NOTES
Calculation
of instantaneous
birth rate1
Abstract-A
formula
giving the instantaneous birth rate is derived for egg bearing
species or species which bear their young. It
is shown that the instantaneous birth rate can
be calculated when the egg ratio or pregnancy
rate and the development
time are known.
The new method is compared with earlier ones
by Edmondson and Caswell: the latter applies
incorrectly
an earlier formula by Leslie while
Edmondson’s
formula
gives biased results,
the bias depending
on the length of the
development
time.
The above formula is exact under the
specific assumptions also made in this
paper, but it fails to account for the fact
that the egg ratio per development time is
not an appropriate measure for the finite
per capita birth rate nor is it an estimate
of it except under special circumstances.
Solving for the finite per capita birth rate,
p, from equation 2 we get
p= b(er-1)/r,
Eggs present on female specimens of a
planktonic species may be used to estimate
an instantaneous birth rate or, as it has also
been termed, the instantaneous rate of reproduction for the species. A method was
described by Edmondson (1960) and has
been used by a number of workers subsequently. Later Edmondson (1968: equations 11 and 13) introduced a correction to
account for the duration of the development
of eggs. The correction, however, has been
omitted in subsequent publications
(e.g.
Edmondson 1971) and also ignored by
others (e.g. Caswell 1972).
Edmondson ( 1960, 1971) calculated a
finite per capita birth rate, p, from egg
ratio by dividing the latter by development
time. This was then used to calculate the
instantaneous rate, b, through a simple
formula
b=ln(l+p).
(1)
Caswell (1972) pointed out that the above
formula relating the finite rate to the instantaneous rate is not correct. The relationship between the finite and instantaneous rate depends in fact on the overall
instantaneous rate of increase, r. He also
noted that a correct relationship between
the two had been given by Leslie ( 1948)
as
b=rP/(er-1).
(2)
‘The publication
of this paper was facilitated
by financial
support from the National
Research
Council of Canada.
LIMNOLOGY
AND
OCEANOGRAPHY
(3)
while later on we will show that
egg ratio/D
= ( ebD- 1)/D,
(4)
where D is development time. Only when
bD and r are so small that we may write
er - 1 - r and ebD - 1 - bD do the finite
per capita birth rate, p, and the “egg ratio/
D” agree. In such a case, both ,B and the
“egg ratio/D”
are approximately equal to
b, the instantaneous birth rate. For moderately large values of r and bD, the finite
per capita birth rate and the “egg ratio/D”
can take quite different values.
I shall present the derivation of formula
4 in fair detail. It has in fact been given
earlier by Edmondson
( 1968: equation
13). This earlier derivation is somewhat
unclear and the formula has not been used
by Edmondson himself or by others in later
publications.
I will make the usual assumptions: constant birth rate, constant death rate, and
fixed development time. These imply that
we can write
Nt = Noert
and
r=b-d,
where Nt = population size at time t, r =
intrinsic rate of growth, b = instantaneous
birth rate or instantaneous rate of reproduction, and d = instantaneous rate of mortality. Note that the definition of b assumes
that number of births in time t, t + dt =
bN,dt.
We have to make a distinction between
egg laying and hatching. In developing the
formulae we assume that the eggs are car-
692
JULY
1974,
V. 19(4)
Notes
ried by the parental female till hatching
at which time free swimming progeny are
liberated.
Eggs are therefore subject to
the same mortality rate as adults. The birth
rate refers to the rate of hatching although
eggs are layed (i.e. deposited in the egg
pouch of the female) some time before
that. We denote I = instantaneous rate of
laying eggs, D = fixed duration of development of eggs, and Ct = total number of
eggs counted at time t. If we have a
measure of the population at two times,
tl and t2, r is obtained from
r=
t1- t2
*
ct =
The integration
(summation)
need be
taken only from t - D to t as any egg laid
before t - D would have hatched at time
t. The terms in the brackets in the above
may now be evaluated
t’
No. of e gs laid
at an ear i:ier time
from t’ to
+ dt’ >
1 - e-b0
ct = lNt------*
b
In other words
No. of eggs/animal
D
= No@
t’ t’ dt’
from the assumption
rate, it follows that
&I,
of constant
( t’ t e-d(t-t’)
Survival
from
to
=
1
.
Ct
NtD
1
=bD(l
-e-bD).
(5)
If bD is small, say less than 0.3, the term
(1 - e-bD)/bD can be approximated by 1 bD/2 with relative error less than 2%, i.e.
Ct/NtD
w Z(1- bD/2).
(6)
The relationship
between the rate of
laying eggs, I, and the instantaneous birth
rate, or, which is the same in our terminology, the instantaneous rate of hatching, b,
is derived as follows
( No. hatching at time t, t + dt ) = bNtdt
= (No. laid at time t - D, t + dt - D) x
(Survival rate)
= (ZNtdodt) (e-dD),
giving
b z le-bD
or
-Substituting
1 = bebD.
(7)
I from equation 7 to 5, we get
-=ct
NtD
= N te -r(t-t’)
No. of e gs laid
= lNtg-r(f-t’)
at an ear kier time
to
+
from
>
=-
or
and hence
Hence
Or
t-De-b(t-t’) dt’.
b = le-dDNt-D/Nt
= IN,< dt’,
where t’ refers to some time before t, t D < t’ < t, and dt’ to a small time interval.
From the assumption of constant rate of
growth, it follows that Ntt may be written
as
Similarly
mortality
= lNt
In Ntl - In Ntz
The r above may be positive, negative, or
zero. To determine b and d in r = b - d,
we assume that Ct, number of eggs, has
been counted for a sample of known size.
This sample size, for shortness, we also
denote by N,. We get
Nt,
t
t-D
&f
Ct
=IN,
ste-r(t-t’)es
693
bebD 1- e-bD
--mebD- 1
=-.
D
Again if bD is small, say, less than 0.3, we
can approximate
the above by writing
ebD = 1 + bD + %(bD)2, giving
Ct/NtD - b[l+ ?‘z(bD)].
(9)
Comparing this with equation 6 we note
that the egg ratio per development time
Notes
694
Table 1. Comparison of different estimates of
the instantaneous birth rates when the egg ratio
G/N, = 1.
Development
time
D
0.5
1
2
t
5
10
Instantaneous
Equation
11
birth
rate
Equation
1.100
0.693
0.406
0.288
0.223
0.182
0.095
10
1.386
0.693
0.346
0.231
0.173
0.139
0.069
underestimates 1, the instantaneous rate of
laying eggs, about g ( bD ) ( 100) % and
overestimates b, the instantaneous rate of
birth, by about the same amount. The
exact solution is obtained from equation
8 as
Zn [ (Ct/Nt) + 11/D = b.
(10)
Algebraically,
this is equivalent to equation 4 quoted earlier. It is, I believe, the
correct formula
for estimating
instantaneous birth rate, b, from egg counts Ct
or egg ratios Ct/Nt. It is the same as equation 13 in Edmondson (1968). Equation
10 is compared with that given by Edmondson ( 1960, 1971)) in our terminology, as
In [ ( Ct/Nt D) + l] = b.
(11)
The finite birth rate, p, is defined as the
number of births in t, t + 1 divided by N(t).
It is readily calculated from Leslie’s formula, equation 3, once the instantaneous
birth rate, b, and the intrinsic rate of
growth, r, are known. We note that p
depends on r although the instantaneous
birth rate can in fact be calculated without
knowledge of the growth rate, r.
The values of the instantaneous birth
rate given by Edmondson’s formula, equation 11 and by the corrected formula, equation 10 are compared in Table 1. The Ct/
Nt ratio has been kept at 1.
In general the formula given by Edmondson underestimates the value of the
instantaneous birth rate when D < 1 and
overestimates it when D > 1 irrespective of
the value of the egg ratio. The magnitude
of the bias will however depend both on
D and Ct/Nt when D # 1. In Table 1
the differences range from 0 to 38% for the
range of values of D. When equations 10
and 11 are applied to data given by Wright
( 1965: table 1), the discrepancies between
the calculated instantaneous rates range
from 0 to about 44%, with equation 11
always underestimating
the instantaneous
birth rate.
Strictly speaking equation 10 applies only
when the assumptions made in the derivation are valid. The main assumption was
that the steady state situation prevails. In
practice no doubt the formulae would be
applied mostly to nonsteady states when
the temperature and hence the development time and perhaps other parameters
as well are changing. How this will bias
the results I do not know.
These formulae may also be applied to
species which bear their young, such as
humans or other mammals. In this case,
the egg ratio has to be replaced by pregnancy rate per capita and not by pregnancy
among the mature females.
Jyri E. Paloheimo
Department of Zoology and
Institute of Applied Statistics
University of Toronto
Toronto, Ontario M5S 1Al
References
CASWELL, H. 1972. On instantaneous
and finite
birth rates. Limnol. Oceanogr, 17: 787-791.
EDMONDSON, W. T. 1960. Reproductive
rates of
rotifers in natural populations.
Mem. 1st. Ital.
Idrobiol. 12 : 21-77.
-.
1968. A graphical model for evaluating
the use of the egg ratio for measuring birth
and death rates. Oecologia 1: l-37,
1971. Reproductive
rate determined indi;ectly
from egg ratio, p. 165-166. In W. T.
Edmondson and G. G. Winberg [eds.], Secondary productivity
in fresh waters. IBP Handbook 17. Blackwell.
LESLIE, P. H. 1948. Some further notes on the
use of matrices in population
mathematics.
Biometrika 35 : 213-245.
WRIGHT, J. C. 1965. The population dynamics and
production
of Daphnia in Canyon Ferry ResMontana.
Limnol.
Oceanogr.
10 :
ervoir,
583-590.
Submitted: 12 March 1973
Accepted: 15 February 1974