnoccur because the birth rate is not constantover the interval. Over intervals longer than/?p, changes in egg ratios also contribute to theerror.

Taylor & Slatkin's formula worked betterover intervals longer than the duration of theegg stage, such as the 7-day duration of ourexperiments and the 7.5- and 10-day intervalsin our simulations, because the average birthrate approached the stable birth rate as thepopulation structure stabilized. If the adult eggratio is initially close to its stable value, thismethod predicts what the birth rate will be-come as the population structure stabilizes. Inthe simulations it worked better for the lifehistory with constant age-specific fecundity,because the initial adult egg ratio was un-affected by the age composition of the adultpopulation and was consequently exactly at itsstable value.

Among the methods which do not includedata on the proportion of juveniles and adultsPaloheimo's method was the best at the timeinterval used in our experiments. Seitz'method gave results very similar toPaloheimo's for the experimental populations.Because Seitz' calculations are more elaborateand require estimates of more parameters,Paloheimo's method is generally preferable.

These results suggest that Paloheimo's for-

mula is the best choice for estimating birth anddeath rates when the interval between samplesis close to the duration of egg development(see also Keen & Nassar, 1981: Polishchuk.1980). Both the experiments and the simula-tions indicate Taylor & Slatkin's method mightwork better at longer sampling intervals, parti-cularly if disturbances in the age structure ofthe population are episodic. Taylor & Slatkin'smodified formula seems to have no valueunder the conditions tested, and othermethods based on direct inference of birthrates from egg age distributions are probablysimilarly unreliable when projected over inter-vals longer than some fraction of the durationof the egg stage.

Our tests simulated a single initial episode ofdisturbance to the age structure of a popula-tion, but the age structure will tend to stabilizeafter a disturbance under any conditions. Therate at which this occurs generally increaseswith the birth rate (not the net populationgrowth rate). In rapidly reproducing popula-tions, newborn juveniles quickly dominate thepopulation structure. Because the structurechanges more slowly for populations with lowbirth rates, Taylor & Slatkin's formula may notbe the best choice at longer time intervals.However, birth rate estimates have much smal-ler range of error for populations with lowbirth rates (Taylor & Slatkin, 1981), and deathrate estimates are consequently much morereliable; thus the choice of methods is not socritical.

Other investigators have used computersimulations to test properties of birth anddeath rate estimates. Seitz (1979) tested birthrate estimates using the formulae of Edmond-son, Caswell and Paloheimo. and two methodsthat he derived. (His two-stage model ismethod 4 in Table 1,) He simulated populationgrowth in a variable environment. For thetwo-stage models (eggs and animals; juvenileand adult animals not treated separately), hisresults were similar to ours. Ranked by de-creasing reliability, the methods were: Seitz',Paloheimo's, Edmondson's and Caswell's.Seitz' and Paloheimo's methods gave closeresults. Full details of the simulations andresults were not given, but he concluded that:sampling interval affected the error in theestimates; the estimates were worse whendeath rates were high, especially if they were

Cladoceran birth and death rates estimates 371

higher for eggs and adults than for juveniles;and in general deviations from steady stateconditions may produce great errors.

Polishchuk (1980, 1982) used computersimulations to test the methods of Leslie(1948) as modified by Polishchuk (1980). EIs-ter (1954) and Paloheimo (1974). He foundthat Paloheimo's method estimated the birthrate most accurately and that all of themethods worked best at a sampling intervalequal to the duration of egg development.

Differential age- or stage-specific mortality,which we did not test here, also contributes toerrors in birth and death rate estimates. Otherproblems to consider when choosing methodsand interpreting estimates include the effectsof vertical stratification or migration of thepopulation (Prepas & Rigler, 1978; Keen.1981) and sampHng errors in estimates ofparameters (DeMott, 1980; Keen & Nassar,1981; Taylor & Slatkin, 19K!).

The birth and death rate estimates that wehave tested here are frequently used by aquaticecologists. Error of the magnitude that ourresults show could significantly affect conclu-sions about factors affecting birth rates, espe-cially when birth rates are high and environ-mental conditions are rapidly changing. Itcould drastically alter conclusions about pro-cesses affecting death rates. Inference aboutpopulation processes that depend on theseestimates must be made with caution.

As a consequence of this study we give thefollowing practical recommendations. It is veryuseful to monitor the ratio of adults to juve-niles. Large errors can be expected when thechange in this ratio is very rapid, particularlyat high birth rates. Any individual estimate ofbirth and death rate should be treated withcaution, although a cladoceran population canachieve a near stable age distribution ratherquickly following a perturbation. However,trends over several estimates or correlationsbased on a longer data set are more reliable(see e.g. DeMott, 1983), For general use. thePaloheimo formula is simple and reasonablyaccurate for slowly changing age-structures.When age-structure is changing quickly, theTaylor & Slatkin method is most accurateprovided that the juvenile developmental timeis exactly known.

We hope that plankton ecologists will con-tinue to criticize and test these methods.

Although they are an imperfect set of tools,they are among the most useful that we havefor studying processes in planktonic communi-ties.

Acknowledgments

We thank Winfried Lampert for encouragingdiscussions and helpful criticism. We appreci-ate helpful comments from Scott Cooper,William R. DeMott. Diana Engle, MonaMort. Ace Sarnelli and Alfred Seitz,

B. E. Taylor was supported by contractDE-AC-()9-76SRnO819 between the UnitedStates Department of Energy and the Instituteof Ecology of the University of Georgia duringmost of her work on this project.

References

Argentesi F,, de Bernardi R, & di Cola G, (1974)Mathematical models for the analyses of popula-tion dynamics in species with continuous recruit-ment, Memorie deU'hiitulo di Idrobiologia. 31,24.S-275.

Caswell H. (1972) On instantaneous and finite birthrates, Limnologv and Oceanography. 17, 787-791.

Cooper W.E- (1965) Dynamics and production of anatural population of a freshwater amphipod,Hyalella azteca. Ecological Monographs, 35, 377-394.

DeMott W,R. (1980) An analysis of the precision ofbirth and death rate estimates for egg-bearingzooplankters, Evohuion and Ecology of Zoo-plankton Communities (Ed, W. C, Kerfoot), pp,337-345. University Press of New England,Hanover. New Hampshire,

DeMott W.R, (1983) Seasonal succession in a natu-ral Daphnia assemblage. Ecological Monographs,S3, 321-340,

Dorazio R.M. & Lehman J,T. (1983) Optimal repro-ductive strategies in age-structured populationsofzooplankton. Freshwater Biology. 13. 157-175.

Edmondson W,T, (19WI) Reproductive rates of rotif-ers in natural populations. Memnrie ttnWhtitutodi Idrohiologia. 2, 21-77,

Edmondson W T, (1%8) A graphical model forevaluating the use of the egg ratio for measuringbirth and death rates, Oecologiu. 1, 1-37.

Edmondson WT, & Winberg G,G, (Eds) (1971) AManual on Methods for the Assessment of Secon-dary Productivity in Fresh Waters. IBP Hand-book No. 17, Blackwell Seientific Publications,Oxford,

Elstcr H.-J, (1954) Ober die Populationsdynamikvon Eitdiaptomus gracile Sars und Heterocope

372 W. Gabriel, B. E. Taylor and S. Kirsch-Prokosch

borealis Fischer im Bodensee-Obersce. ArchivfUr Hvtlrohiotogie, Suppl. 20, 54(>-614.

Gehrs C.W. & Robertstin A. (1975) Use of lifetables in analysing and the dynamics of copcpodpopulations. Ecology, 56. 665-<i72.

Hairstan, N.G.. Jr & fwombly S. (1985) Obtaininglife table data from cohort analyses: a critique ofcurrent methods. Linmologv and Oceanography.30, 886-893.

Hall D.J. (1964) An experimental approach (o thedynamics of a natural population of Daphniagaleata mendotae. Ecology. 45, 94-112.

Haney J.F. & Hall D.J. (1973) Sugar coated Daph-nia: a preservation technique for Cladocera.Limnology and Oceanography. 18, 331-333.

Keen R (1981) Vertical migration, hatching rates,and distnhution of egg stages in freshwaterzooplankton. Journal of Thermal Biology. 6,349-351.

Keen R. & Nassar R. (1981) Confidence intervalsfor birth and death rates estimated with theegg-ratio technique for natural populations ofzooplankton. Limnology and Oceanography, 26.131-142.

Keyfitz N. (1977) Introduction to the Mathematics ofPopulation, with Revisions. Addison-WesleyPublishing Company. Reading. Mass.

Kirsch S. (19S2) Untersuchungen zum EinfluB vonStorfaktoren auf die Aussagekraft populations-dynamischer Modelle: Tesi mit Laborpopula-tionen von Daphnia pulicaria. Diplum-T^eses,Gottingcn.

Lampert W. (1975) A laboratory system for thecultivation of large numbers of Daphnia undercontrolled conditions. Archiv fiir Hydrobiologie.48, 138-140.

Leslie PH. (1948) Some further notes on the use ofmatrices in population mathematics. Biometrika,35, 213-245.

Lynch M. (1982) How well does the Edmondson-Paloheimo model approximate instantaneousbirth rates? Ecology. 63, 12-18.

Lynch M. {19S3) Estimation of size-specific mortal-ity rates in zooptankton populations by periodicsampling. Limnology and Oceanography. 28,533-545.

Muck P. & Lampert W. (1')8<I) Feeding of freshwater liltcr-fceders at very low food concentra-tions: Poor evidence for "threshold feeding' and'optimal foraging' in Daphnia longispina andEudiaptomus gracilLi. Journal of Plankton Re-search. 2, 3fi7-379.

Paloheimo J.E. (1974) Calculation of instantaneousbirth rates. Limnology and Oceanography. 19,692-694.

Polishchuk L.V. (1980) A comparison of differentmethods of estimation of birth and death rates inplankton animals. Zhurnal Obshchei Biologii, 41,125-137 (in Russian with English summary|.

Polishchuk L.V. (1982) The 'negative mortality" ofplanktonic animals and sampling schedule. Zhur-nal Obshchei Biohgii, 43, 411-418 [in Russianwith English summary].

Polishchuk L.V. & Ghilarov A.M. (1981) Compari-son of two approaches used to calculate zoo-plankton mortality. Limnology and Oceanogra-phy. 26, 1162-1168.

Pollard, J.H. (1973) Mathematical Models for theGrowth of Human Populations. Cambridge Uni-versity Press.

Prt-pas E. & Rigler F.H. (1978) The enigma ofDaphnia death rates. Limnology and Oceanogra-phy. 19, 970-988.

Ricker W.E. (1975) Computation and interpretationof biological statisttcs ot fish populations. Hsher-ies Research Board of Canada, Bulletin. 191.

Rigler F.H, & Cooley J.M. (1974) The use of fielddata to derive population statistics of multivoltinecopepods. Limnology and Oceanography. 19,636-655.

Rigler F.H. & Downing J.A. (1984) The calculationof secondary productivity. A Manual on Methodsfor the Assessment of Secondary Productivity inFresh Waters (Eds F. H. Rigler and J. A.Downing), 2nd edn. pp. 19-58. IBP HandbookNo, 17. Blackwell Scientific Publications, Ox-ford.

Romanovski Y.E. (1984) Prolongation of post-embryonic development in experimental andnatural cladoceran ptJpulations. huernaiionaleRevue der Gesamten Hydrohiologie, 69, 149-157.

Seitz A. (1979) On calculation of birth rates anddeath rates in fluctuating populations with con-tinuous recruitment. Oecologia, 41, 343-359.

Taylor B.E. & Slatkin M. (1981) Estimating birthand death rates of zooplankton. Limnology andOceanography, 26, 143-158.

ITirelkeld S.T. 0^79) Estimating cladoceran birthrates: the importance of egg mortality and theegg age distribution. Limnology and Oceanogra-phy. 24. 601-612.

Winberg G.G. (1971) Methods for the Estimation ofProduction of Aquatic Animals (Transl. by A.Duncan). Academic Press, London.

Wright I.e. (1965) The population dynamics andproduction of Daphnia in Canyon Ferry Reser-voir, Montana, Limnology and Oceanography,10, 585-590.

(Manuscript accepted 6 April 1987)