787
NOTES
We have used this method only to
pare samples for examination with an
tron microscope; however, it should
be useful in preparing samples for
microscopy.
JOHN E. HARRISI
preelecalso
light
THonkks R. MCKEE
RAmtom
C. WILSON,
JR.?
School of Geosciences,
Texas AbM University,
College Station 77843.
U.
GRANT
WHITEHOUSE
College of Science,
Texas AFL-211University,
College Station 77843.
’ Present
address:
National
Oceanic and Atmospheric
Administration,
Environmental
Data
Service,
National
Oceanographic
Data Center,
Rockville,
Maryland
20852.
’ Present address: Department
of Geology, University of Utah, Salt Lake City.
ON INSTANTANEOUS
AND FINITE
ABSTRACT
A formula relating instantaneous and finite
birth rates, originally
derived by Leslie, is
presented and compared with the formula of
Edmondson.
It is shown that Edmondson’s
formula
is a special case of Leslie’s, valid
when there is no mortality.
The bias involved in the use of Edmondson’s formula in
cases where mortality
is nonzero is examined.
Since Leslie’s fomiula
requires no new assumptions or data, it should be used in place
of Edmondson’s
formula.
The purpose of this note is to correct
an often cited formula, first proposed by
Edmondson (1960), relating instantaneous
and finite per capita birth rates.
Repeated censusing of a population can
often be thought of as discrete sampling
of a continuous population growth process.
(Actually
the population growth process
1 This research
ence Foundation
fellowship.
was supported
Grant GI-20
REFERENCES
BIHKS, L. S., J. S. GROSSO, R. J. LABRIE, J. W.
SASULIN, AND D. J. NAGEL. 1966. Characterization
of particulate
matter in the ocean.
U.S. Nav. Res. Lab. Rep. 6398. 20 p.
1956.
Electron
BORASKY, R., AND B. MASTEL.
microscopy of magnesium oxide particles collected on membrane filters.
U.S. At. Energy
Comm. Rep. HW-46722.
13 p.
1966.
The use of membrane
ECKHOFF, R. K.
filters in the preparation
of samples for electron microscopic examination
of the sizes and
Microsc. Crysshapes of sub-sieve particles.
tal Front 15: 136-142.
1954.
Preparation
of aerosols
KALWS,
E. H.
J. Appl. Phys. 25:
for electron microscopy.
87-89.
AICISTYRE, A., AND W. H. B&
1967.
hlodern
coccolithophoridae
of the Atlantic Ocean-l.
Deep-Sea Res. 14: 561-597.
1968.
WALKER, G. H., AND B. L. W1LLIAbiS.
Improved
method of removing plastic backing films from carbon electron microscope
replicas.
Rev. Sci. Instr. 39: 1968-1970.
WILSOS, R. C.
1970,
The mechanical
properties of the shear zone of the Lenris Overthrust,
Glacier
National
Park,
hlontana.
Ph.D. thesis, Texas A&11 Univ. 89 p.
by National
Sciand a graduate
BIRTH
RATE?
itself is discrete, but usually on a scale
enough finer than the sampling process
that it can be considered continuous.)
By
techniques such as the egg-ratio method
( Edmondson 1960, 1968) such sampling
can provide estimates of the number of
offspring produced during a sampling interval. The number of deaths in a sampling interval is difficult or impossible to
measure in most cases. If this datum were
available, the conclusions to follow could
be applied to finite and instantaneous
death rates, as well as birth rates.
In expressing the generation
of these
offspring as a rate, the investigator chooses
OIX of two models of population
growth
One apduring the sampling interval.
proach is to consider the population to
change in discrete steps, with all the births
occurring
at 011e instant within the sampling interval. These births are converted
to a per capita basis by comparing them
with the population
size at the instant
788
NOTES
when the population change is supposed
to occur. This is often the population at
the beginning of the census period, but
sometimes an average of the population
sizes at the ends of the sampling period is
used. This calculation produces the finite
per capita birth rate, p.
Alternatively,
the investigator may wish
to consider births to occur continuously
over the entire sampling period. To express this as a per capita rate, the births
at each instant must be compared to the
population size at that instant. Since the
population is changing continuously over
the interval, this instantaneous per capita
birth rate, b, will in general differ from
the corresponding finite rate.
Leslie (1948) d erived a relation between
the two rates, as follows. Let N( t ) and
N( t + 1) be the population sizes at times
t and t + 1 and let B(t) and D(t) be the
number of births and deaths occurring in
the interval [t, t + 11. The finite birth and
death rates are defined by p = B (t )/iV( t )
and 6 = D(t)/N(t).
The finite growth
rate x = er is defined so that N( t + 1) =
AN(t). The instantaneous birth and death
rates are b and d. Then
Ayt + 1) = iv(t) + B(t) -D(t),
and dividing
gives
(1)
both sides of ( 1) by N ( t )
h=l+P-&
(2)
Assuming that the population grows continuously at the rate r over the interval
[t, t + 11, the total number of births is
/ 2,
B(t) = Jtt+lbN(T) d;7.
\“I
But N(T) = N(t)er(T-t),
so replacing
by another dummy variable, s, we
write
B(t) = Jr bN(t)ers ds
= bN(t) i:ers ds.
T- t
can
Dividing both sides of (5) by N(t)
performing the integration gives
and
p = LW9 W-
(6)
U,
(4)
(5)
or
b = rp/(er - 1).
(7)
When r = 0, it follows from ( 5) that b
= P*
Equation (7) is the correct formula relating instantaneous and finite rates under
the assumptions :
i) During the interval [t, t + l] the
population is growing continuously
at the constant rate r.
ii) During the interval [t, t + I] there
is a constant instantaneous
per
capita birth rate, b.
Together these assumptions imply that the
instantaneous death rate d is also constant
on [t, t + l] since r = b - d. These assumptions are independent of those required by
the method used to estimate p. Equation
(7) was also presented by Birch (1948)
and Andrewartha and Birch ( 1954).
An analysis similar to the one above
yields the instantaneous death rate
d= r6/(er-1).
(8)
As a check on the consistency of (7) and
( S), note that
b-cl=[r/(er-l)](p--8).
(9)
Since, from (2)
p-6=er-1,
(10)
e-qua-ton (9) simplifies to b - d = r, as it
should.
Edmondson ( 1960) presented the equation
b = In (1-t ,@)
(11)
as relating the instantaneous and finite
birth rates under conditions of no mortality. Under these conditions (11) is correct,
as can be seen by substituting b - d for r
in (7), setting cl = 0, and solving for b.
Under analogous conditions of no births,
simplification
of ( 8) shows that
d=-In
(l-6).
Using (11) and ( 12) yields
b-d=ln(l+p)
+ln(l-a),
(12)
(13)
which equals r only when p, F, and d are
all zero.
789
NOTES
t
**+..*CORRECT
----INCORRECT
t
02
-I
5
r
5
15
FIG. 1. The bias caused by the use of equation
( 11) instead of (7 ) to calculate b is plotted on a
log scale as a function of the observable population variables
T and p. The bias is given by &/&,
where 6, is estimated by ( 11) and k by ( 7 ) . For
small values of T and p the bias is given approxiof r and p above
mately by e(r-p)‘P. Combinations
the dotted line result in negative estimates of the
death rate d. To show the form of the bias, values of T and p have been plotted which exceed
those commonly found in natural populations.
Time
birth rate b for
FIG. 2. The instantaneous
Daphnia
schoedleri
as calculated
correctly
(by
equation
7 ) and incorrectly
(by equation
11) .
Data from Wright
( 1965).
Unfortunately,
equation ( 11) has often
been used to estimate b, and then d = b-r
in cases where d is not zero. The intuineous death rate assumes just this kind of
tively reasonable idea behind this was to linearity.
measure the death rate by the difference
Among the works which have used equabetween the potential growth rate in the tion ( 11) in this way are papers by Edabsence of any mortality, In ( 1 + p), and mondson ( 1960, 1965, 1968)) Edmondson
r, the observed growth rate (Edmondson
et al. ( 1962)) Hall ( 1964)) Tappa ( 1965 ) ,
1968, p. 2). The derivation of ( 11) from
Wright
( 1965 ) , and Hillbricht-Ilkowska
(7) should make it clear that this proceand Pourriot ( 1970) ; a recent review artidure will not give the correct value of b, cle by Mann ( 1969) ; and books by Hutchand hence of d. To see just where intuiinson ( 1967), Winberg ( 1971)) and Edtion is wrong in this case, recall that A, p, mondson and Winberg ( 1971).
and 6 must satisfy equation ( 1). For these
In Fig. 1 the bias involved in estimating
finite rates, the difference between potenb by ( 11) instead of ( 7) is plotted as a
tial growth in the absence of mortality, 1 +
function of the two observable population
p, and the observed growth rate, A, does
variables, p and r. Not all values of p and
equal the death rate, 8. To deal with inr are jointly possible; any above the dotted
stantaneous values as described above reline in Fig. 1 result in a value of b that is
quires one to take logarithms of both sides
<r, which implies a negative death rate.
of ( 1). But because the log function is
(Note that when p < er - 1,
not linear, r#ln(l+p)-ln(6).
The use
b=[rp/(er-l)]<r.
of In (1 + p) - r to estimate the instanta-
790
NOTES
Since cl = b - r, this implies that ~7< 0.)
Although
random variation
involved
in
sampling, or phenomena not included in
the model, such as immigration, will occasionally result in such values (it happens
three times in the example presented below), such occurrences should be rare
and the bias will almost always be <l;
i.e. equation ( 11) underestimates the true
birth rate and hence underestimates the
death rate.
To show the form of the bias, the scales
for both r and p in Fig. 1 have been extended to values beyond those likely to be
encountered in natural situations. To evaluate the significance of the bias in a real
situation the data on Daphnia schoederi
presented by Wright (lk5)
were reanalyzed. Figure 2 shows the two estimates
of the instantaneous birth rate, Over much
of the year the estimates are nearly identical, but equation ( 11) underestimates 72
considerably in the middle of the year. On
three dates (10 May, 6 and 20 August) the
bias was >l, and on all three dates b (by
either estimate) is <r. Eliminating
these
three dates leaves an average bias of 0.895,
i.e. the use of ( 11) underestimated 71 by
an average of 10.5%. The maximum underestimates were 25% (bias = 0.75) on 9 July
and 32% (bias = 0.68) on 27 August. Since
d = b - r, Wright’s values of cl are correspondingly underestimated.
Fortunately, Wright’s biological conclusions do not seem to be damaged in the
least by the correction. He hypothesized
that the midsummer peak in the death
rate is due to predation by Leptodora;
the correction increases this peak, without
changing the form of the graph significantly. Although the analysis has not been
carried out it appears that the same result
will hold for Hall’s (1964) work OII Daphnia galeata menclotae where a midsummer
peak in mortality was also attributed to
predation.
Presumably the same will be
true of conclusions based on birth rate
patterns in other works that have used
( 11) to estimate b, in spite of the errors
in the birth rate values.
From the practical point of view it is
important to compare the assumptions and
data requirements of the two formulae.
Both (7) and (11) involve the same assumptions; i.e. the population growth rate
r and the instantaneous birth rate 12 are in
fact constant over the interval [t, t + 11.
The finite per capita birth rate p is required by both formulae; it is customarily
estimated by the egg-ratio method or one
of its modifications
(Edmondson
1960,
1968). In addition the population growth
rate, r, is required by (7). This, however,
is easily calculated (under the assumption
that it is in fact constant over the interval
[t,t+ll)
by
r=ln[N(t+l)]-hi[N(t)].
(14)
This piece of data is usually available in
studies using ( 11) anyway, since it is used
in the calculation of n.
I would like to acknowledge Dr. D. Hall
for helpful discussions of the problem.
Dr. W. T. Edmondson provided several
valuable suggestions, for which I am more
than grateful.
HAL
CASWELL
Department of Zoology,
Michigan State Unizjersity,
East Lansing 48823.
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J. Anim.
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Ecol. 17: 15-26.
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EDXIONDSOK, W. T. 1960. Reproductive
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Ecol. Monogr. 35: 61-111.
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Ecology 43: 625-633.
AXD G. G. WINBERG. [EDs.].
1971.
SeLondary production
in fresh waters.
IBP
358 p,
Handbook
17. Blackwell.
An experimental
approach
HALL, D. J. 1964.
NOTES
to the dynamics of a natural population
of
Daphnia
galeata mendotae.
Ecology
45 :
94-112.
HILLBRICHT-ILKOWSKA,
A., ASD R. POURRIOT.
1970. Production of experimental
populations
of Brachionus calyciflomcs Pallas (Rotatoria)
exposed to artificial
predation
of different
rates. Pol. Arch. Hydrobiol.
17: 241-248.
HUTCHINSON, G. E. 1967. A treatise on limnology, v. 2. Wiley.
1115 p.
LESLIE, P. H. 1948. Some further notes on the
use of matrices in population
mathematics.
Biometrika
35: 213-245.
791
1969.
The dynamics of aquatic
MANN, K. H.
ecosystems.
Advan. Ecol. Res. 6: 1-81.
TAPPA, D. W. 1965. The dynamics of the association of six limnetic species of Daphnia in
Aziscoos Lake, Maine.
Ecol. Monogr.
34:
395423.
WINBERG, G. G. 1971. Methods for the estimation of production
of aquatic animals.
Academic. 175 p.
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dynamics
and production
of Daphnia in Canyon Ferry
Reservoir, Montana.
Limnol. Oceanogr. 10 :
583-590.