J. N. Am. Benthol. Soc. 1990, 9(1):9-16

© 1990 by The North American Benthological Society

Empirical evidence for differences among methods for

calculating secondary production

C. Plante and J. A. Downing

Departement de Sciences biologiques, Universite de Montreal,

C.P. 6128, Succursale 'A', Montreal, Quebec, Canada H3C 3J7

Abstract. The hypothesis that different secondary production estimation methods yield unbiased

and equally precise estimates is tested using published data from 66 benthic invertebrate populations

from lentic habitats. Tests are performed by Kruskal-Wallis one-way analysis of the residuals of a

published empirical equation accounting for the important covariables biomass, body-mass, and

water temperature. While no method was found to be significantly biased, the size-frequency method

was less precise than the Allen curve, growth increment summation or instantaneous growth

methods, yielding estimates about three times farther from the probable production values than

other methods. Imprecision of inferred cohort production interval (CPI) is suggested as one source

of error.

Key words: secondary production, calculation methods, precision, bias, benthos, cohort produc

tion interval.

Secondary production measurements are nec

essary for studying the transfer of energy and

material in natural ecosystems and managing

aquatic resources (Downing 1984). Comparing

secondary productivity of diverse aquatic eco

systems is useful in forming general theories of

aquatic productivity. Such theories will be tract

able only if measures of secondary production

are comparable among ecosystems. Several pro

cedures based on common concepts of popu

lation dynamics are currently used to make such

estimates (Benke 1984, Downing 1984). The most

commonly used methods for estimating inver

tebrate population production are Allen curves,

growth increment summation, instantaneous

growth, and size-frequency. Each technique

makes different simplifying assumptions about

such factors as patterns of growth and mortality

(Benke 1984, Rigler and Downing 1984). Im-

precisions in these assumptions can be trans

lated into bias in resulting estimates.

Side-by-side comparisons of secondary pro

duction methods suggest that under different

conditions some techniques yield different es

timates. Several authors (e.g.. Waters and Craw

ford 1973, Benke 1976, Riklik and Momot 1982,

Lauzon and Harper 1986) have found that dif

ferent calculation methods yield differences

from 1 to 25% in estimated production. Simu

lation studies corroborate this finding and show

that different methods of production calcula

tion must give rise to differences in estimated

production rates. Cushman et al. (1977) com

pared the removal summation, instantaneous

growth and size-frequency methods and con

cluded that all methods yield underestimates if

their assumptions are not consistent with the

characteristics of the population. Cushman et

al. (1977) suggested that the removal summa

tion method is the most robust and that biases

can be reduced by increasing sampling inten

sity. Morin et al. (1987) compared the size-fre

quency, the Allen curve, the growth increment

summation, and the instantaneous growth

methods for populations with different patterns

of growth, mortality and recruitment, using dif

ferent degrees of sampling intensity. They dem

onstrated that the size-frequency method should

underestimate population production, espe

cially where hatching is perfectly synchronous.

All methods were found to underestimate pro

duction if the sampling interval did not cover

intense periods of productivity. In addition,

Morin et al. (1987) showed that sampling errors

can result in both bias and imprecision in pro

duction estimates, especially in the case of the

size-frequency method.

All calculation methods give errors under

certain circumstances. Such calculation errors

may be insignificant because confidence inter

vals around individual production estimates are

broad (Morin et al. 1987). No study to date has

analyzed production data to see whether dif

ferent production calculation methods actually

give systematically biased or excessively vari

able estimates under actual field conditions. The

10 C. Plante and J. A. Downing [Volume 9

objective of this study was to compare pub

lished field production estimates, made with

different methods, to see whether any tech

niques yield significantly biased or significant

ly more variable estimates than others.

Methods

Several factors such as biomass, body-size, rate

of growth, voltinism, temperature, and food

availability and quality are known to be im

portant covariables of population production.

Direct comparisons of average annual produc

tion of populations differing in these charac

teristics would be inappropriate. In this study,

therefore, we approached this problem by col

lecting published data on the secondary pro

duction of lentic invertebrate populations from

a diverse array of populations and environ

ments. We then employed these population and

environmental characteristics as covariables in

an analysis of covariance to test the hypothesis

that different techniques for estimating second

ary production yield equivalent production es

timates for populations of equal biomass and

size, living at similar temperatures.

Plante and Downing (1989) developed an

equation to predict the productivity of lentic,

aquatic invertebrate populations based on mul

tiple regression analyses of published second

ary production estimates of 137 populations of

benthos and zooplankton from 50 lakes, res

ervoirs, and ponds. The best regression fit was:

log10P = 0.05 + 0.79 log10B

+ 0.05T - 0.16 log10W0 (1)

(R2 = 0.79, n = 138, F = 165, p <k 0.001) where

P is the annual secondary production (g dry

mass/m2/yr), B is the mean annual biomass (g

dry mass/m2), T is the mean annual surface tem

perature (°C), and Wm is the maximum individ

ual body mass (mg dry mass/individual). The

equation characterizes covariation of P, 6, T and

Wm equally well for both benthos and zoo-

plankton (Plante and Downing 1989). It pre

dicts the most probable level of P given the

population biomass, body-size, and the ambient

temperature. Because this equation was pro

duced from data covering the range of possible

ecological conditions in lentic habitats (Plante

and Downing 1989), it can be used to remove

the effect of the important covariates, B, T and

Wm from the published production estimates. If

the data collected using all computation meth

ods are equivalent, analysis of variance should

reveal no significant difference among the av

erage distances between observed production

and predictions from Equation 1. Further de

tails of such residual analyses, corresponding

to the analysis of covariance, are presented by

Draper and Smith (1981) and Gujarati (1978).

This study analyzes differences in residuals

in published data on benthos production esti

mated using three methods: the size-frequency

method (SF), the Allen curve and growth in

crement summation (AC-GS), and the instan

taneous growth method (IG). Allen curve and

growth increment summation were examined

together because both are based on the rela

tionship between the number of organisms and

individual body mass in a cohort (Rigler and

Downing 1984). The data set analyzed consists

of all of the 66 benthic invertebrate production

estimates used to compute Equation 1. These

populations come from 22 different ecosystems

(Table 1). They span a range of production from

0.03 to 66.40 g dry mass/m2/yr, a range of mean

annual biomass from 0.002 to 10 g dry mass/

m2, a range of body-size from 0.01 jug to 60 mg

dry mass, and were found in environments with

average annual surface water temperatures

ranging from 4 to 18°C (Table 1). Of these es

timates, 20 were made with the size-frequency

technique, 26 were made with the Allen curve

or growth increment summation methods, and

20 were made with the instantaneous growth

technique.

The residuals from Plante and Downing's

(1989) regression equation (Equation 1) (ob

served log P — predicted log P) were calculated

for each of 66 populations and two hypotheses

were tested. The hypothesis that all three es

timation procedures yielded equal residuals was

tested using Kruskal-Wallis one-way analysis

(Conover 1971). Rejection of the null hypoth

esis would indicate that one or more of the cal

culation methods yield biased production es

timates. Precision of estimation was examined

by testing the hypothesis that all three esti

mation procedures yielded equal absolute val

ues of residuals. Rejection of the second null

hypothesis would indicate that one or more of

the calculation methods yielded estimates that

were significantly farther from the most prob

able value.

1990] Calculating secondary production 11

Results and Discussion

Figure 1 shows that observed production rates

rise with predictions with a slope close to 1,

therefore Equation 1 accounts efficiently for the

covariables B, T and Wm for data on benthic

invertebrate production. Figure 1 also suggests

that some methods tend to yield different ob

servations from others. For example, estimates

using the size-frequency method often seem to

lie above the others. Figure 2 shows frequency

histograms of calculated residuals from Equa

tion 1 separated by method. Although Figure 2

shows that residuals for the size-frequency

method cluster less tightly around the predict

ed values, Table 2 shows that there is no statis

tically significant {p > 0.05) tendency for the

size-frequency method to yield a systematically

positive or negative bias. This analysis suggests

that if a large number of production estimates

is made, none of these techniques will engen

der systematic bias in the average production

estimate obtained.

On the other hand, several individual pro

duction estimates made with the size-frequency

method seem to fall far from the value expected

on the basis of biomass, body-size and temper

ature. The median absolute value of the resid

uals for the size-frequency method (Fig. 2) is

0.39 compared with medians of 0.14 and 0.06

for the Allen curve-increment summation and

instantaneous growth techniques, respectively.

This difference means that, on average, size-

frequency estimates of production are about 3

times farther (inverse log of the difference be

tween the median of the methods) from the

most probable production value than the other

two classes of methods. Table 3 shows that the

absolute values of the residuals of estimates

made with the size-frequency method are sig

nificantly (p = 0.0002) greater, on average, than

the absolute values of the residuals obtained

using the other two methods. A Kruskal-Wallis

test shows no significant difference among the

absolute values of the residuals obtained using

the Allen curve, increment summation or in

stantaneous growth methods (p = 0.17). Al

though our observations are independent mea

surements of production made on autonomous

populations, more than one population was in

cluded for several lakes, suggesting some lack

of statistical independence. We believe, how

ever, that results shown in Table 3 are so highly

significant that some lack of independence poses

no practical problem to interpretation. Al

though Table 2 shows that the size-frequency

method yields no bias if a large number of pro

duction estimates are considered, Table 3 shows

that, on average, individual size-frequency pro

duction estimates were significantly farther from

the most probable production value than esti

mates made with other methods.

Our results appear to contradict the simula

tion studies of Morin et al. (1987). Their work

suggests that the size-frequency method should

yield severe underestimates of secondary pro

duction when cohort synchrony is perfect, and

that all production estimation methods should

yield approximately equal precision for a given

sampling effort. We found no detectable differ

ence in the bias of various techniques but found

that the size-frequency method is the least pre

cise. It is likely that the degree of bias found in

cases where cohort synchrony is not perfect (—30

to +10%) would not be detectable in our anal

yses owing to the errors associated with esti

mates of biomass, numbers, and environmental

characteristics. The relatively low precision of

the size-frequency method in actual field ap

plications may be due to factors not examined

by Morin et al. (1987). For example, Benke (1984)

shows that all size-frequency estimates of sec

ondary production must be corrected to annual

production by multiplying the estimate by

365/CPI, where CPI is the average cohort pro

duction interval. This correction was applied in

all studies except the size-frequency estimates

of Tudorancea et al. (1979) where CPI correction

would have resulted in an even greater depar

ture from the most probable production value.

Size-frequency estimates are usually made when

complete population life-history data are un

available or impossible to collect, e.g., where

successive cohorts are asynchronous and over

lap considerably. Except in situations where

there is no synchrony of reproduction, if enough

population data were collected so that the CPI

and the growth pattern of each cohort could be

known with precision (knowledge of growth

patterns is necessary to find appropriate size

classes, Benke 1984), then one would possess

sufficient data to apply cohort methods such as

the Allen curve or growth increment summa

tion technique. Thus, such information is rarely

well known where the size-frequency method

is applied.

Table 1. Data used to test for empirical differences in bias and precision of secondary production estimates in populations of aquatic insect larvae made

using the size-frequency (SF), Allen curve and growth increment summation (AC-GS), and instantaneous growth (IG) methods. Data are listed in decreasing

order of absolute values of the residuals from Equation 1. The taxonomic group (Group) is indicated as I for insects, M for molluscs, C for crustaceans, and A

for annelids. Resid. is the residual from Equation 1 (log observed — log predicted). Wm is the maximum individual body mass (mg dry mass), B is the annual

mean biomass (g dry mass/m2), T is the mean annual water temperature (°C at surface), P is the secondary production (g dry mass/mVyr), and P is the secondary

production predicted from Equation 1 (g dry mass/m2/yr). Some temperature data were obtained from other sources (see Plante and Downing 1989).

Taxon

Ceratopogonidae

Amnicola limosa

Chironomus sp.

Pisidium spp.

Chaoborus punctinatus

Tanytarsus gracilendicus

Tendipes decorus

Procladius sp.

Valvata tricarinata

Parartemia zietziana

Harnischis curtilamellata

Tanytarsus spp.

Parachironomus mancus

Procladius freemani

Chironomus anthracinus

Tinodes waemeri

Chironomus islandicus

Chironomus sp.

Tanytarsus barbitarsus

Cryptochironomus spp.

Pisidium sp.

Chironomidae

Procladius simplicistilus

Chironomus anthracinus

Zavrelymia melanura

Chironomus plumosus

Erpobdella testacea

Psilotanypus ruforittatus

Chironomus anthracinus

Tanytarsus inopersus

Asellus obtusus

Asellus aquaticus

Water-body

Lake Norman

Lake Manitoba

Lake Manitoba

Lake Manitoba

Lake Norman

Myvatn

Texas pond

Texas pond

Lake Manitoba

Pink Lake

Lake Manitoba

Lake Norman

Eglwys Nunydd

Lake Manitoba

Loch Leven

Lake Esrom

Myvatn

Lake Norman

Lake Werowrap

Lake Norman

Lac de Port-Bielh

Eglwys Nunydd

Loch Leven

Lake Esrom

Lac de Port-Bielh

Eglwys Nunydd

Lake Esrom

Eglwys Nunydd

Lake Memphremagog

Eglwys Nunydd

Bob Black Pond

Eglwys Nunydd

Group

I

M

I

M

I

I

I

I

M

C

I

I

I

I

I

I

I

I

I

I

M

I

]

]

]

i

]

\

C

C

logWm

-1.374

2.187

0.784

1.265

-1.337

0.397

-0.275

-0.102

2.765

0.602

-1.589

-2.868

-0.952

-1.26

0.146

0.477

1.778

0.172

-1

-1.929

0.021

0.212

0.118

0.301

-0.4

0.643

1.415

-0.473

-0.068

-0.934

2.27

1.748

logB

-2.454

-0.409

-1.301

-0.721

-1.745

0.578

-0.971

-1.347

-0.796

-0.248

-0.886

-1.347

-1.699

-0.569

0.815

0.244

0.671

-2.409

0.907

-2.081

-1.481

0.537

-1.166

0.959

-0.959

-0.260

-0.545

-0.208

-0.141

-0.770

-0.796

-0.699

T

18.0

13.0

13.0

13.0

18.0

5.0

16.8

16.8

13.0

16.0

13.0

18.0

12.8

13.0

9.0

9.0

5.0

18.0

13.2

18.0

4.0

12.8

9.0

9.5

4.0

12.8

9.5

12.8

12.8

12.8

16.5

12.8

logP

-1.979

0.896

0.252

0.612

-0.995

1.28

0.778

0.38

0.139

1.053

-0.318

0.684

-0.958

0.004

1.409

0.905

0.826

-0.696

1.822

-0.148

-1.301

1.296

-0.779

1.401

-0.769

0.545

-0,433

0.303

0.301

-0.04

0.019

-0.027

logP

-0.829

-0.035

-0.518

-0.137

-0.277

0.605

0.105

-0.218

-0.430

0.495

0.179

0.276

-0.571

0.377

1.039

0.538

0.462

-1.035

L508

-0.449

-1.009

1.007

-0.516

1.154

-0.532

0.313

-0.204

0.528

0.517

0.158

-0.170

-0.205

Resid.

-1.150

0.931

0.770

0.749

-0.718

0.675

0.673

0.598

0.569

0.558

-0.497

0.408

-0.387

-0.373

0.370

0.367

0.364

0.339

0.314

0.301

-0.292

0.289

-0.263

0.247

-0.237

0.232

-0.229

-0.225

-0.216

-0.198

0,189

0,178

Method

SF

SF

SF

SF

SF

AC-GS

SF

SF

SF

AC-GS

SF

SF

IG

SF

AC-GS

AC-GS

AC-GS

SF

AC-GS

SF

AC-GS

IG

AC-GS

AC-GS

AC-GA

IG

IG

IG

AC-GS

IG

SF

IG

Ref,

1

2

2

2

3

4

5

5

2

6

2

7

8

2

9

10

4

7

11

7

12

8

13

14

12

8

15

8

16

8

17

8

n

|

SJ

AND;>

*

fr-4

I

1™!

c?

ume(■«

Table 1. Continued.

Taxon Water-body Group log Wm logB logP logP Resid. Method Ref.

Erpobdella octoculata

Procladius crassinervis

Stempellina spp.

Psilotanypus rufovittatus

Brachicerus sp.

Cladotanytarsus spp.

Orconectes virilis

Chironomus ptumosus

Criptocopus ornatus

Psectrocladius sordidellus

Chironomus comtnutatus

Limnochironomus pulsus

Procladius choreus

Stictochirus rosenscholdi

Procladius choreus

Sialis lutaria

Limnochironomus pulsus

Parartemia zietziana

Orconectes virilis

Hexagenia limbata

Crangonyx gracilis

Orconectes virilis

Orconectes virilis

Tanytarsus holochlorus

Tanytarsus lugens

Penlaneura monilis

Procladius barbatus

Orconectes virilis

Hexagenia limbata

Hexagenia limbata

Glyptotendipes paripes

Clyptotendipes parites

Polypedilum nubeculosum

Microtendipes sp.

Lake Esrom

Loch Leven

Lake Norman

Loch Leven

Texas pond

Lake Norman

Dock Lake

Federsee

Waldsea

Lac de Port-Bielh

Lac de Port-Bielh

Loch Leven

Eglwys Nunydd

Malsj0en

Loch Leven

Lac de Port-Bielh

Eglwys Nunydd

Lake Cundare

North Twin Lake

Savanne Lake

Bob Black Pond

South Twin Lake

Shallow Lake

Eglwys Nunydd

Eglwys Nunydd

Loch Leven

Malsjoen

West Lost Lake <

Savanne Lake

Savanne Lake

Eglwys Nunydd

Loch Leven

Loch Leven

Eglwys Nunydd

[

I

[

[

[

C

[

[

I

I

c

c

t

c

c

c

c

1.079

-0.023

-4.593

-0.73

-0.236

-2.669

2.477

1.146

-0.198

-0.4

0

-0.698

-0.261

-0.198

-0.417

1.176

-0.458

0.602

2.477

1.255

0.418

2.477

2.477

-0.634

-0.75

-0.899

0,284

2.477

1.255

1.255

-0.473

0.556

-0.397

-0.107

-0.229

-0.312

-2.721

-1.604

-0.561

-1.959

0.458

0.931

-1.891

-0.538

-0.569

-0.836

-0.284

-0.924

-0.567

-0.420

-0.745

-1.019

0.788

-0.638

-0.495

0.972

0.471

-0.620

-0.495

-1.747

-0.646

0.970

-0.638

-0.638

-0.284

-0.185

-0.845

-0.553

9.5

9.0

18.0

9.0

16,8

18,0

12.8

11.0

10.3

4.0

4.0

9.0

12.8

7.0

9.0

4.0

12.8

17.0

13.7

11.5

16.5

13.7

12.8

12.8

12.8

9.0

7.0

13.7

11.5

11.5

12.8

9.0

9.0

12.8

0.254

0.024

-0.686

-0.58

0.278

-0.103

0.72

0.953

-1.092

-0.319

-0.402

-0.23

0.532

-0.468

-0.047

-0.29

0.041

0

0.847

-0.097

0.308

1.083

0.641

0.19

0.38

-0.787

-0,259

1.021

-0.168

-0.168

0.488

0.204

-0.193

0.209

0.097

0.178

-0.537

-0.728

0.421

-0.237

0.592

1.078

-0.970

-0.201

-0.287

-0.129

0.435

-0.381

0.039

-0.354

0.103

-0.060

0.898

-0.148

0.355

1.043

0.602

0.229

0.345

-0.815

-0.237

1.042

-0.148

-0.148

0.468

0.188

-0.183

0.199

0.157

-0.154

-0.149

0.148

-0.143

0.134

0.128

-0.125

-0.122

-0.118

-0.115

-0.101

0.097

-0.087

-0.086

0.064

-0.062

0.060

-0.051

0.051

-0.047

0.040

0.039

-0.039

0.035

0.028

-0.022

-0.021

-0.020

-0.020

0.020

0.016

-0.010

0.010

IG

AC-GS

SF

AC-GS

SF

SF

IG

AC-GS

SF

AC-GS

AC-GS

AC-GS

IG

AC-GS

AC-GS

AC-GS

IG

AC-GS

IG

SF

SF

IG

IG

IG

IG

AC-GS

AC-GS

IG

AC-GS

IG

IG

AC-GS

AC-GS

IG

15

9

7

9

5

7

16

18

19

12

20

9

8

21

13

22

8

6

23

24

17

23

16

8

8

9

21

23

24

24

8

13

9

8

£

c

c

2

P

*<

|

References:!. Bowen(1983),2. Tudorancea et al. (1979), 3. Eaton (1983), 4. Lindegaard and Jonasson (1979), 5. Benson etal. (1980), 6. Marchant and Williams

(1977), 7. Wilda (1983), 8. Potter and Learner (1974), 9. Charles et al. (1974), 10. Dall et al (1984), 11. Walker (1973), 12. Laville (1972), 13. Charles et al.

(1976), 14. Jonasson (1975), 15. Dall (1980), 16. Dermott et al. (1977), 17. Martien and Benke (1977), 18. Frank (1982), 19. Swanson and Hammer (1983),

20. Lavilie (1975), 21. Aagaard (1978), 22. Giani and Laville (1973), 23. Momot and Gowing (1977), 24. Riklik and Momot (1982).

14 C. Plante and J. A. Downing [Volume 9

ION 1—- o Q1- O (Z Q.- Q UJ0. m O-1- o o -2-

AAAAASF OODDDAC—GSD ■■■■■IGn 1:1nA&V

A4»* 9^DA /-A /A-2

LOG "predicted PRODUCTION2Fig. 1. Relationship between observed secondary

production of aquatic invertebrates and the produc

tion predicted using Equation 1. The solid line

indicates a 1:1 relationship. SF indicates that the es

timate was made using the size-frequency method,

AC-GS indicates the Allen-curve or increment sum

mation method and IG indicates the instantaneous

growth method.

Errors in annual production estimates by the

size-frequency method will be proportional to

differences between real and assumed time spent

by a cohort to complete its growth. For example,

some of the large residuals in Table 1 were found

for larval chironomid populations in Lake Nor

man by Wilda (1983). The CPI for these popu

lations was inferred from the laboratory-de

rived development equation of Mackey (1977).

Table 2. Kruskal-Wallis test (Conover 1971) for

bias in various production estimation methods. The

analysis was performed on the residuals from Equa

tion 1 using estimation methods as treatment groups,

p is the approximate Chi-square probability.

Method

Size-frequency

Allen curve and increment summation

Instantaneous growth

Kruskal-Wallis statistic = 1.44

Size-frequency

Allen curve, increment summation

and instantaneous growth

Kruskal-Wallis statistic = 1.41

Num

ber of

cases

20

26

20

Mean

rank

37.7

32.1

31.0

p = 0.49

20

46

37.7

31.7

p = 0.24

>-o

aLd

-1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8

RESIDUALS

Fig. 2. Frequency histogram of the residuals (log

observed — log predicted) from Equation 1 grouped

by production estimation technique. Abbreviations

are as in Figure 1.

Table 3. Kruskal-Wallis test (Conover 1971) for

differences in precision of various production esti

mation methods. The analysis was performed on the

absolute value of the residuals of Equation 1 using

estimation methods as treatment groups, p is the ap

proximate Chi-square probability.

Method

Num

ber of Mean

cases rank

Size-frequency 20 46.9

Allen curve and increment summation 26 30.7

Instantaneous growth 20 23.6

Kruskal-Wallis statistic = 15.6 p = 0.0002

Size-frequency 20 46.9

Allen curve, increment summation

and instantaneous growth 46 27.7

Kruskal-Wallis statistic = 14.1 p = 0.0002

1990] Calculating secondary production 15

Wilda (1983) assumed that these chironomids

were producing the equivalent of 18.5 or 22

consecutive cohorts per year, a figure that was

probably too large (T. J. Wilda, Duke Power

Company, personal communication). If one as

sumes that the cohort P/B is 5 (Waters 1977),

then Equation 1 can be used to approximate this

number of consecutive cohorts produced in one

year (Plante and Downing 1989). The actual

number of cohorts formed annually probably

ranged from about 6 for Chironomus sp. to 36 for

Stempellina spp. if cohorts were consecutive, re

sulting in errors in annual secondary produc

tion from +330% to —55%. Other sources of

error certainly exist, but our analyses underline

the difficulty of estimating secondary produc

tion without good life-history data, realistic

measures of growth rate, or accurate measure

ments of larval development time.

In conclusion, the biases suggested by sim

ulation studies do not appear to be a major prob

lem in actual data. Use of the size-frequency

method, or characteristics of populations that

are studied using the size-frequency method,

result in estimates that can be much farther from

probable production values than the estimates

found using Allen curve, growth increment

summation and instantaneous growth tech

niques. Much of this imprecision may arise from

incorrect CPI correction, but could also stem

either from a lack of synchrony in developing

cohorts (Morin et al. 1987), the need for cor

rection factors such as Pc/Pa where growth is

non-linear with time (e.g., Menzie 1980), or the

insufficiency of the method's assumptions about

several other factors (Hamilton 1969, Rigler and

Downing 1984). The size-frequency technique

is most accurately applied when sufficient data

exist so that proper CPI corrections can be ap

plied. If only order-of-magnitude production

estimates are required, these can be made using

Equation 1 with only measurements of annual

mean biomass, body-mass, and temperature. If

more exacting measures are required, we agree

with Morin et al. (1987) that, whenever possi

ble, sufficient data should be collected to apply

the Allen curve, increment summation or in

stantaneous growth methods.

Acknowledgements

Financial support for this research was pro

vided by an operating grant to J. A. Downing

from the Natural Sciences and Engineering Re

search Council of Canada, and a team grant from

the Ministry of Education of the Province of

Quebec (FCAR). We thank the members of the

Groupe d'Ecologie des Eaux douces, A. Morin,

A. C. Benke, and two anonymous referees for

their comments and criticisms.

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Received: 25 August 1989

Accepted: 28 November 1989