Notes
References
H. B. 1961. The invertebrate
fauna of a
Welsh mountain stream. Arch. Hydrobiol.
57:
344-388.
AND M. J. COLEMAN.
1968. A simple meth-3
od of assessing the annual production of stream
benthos. Limnol. Oceanogr. 13: 569-573.
-,
AND F. HARPER. 1972. The life history of
Gammarus Zacustris and G. pseudolimnaeus
in
southern Ontario. Crustaceana Suppl. 3, p. 329341.
MACKEY,
A. P. 1977. Growth and development
of
larval Chironomidae.
Oikos 28: 270-275.
REISEN,
W. K. 1975. Quantitative
aspects of Simulium uirgatum Coq. and S. Species life history in a southern Oklahoma stream. Ann. Entomol. Sot. Am. 68: 949-954.
WATERS,
T. F. 1977. Secondary production
in inland waters. Adv. Ecol. Res. 10: 91-164.
ZWICK, P. 1975. Critical notes on a proposed method to estimate production.
Freshwater Biol. 5:
65-70.
HYNES,
BECKER, C. D. 1973. Development
of SimuZium
(Psilozia) uittatum
Zett. (Diptera:
Simuliidae)
from larvae to adults at thermal increments
from 17.0 to 27.0 C. Am. Midl. Nat. 89: 246251.
BENKE, A. C., AND J. B. WAIDE. 1977. In defense
of average cohorts. Freshwater Biol. 7: 61-63.
CUSHMAN, R. M., J. W. ELWOOD, AND S. G. HILDEBRAND. 1975. Production dynamics of AZloper-la mediana Banks (Plecoptera: Chloroperlidae)
and Diplectrona
modesta
Banks
(Trichoptera:
Hydropsychidae)
in Walker
Branch, Tennessee. Oak Ridge Natl. Lab. Environ. Sci. Div. Publ. 785. 66 p.
FAGER, E. W. 1969. Production of stream benthos:
A critique of the method of assessment proposed by Hynes and Coleman (1968). Limnol.
Oceanogr. 14: 766-770.
GILLESPIE, D. M., AND A. C. BENKE. 1979. Methods of calculating
cohort production from field
data-some
relationships.
Limnol.
Oceanogr.
24: 171-176.
HAMILTON, A. L. 1969. On estimating annual production. Limnol. Oceanogr. 14: 771-782.
Limnol.
Oceanogr.,
24(l),
@ 1979, by the American
171
1979, 171-176
Society of Limnology
and Oceanography,
Submitted: 2 May 1978
Accepted: 14 June 1978
Inc.
Methods of calculating cohort production
field data-some
relationships
Abstract-Four
commonly used methods of
directly computing cohort production
are increment-summation,
removal-summation,
instantaneous
growth,
and the Allen curve.
These may be used in either instantaneous
exponential
form or discrete linear form. The
instantaneous
exponential
forms of the four
methods are shown to be exactly equivalent.
In discrete linear forms, the increment-summation and removal-summation
methods are
exactly equivalent,
while the instantaneous
growth and Allen curve methods are approximations of the other two. Simplified
computational equations
are given for calculating
production by the various methods.
Several methods have been used to estimate production
of aquatic invertebrates from field data, and most are limited to populations
in which cohorts can
be distinguished.
The population in each
case is sampled at specified
intervals
throughout
its life history, during which
time no further recruitment
occurs, but
from
population
mortality
and individual
growth occur simultaneously.
Changes in
number,
standing
stock biomass, and
mean individual
weight between sampling intervals can then be used in different ways to estimate production
for
each interval and ultimately
to estimate
cohort production.
The methods most
commonly applied to such data are the
increment-summation,
removal-summation, instantaneous
growth, and Allen
curve (Waters 1977). For the incrementsummation
method,
the mean of the
numbers of organisms at two successive
dates is multiplied
by the change in
mean individual
weight to obtain production for the interval, and the sum of
all intervals is cohort production.
For the
removal-summation
method, production
lost in each interval is calculated as the
product of change in numbers between
sampling dates and the average of the
I72
Notes
mean individual
weights at the two dates.
The sum of the losses for all intervals
equals total cohort production.
Production lost plus change in biomass during
an interval is the total production during
that interval.
For the instantaneous
growth
method,
production
between
dates is calculated as the product of mean
of the biomasses for both dates and the
instantaneous rate of increase of mean individual
weight of the organisms. Total
cohort production
is the sum of the separate estimates. The Allen curve is a plot
of number
against
mean individual
weight.
A curve can be fitted to the
points, and the area under the curve, obtained planimetrically
or otherwise, will
estimate total cohort production.
Our main purpose here is to describe
the mathematical
and graphical relationships among these methods for both continuous exponential
and discrete linear
models of population growth and mortality. Although
several
workers
have
shown mathematical relationships
among
some of these methods (Chapman 1971;
Winberg 1971; Winberg et al. 1971; Crisp
1971; Tanaka 1976, 1977), certain relationships have not been described. Others have applied different methods to the
same field data, usually with good agreement among methods (e.g. Waters and
Crawford 1973; Benke 1976), and Cushman et al. (1978) have done the same
thing with computer-simulated
populations. Recent papers by LeBlond and Parsons (1977) and by Ricker (1978) have
dealt with the continuous
exponential
case, which
was first formulated
by
Clarke et al. (1946).
We thank those students
and colleagues who contributed
to the ideas discussed here, including
J. B. Favor, R. L.
Henry, D. L. Stites, K. A. Turgeon, W. T.
Momot, and J. B. Waide. A review by W.
E. Ricker helped to clarify certain sections.
Continuous
exponential
production
model-Cohort
production
may be treated as a continuous exponential
function,
either between successive samples or as
a curve fitted to a series of samples. It is
usually assumed that change in popula-
tion number and individual
growth are
both exponential.
Since (by definition)
there is no recruitment,
change in number is negative, and exponential mortality
can be expressed as
dNldt
= -mN
(la>
where N is population number, t is time,
and m is the per capita mortality
rate,
which upon integration yields
N, = N,epmt
0)
where No and Nt are initial
and final
numbers for the interval between sampling dates. Exponential
growth can similarly be expressed as
dW/dt = gW
where W is mean individual
weight,
a growth rate per unit of weight, and
(24
g is
Wt = Woegt
W
where W,, and Wt are initial and final
weights for the interval.
Instantaneous
growth and mortality rates can thus easily
be obtained between sample dates given
N and W at each date.
Rate of change in standing stock biomass, which is the product of number and
mean individual
weight, can be obtained
from la and 2a:
dB
-c-z
dt
d(NW)
dt
WdN+N!!!!!
dt
dt
= -mNW+gNW
= NW(g - m) = B(g - m) (3a)
where B is standing stock biomass and
(g - m) is the per unit rate of biomass
change (equivalent
to g of LeBlond and
Parsons 1977). The definite integral form
is
Bt = B,,e(g-m)t
W
where B, and B, are initial and final population biomasses.
The total rate of production, dP/dt, may
be considered as the rate of addition to
biomass plus the rate of loss of biomass
to predators or other causes of mortality,
the latter being expressed as mortality
rate times mean biomass of animals lost
during each short time interval:
I73
Notes
dP -dB -wdNdt
dt
dt
(4)
P=AB+
$--mB,[e’V-m)t
- 11.
(9)
This reduces to
(the negative sign appearing because dNl
dt is negative). The last term of Eq. 4 is
the instantaneous
equivalent
of the “removal” in the removal-summation
method. Substituting
Eq. la and 3a into 4:
dP/dt = NW(g - m) + mNW = NWg
and substituting
dP/dt = N(dWldt).
(5)
equivalent
of
in the incre-
This is the instantaneous
the production
increment
ment-summation
method.
The production
during some time interval, t, can be found by integrating
either 4 or 5 over the interval in question.
Taking Eq. 5 first, the easiest approach is
to substitute
Eq. lb and 2b into the
expression and integrate. This yields
- 11,
(6)
which, since e tg-m)t = B,IB, (Eq. 3b), further reduces to
W
+ 0, g + m>,
(10)
and, from
Eq.
1’Hospital’s Rule,
9,
again
invoking
(AB = 0, g = m).
P = mB,t
from Eq. 2a,
P= &B”[“(g-m)t
P = AB + ABA
g-m
(11)
Equations 10 and 11 are equivalent to the
equations derived by LeBlond and Parsons (1977) except for their different definition of g. The second part of Eq. 10 is
equivalent to removal and can be used in
the removal-summation
method for summing production
losses over a series of
intervals.
Note that Eq. 8 and 11 are
equivalent,
since by definition
g = m,
and that Eq. 7 and 10 can also be equated. In other words, the increment-summation and removal-summation
are exactly equivalent
for the exponential
model, as indicated
by Ricker’s (1978)
Eq. 1 and 2.
The instantaneous growth rate method
of calculating production,
P=gAB
W + 0, g f m> (7)
g-m
P = gtB*
(12)
where AB = Bt - B,. The restriction
is
where B* is mean standing stock biomass
necessary because Eq. 7 is undefined
if
during an interval, is equivalent to Eq. 6
AB = 0, or g = m, which, from Eq. 3b,
if biomass change is exponential,
since
are equivalent.
However, Eq. 6 may be
treated as a fraction with both numerator,
Be = BO[ecgernjt- 11
gB,[eQpmM - 11, and denominator,
g - m,
k - m>t
functions of (g - m) that approach zero as
(g - m) approaches
zero. Then,
by (see Clarke et al. 1946; Chapman 1971).
I’Hospital’s
Rule (explained in most cal- This method is thus exactly equivalent to
the previous two.
culus texts),
Neess and Dugdale (1959) presented
Limit (Eq. 6) = gB,t,
an algebraic
form of the Allen curve
k - 4-0
which is presented here as modified by
and, for practical purposes
Gillespie
(1969), and using our present
notation, as
P = gB,t
(AB = 0, g = m).
(8)
Equations
7 and 8 are computational
forms that can be used in the incrementsummation method, assuming exponential growth and mortality, where g and m
are calculated from Eq. lb and 2b.
The same thing is done with Eq. 4. After substituting
and integrating
we have
P = N,W,,’
( >
1-F
[
w,
-
w()
(
1-f
)
1
.
(13)
This can be shown to be equal to Eq. 6,
174
Notes
or less equal, a functional
regression
gives the best result (see Ricker 1973).
Discrete
linear production
modelFigure 1 helps to illustrate relationships
among production
methods for a linear
production model. Note that BO = N,,W, =
X + V, and B, = NtWt = V + 2. Also, X is
the amount of original biomass lost during t, Y is the amount of new production
lost during t, and 2 is the amount of new
production added as biomass to form B,.
If we define AN = N, - Nt, then
Ak
E,v
7
x = whw,
NJ
Y = %(AN)(Wt - W,),
(15)
(16)
and
2 = Nt(Wt - W,).
Mean
Weight
Fig. 1. Hypothetical
Allen curve fitted to a series of data points with a geometric representation
of production equations. Further explanation
given
in text.
thus demonstrating
the exact equivalence of this form of the Allen curve to
the other three methods. The major difference is that only the ratio m/g is required, rather than m and g independently. Clearly,
if m and g are both
known, use of Eq. 13 is unnecessary. Furthermore, an average m/g can be calculated over the entire life of the cohort by
a least-squares fit of data to the equation
In Nt = In N, - :(ln
W, - In W,),
(14)
which plots as a straight line with slope
-(m/g). This regression may be extrapolated to find N,, or, by rearranging, WO, if
either is unknown.
W. E. Ricker (pers. comm.) has pointed
out to us that a major difficulty
arises in
estimating
m/g by regression, since N
and W are both subject to error. If W is
subject to much less error than N, Eq. 14
estimates m/g fairly well. If N is subject
to much less error than W, the regression
of In Wt on (In N,, - In NJ provides a better estimate of g/m. If the errors are more
(17)
The removal during t (“elimination”
of
Winberg 1971; “yield”
of some investigators) is given by
E=X+Y,
08)
which, from 15 and 16, can be shown to
be equivalent to
E = W*AN
(1%
where W* = (W, + W,J/Z. This (or its continuous analog) is used in the removalsummation
method of summing losses
over a series of intervals to determine total cohort production.
The production
increment
during a
sampling interval is given by
P=Y+Z,
(20)
which, by 16 and 17, and defining AW =
Wt - W,, and N* = (Nt + N,J/2, is equivalent to
P = N*AW,
(21)
which is in effect the discrete analog of
Eq. 5 and can be used in the incrementsummation method. It is apparent from
Fig. 1 that:
B,=B,-X+Z
Bt-B,,=Z-X
AB=Z-X.
(22)
Notes
Therefore,
subtracting
Eq. 22 from 20,
P-AB=Y+Z-(Z-X)
=Y+X
P=AB+E
= AB + W*AN
(23)
(by Eq. 18 and 19). Equation 23 is the
discrete analog of Eq. 4 or 10 which show
that production
is equal to change in
standing stock biomass plus the production lost to mortality.
If cohort production
is calculated by
summing production
losses from Eq. 19
(removal-summation)
or by summing production increments
from either Eq. 21
(increment-summation)
or 23, the results
will be exactly equal. Furthermore,
they
are equal to an Allen curve estimate
made by connecting
data points with
straight lines and taking the area under
the curve.
For the instantaneous
growth method,
in which
P = gtB*, a linear approximation is often used, where B* = (B, +
B J/2 (detailed
discussion
in Chapman
1971). For small sampling intervals,
B*
z N*W* and AW/W* = gt, so that gtB*
= N*W*(AW/W*)
and gtB* = N*AW.
Thus, the instantaneous
growth method
and increment-summation
method are
approximately
equal.
Conclusions-If
the assumption of exponential growth and mortality is approximately true, the use of a continuous exponential
model yields
a somewhat
better estimate of production.
In effect,
the integrated exponential equations given here give the area under a smoothed
Allen curve between points, while the
discrete equations assume a straight line
between points (in Fig. 1, the crescentshaped area of Y represents the difference). If the sampling intervals are small
relative to growth and mortality, the discrepancy is probably insignificant
when
applied to field data. If growth and mortality are to be calculated, it is just as easy
to use the continuous model, but it saves
computation
time to use the discrete
equations and calculate directly from tables of field data if production alone is to
175
be determined. If intervals between samples are relatively
large, the continuous
equations
are significantly
better, although their use involves the tacit assumptions that growth and mortality are
constant or that the relationship
between
them remains constant over the interval
[i.e. a constant ratio m/g, which also implies that g/(g - m) and ml(g - m) are
constant].
In general, the increment-summation
method,
which
involves
determining
production
over a series of intervals by
Eq. 7 (and, rarely, 8) or by its discrete
analog, Eq. 21, and summing over the life
of the cohort, appears to be the simplest
approach. The removal-summation
method, summing losses in a similar way, may
be advantageous for some purposes and
produces the same result. LeBlond and
Parsons (1977) suggested that the continuous exponential
form of the removalsummation method is easier to use than
other methods. As shown by our analyses
and by Ricker (1978), there is little difference in the effort required to use continuous exponential
forms of the above
two methods. If a continuous
model is
used, the computational
form of the instantaneous growth method is more cumbersome than the first two, but for the
discrete model it is almost as simple as
the increment-summation
method. Based
upon our analysis, the Allen curve may
seem superfluous
for most situations,
since arithmetic
methods are simpler.
However,
in certain situations, particularly when few samples per cohort are
available or sampling error is large, the
Neess-Dugdale
equation or a smoothed
curve hand-fitted
to data points may be
appropriate.
The methods discussed above apply to
populations
in which cohort development is highly synchronous. We have not
considered
a population
of mixed size
classes, but in such a situation the continuous model may still be applied if
m/g is approximately
constant for all size
classes. This is true since total AR is simply the sum of AR for each of the constituent size classes, and, from Eq. 7
Notes
176
~ g
g-m
AB(tota1) = ~ g
g-m
+ ~ g
g-m
AB(l)
AB(2)
+ *** + ~ g
g-m
AB(n)
for size classes 1, 2, . . . , n. The major
problem in this case is that independent
estimates of m and g are generally required.
D. M. Gillespie
A. C. Benke
School of Biology
Georgia Institute of Technology
Atlanta 30332
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instantaneous-growth,
removal-summation,
and
Hynes/Hamilton
estimates of aquatic insect
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LEBLOND, P. H., AND T. R. PARSONS. 1977. A simplified expression
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NEESS,J., AND R. C. DUGDALE. 1959. Computation
of production
for populations
of aquatic midge
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Publ.
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production and biomass in secondary production
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or production to mean biomass, and
errors for estimate due to calculation
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Submitted:
Accepted:
27June 1977
30 June 1978