Transactions of the Society for British Entomology

T. lUWCUt AND CO. LTD., ARBROATH, ANGUS, SCOTLAND.

VOL. 18

PART 5

y

\ 4 197

H ^ 1

.FiVAFtD
UN1V— A3i F *1

TRANSACTIONS

OF THE

SOCIETY FOR BRITISH
ENTOMOLOGY

World List abbreviation : Trans. Soc. Brit. Ent.

CONTENTS.

B. F. J. Manly and M. J. Parr

A New Method of Estimating Population Size, Survivorship, and
Birth Rate from Capture-Recapture Data

Date of Publication, December 1968.

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of Entomology, University Museum, Oxford

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TRANSACTIONS OF THE SOCIETY
FOR BRITISH ENTOMOLOGY

VOL. 18 DECEMBER 1968 PART V

A New Method of Estimating Population Size,
Survivorship, and Birth Rate from Capture-

Recapture Data

By B. F. J. Manly (Dept, of Mathematics)

and

M. J. Parr (Dept, of Biology)

University of Salford

Abstract

A method of estimating sampling intensity (p . ) in capture-

recapture studies is described and hence an estimate of population
size (N . ) is available from

*N. =n. / p.

where n is sample size. Simple formulae are also given for
estimating survivorship (s.^ ) and ingress (6,_^.+J). The

method is illustrated with data obtained from a capture-recapture
study of the Six-spot Burnet moth (Zygaena filipendulae L.).

Working with wildfowl data, Lincoln (1930) developed a mark-
release-recapture method for estimating the total population size.
This method, the Lincoln Index, suffers from the great disadvan¬
tage that it requires rather restrictive assumptions to be made.
Many methods have been developed from the basic Lincoln Index
to allow a chain of estimates of population size and other para¬
meters to be obtained from a series of samples (Southwood, 1966;
Parr, Gaskell & George, 1968). The first of these were the methods
of Fisher & Ford (1947) and Jackson (1948). In recent years
methods based on more realistic assumptions have been derived,
of which those of Seber (1965) and Jolly (1965) seem to have the
most general application in situations where “births”, immigra¬
tions, deaths, and emigrations occur.

Jolly’s (1965) method is probably the least restrictive of any in
general use at the present time but, in common with the others
mentioned here, it assumes that mortality is independent of age.

*Denotes estimates.

82 ' [December

This implies that the probability of an animal surviving through
any period of time is not affected by its age at the start of the
period. For many organisms it is clear that this assumption is
unjustified and in these cases a more suitable method is required.
The method described below does not assume that mortality is
independent of age, and, in fact, requires the minimum of assump¬
tions to be made.

1. Data Format

The data required is obtainable in the following manner.
The capture-recapture experiment lasts from a time t1 until
a later time t . At each of the times t,, t,, . . t

. . t a random sample of animals is taken from the population,
where the time interval between samples need not be constant.
All of the animals seen in the sample at time t. (i = l, 2,
. . . m-1) are marked in such a way that if they are seen in any
later sample then it will be recognised that they were previously
captured at time t . A convenient way of doing this is to use a
different colour of mark for each sample. This form of marking is
usually referred to as date-specific. (An alternative marking
method involves giving each animal an individual mark when it is
first captured. The individual can then be identified whenever it
is recaptured.)

Using this marking technique it is possible to arrange the data
as illustrated in Table 1. An example is given in Section 3. This
table will contain all of the information relevant to population
estimation by the present method. The entries in rows j and k of
Table 1 relate to two hypothetical animals. The jth animal was

Table 1

Suggested Data Format

animal

ti

t2

U

Sampling times

t4 t5 t6 t7 ...

tm

1

X

y

z

X

2

X

z

z

y

y

X

3

X

y

z

z

y

z

X

•

i

X

y

z

z

y

X

l

k

X

z

z

X

l

n

X

y

z

X

x : first or last capture of an animal
y : animal captured

z: animal not captured whilst known to be alive (in practice this symbol

can be omitted)

83

1968 J

first captured at time t2 and last captured at time t7. It was also
captured at times ts and tc, but not captured at times £* and t5.
The fcth animal was captured at times t5 and t6) but at no other
time.

2. Derivation of Formulae

If we let : —

AT. = the total number of animals in the population at time t

s. _^.+2 = the proportion of the population alive at time t still alive

at time t . „ and
i+i 5

b._^. i — the number of animals entering the population in the
interval t . to t . , tt and alive at t. ,

i 1+1’ i+i

then the present method allows the estimation of N for i — 2 , 3,
. . . m-1, s. . , , for i — 1, 2, . . . m-2, and b . . , , for i = 2. 3, . . . m-2.

In what follows it is convenient to write s. . , as s. and b. . , ,
as b.. The terms “alive” and “entering” are to be interpreted in a
wide sense. Animals emigrating from the population are considered
to “die”, whilst those entering (ingress) include births or emer¬
gences.

The important assumptions that we make are

(a) that sampling is random, with all individuals in the
population having an equal chance of capture, and

(b) that marking has no affect on animals.

If the sample size at time t is n. then this will consist of a
certain proportion, p of the total population, N .. This proportion
will be the sampling intensity at that time, and the relationship

Pi=nJNi

will clearly hold. It follows that an estimate of A7, can be obtained,

whenever an estimate of p is available, from

1/

*N =n / *pi. (1)

The proportion of the population alive at time t surviving until

t. , is the survivorship over the period, that we denote by s as
above. Assuming that mortality is unaffected by marking, so that
the survival of the animals in the samples is similar to that of other
animals, it is expected that s .n. of the n animals in the sample at
time t . will survive until t . Since the sampling intensity at time
t is p.+1 it is also expected that a proportion p.+1 of those
animals still alive will be captured again in the sample at time t r

We have then the approximate relationship

r=n,s.p,,r

l l 1^1 + !

*Denotes estimates.

84

[ December

where r . denotes the number of animals captured in the ith and
(i + i)th samples. An estimate of s t is therefore

*si=ri/(n*pl+I) (2)

The number of new entries to the population in the interval $ .
to ti+1 can also be easily estimated. If a survival rate of st
applies over the interval then the population size at time t will

l T * *

be A 1 8 lf apart from new entries. The difference between this size
and the actual size, N . ,, wTill be the new entries alive at t . , , ;

i+i> i+i

bt=Ni+,-*iNt>

suggesting the estimator

*b . = *N . — *s *N. .

i i+i i i

(3)

Equations (1) to (3) show that the estimation of population
parameters is a simple matter, providing that estimates of the
sampling intensities are available. The other quantities involved,
n. and r are obtained from inspection of the data. Fortunately,
an estimator of sampling intensity is available for time t (i = 2, 3,
. . . m-I), providing that the data gives a reasonable number of
animals captured three or more times throughout the experiment.
The derivation of the estimator follows from dividing the popula¬
tion at time t into various classes of animal, according to capture-
recapture patterns. The classes are illustrated in Table 2. It will
be seen that there are four classes involved, and that any particular
animal alive at time t must be in one, and only one, of the classes.
An animal captured at any time before t and also at least once
after t . will be in one of the upper two classes. The importance of
these two classes is that any animal in either class is known to be

Table 2

Animals Alive at Time t<

captured at least once before U
and also at least once after t»

not captured before tr and/or
not captured after ti

at time ti

not

captured

captured

A

B

li

a

.

A

B

totals n

i

•Denotes estimates.

N -n

i »

85

1968]

alive before and after t on the basis of its captures at other times.
It is therefore observed that, of the A,. +B animals known to be
alive before and after t.f a proportion A / (A1. +B1.) were captured
at t ... This proportion gives an estimate of the sampling intensity
(p.) at time t and hence a method of obtaining population estimates
is fully defined. Together with equations (1) to (3) we now have

*p=AJ{AH + Bn). (4)

The quantities A and B are obtainable from the data in the
form of Table 1. Examination of this table will show that

*y = K
?( V + z) = Ah+Bh
and 'Z(x + y) = A +A —n

l H 2iJ p

where Zy denotes the total number of y’s in the column relating to
time t .*\y + z) the total number of y’s and z’s, in the column, and
z( x + y ) the total number of x’s and y’s in the column.

In order that the estimate of sampling intensity is not subject
to large sampling fluctuations it is necessary for A to be modera¬
tely large. It is, in fact, desirable that this quantity should be
larger than about 10. Unfortunately, a value of zero leads to an
infinite estimate of poulation size, unless B is also zero. In the
latter case no estimate of sampling intensity is available.

3. Example

The use of the new method will be illustrated by a short mark-
recapture study of the Six-spot Burnet moth ( Zygaena filipendulae
L.) made at Dale, S.W. Pembrokeshire, in July 1968. The colony
intermingled with a much smaller colony of the Five-spot Burnet
(Z. trifolii Esp.) and occupied the tip of the Dale Fort Peninsula.
Sunny weather throughout the period of study ensured adequate
mixing of the population between samples. One sample was taken
on each of five days.

The insects were given date-specific marks of cellulose ‘dope’
applied to the underside of the hindwings, the sexes not being
distinguished for the purpose of the study. The colours of mark¬
ing were allocated as follows :

19 July

20
21
22

11

11

green (g)
white (w)
blue (b)
orange ( o ).

‘Denotes estimates,

86 ' [December

A sample was also taken on 24 July but, as this was the last
sample, no marking was required. During the five days of the
experiment 141 different insects were seen. The recapture data
were recorded in the following fashion, using the colour abbrevia¬
tions indicated above :

19 July: 57 captured, marked and released.

20 July : 52 captured ; 25 g, 27 unmarked.

21 July: 52 captured; Sg, 9 w, 11 gw, 24 unmarked.

22 July: 31 captured; 2g, 3 w, 4 b, 5gb, \wb, 2gwb, 14 un¬

marked.

24 July: 54 captured; 1 g, 2 w, lb, 5 o, 4gw, 2 gb, 2go, 4 wb,

\wo, 1 bo, 5gbo , 1 gwbo, 19 unmarked.
Using this information Table 3 was constructed, in the format of
Table 1.

The computation of estimates is illustrated by the following
three examples. The estimate of the population size on 21 July,
which we denote by *N:I, was obtained by substituting the rele¬
vant data of Table 3 into equation (1) :

*N=ni/*p=nt(An + Bli)/A,i
*N,, = 52(14 + 12)/ 14 = 96-6

The estimate of the survivorship over the period 20-21 July, which
we denote by *s,g 2l (or *s >n), was obtained from equation (2):

*ai^i+, =rt/(n*Pi+.)

/. *8" =20(14 + 12)/ (52x 14) = 0-7 143

Having calculated the population size estimates for 20 and 21 July,
as indicated above, the estimated number of new insects joining
the population in the period 20-21 July, which we denote by * bsg u

(or *b.0), was obtained from equation (3):

*bt . =*N, , - *s *N{

i — w+f i+t i i

*b,ll 2l =96 6- (0-7 143x90-1) = 32-24

The full series of estimates otained from the data are given in
Table 4.

Table 4

Zygaena filipendulae — population estimates

Date-July

19

20

21

22 1 24

Population size (*Ni)

90-1

96-6

930 | —

Survivorship +

0-7601

1 0*71

43

1

0-6923

—

New entries (*br->j+/)

—

32 24

26-12

1 -

•Denotes estimates.

1968] 87

Table 3

Zygaena filipendulae — Full lay-out of recapture data

date

(continued from fh

~st column)

July

date

19

20

21

22

24

July

Colour : g

w

b

0

19

20

21

22

24

X

y

y

y

X

Colour :

fir

w

b

o

X

y

y

X

X

y

z

X

X

y

X

X

y

z

X

X

y

X

X

y

z

X

X

y

X

X

X

X

y

X

X

X

X

y

X

X

X

X

y

X

X

X

X

y

X

X

z

y

X

X

y

X

X

z

X

X

y

X

X

z

X

X

y

z

z

X

X

z

z

X

X

y

z

z

X

X

z

z

X

X

y

z

z

X

X

y

X

X

y

rr

JLt

z

X

X

X

X

X

X

X

X

X

X

X

X

X

X

z

X

X

X

X

z

X

X

X

X

z

X

X

X

X

z

X

X

X

X

z

X

X

X

X

z

X

X

X

X

z

X

X

X

X

X

X

z

y

y

X

X

X

X

z

y

y

X

X

X

X

z

y

y

X

X

X

X

z

y

y

X

X

X

X

z

y

y

X

no. capt. only

X

z

y

z

X

once

21

13

13

9

19

X

z

y

z

X

n = total sample

X

z

X

i

X

z

z

y

X

size

57

52

52

31

54

X

z

z

y

X

II

^5

lAl

—

15

14

10

—

X

z

z

z

X

i li

X

y

X

I z = B

—

11

12

20

—

X

y

z

X

i li

(continued in next column)

n

25

20

12

15

—

(To save space the details for moths seen on only one day have not been
shown in full, but only the total number given for each day, e.g. 21 for 19
July, See Table 1 and the text for a full explanation of this table).

4. Discussion

The method described in this paper has certain advantages
over other methods of population estimation, and, in particular,
fewer assumptions are made than with the comparable methods of
Fisher & Ford (1947) and Jolly (1965). The most important
assumption not made is that mortality is independent of age. The
effect of making this assumption without justification, and there-

88 - [December

fore using a method requiring it, is not fully understood. It would
probably not be important providing that the total period covered
by any capture-recapture experiment is small compared with the
average life span of the animals in question. However, the present
method can certainly be recommended whenever mortality is
thought to be strongly influenced by age. If mortality is in¬
dependent of age then Jolly’s (1965) method uses the data more
efficiently than does the present method, and his method is then
superior because of this. (In practice the two methods appear to
give rather similar estimates.)

One other advantage of the present method is its simplicity,
with regard to the principles involved as well as the computations
required for estimates. This has an important result: “curious”
estimates can often be explained quite easily in terms of peculiari¬
ties of the data.

The assumption that marking does not affect mortality is not
strictly required in order to obtain valid estimates of population
size. Since sampling intensity is estimated from a group of
animals known to be alive before and after a certain time, the only
effect of changing the mortality rate is to change the size of this
group, but not the proportion captured at a time when all of the
animals in the group are alive. To this extent the estimates of
population size are independent of the mortality rate. The esti¬
mates of survivorship and new entries could be seriously affected
if marking alters the mortality rate.

The method can be used in the form of a triple catch. To do
this only three samples need be taken, and this allows the estima¬
tion of the population size at the time of the second sample,
together with the survivorship from the time of the first to the
time of the second sample.

We have assumed so far that both losses and gains occur in the
population of interest. If it can be assumed that only one, or
none, of these occurs then the method can be changed slightly in
order to take this into account. The important modification occurs
in the dividing of the population into various classes (Table 2) in
order to estimate sampling intensity. As an example we consider
the case when only losses occur. In this case any animal seen after
t( was certainly alive when the sample at time t. was taken. An
estimate of sampling intensity at time t . (i = l, 2, . . . m-1) is there¬
fore given by the proportion of the animals seen at any time after
t. that were in the sample at time t . Population estimates then

follow from equations (1) and (2).

89

1 908 J

No attempt lias been made to give standard errors for the
estimators of this paper. Work is continuing in this direction. It
is likely that there is a slight bias in the estimates, and this ques¬
tion is also under investigation.

Acknowledgements

We are grateful to Dr. L. M. Cook, Dr. It. M. Cormack, Mr. T. J.
Gaskell, Mr. B. J. George, Dr. G. M. Jolly and Professor P. M.
Sheppard, F.R.S., for their comments and helpful criticism of the
first draft of this paper. Thanks are also due to Mr J. H. Barrett,
Warden of Dale Fort Field Centre, for providing field work
facilities, and to Mr. T. M. Stokes who carried out the sampling of
the Burnet moth population on 24 July.

References

Fisher, R. A., and Ford, E. B. (1947). The spread of a gene in natural
conditions in a colony of the moth Panaxia dominula L. Heredity,
1: 143-74.

Jackson, C. H. N. 1948. The analysis of a tsetse-fly population. III. Ann.
Eugen., Lond., 14: 91-108.

Jolly, G. M. 1965. Explicit estimates from capture-recapture data with
both death and immigration-stochastic model. Biometrika, 52 :
225-47.

Lincoln, F. C. 1930. Calculating waterfowl abundance on the basis of
banding returns. U.S.D.A. Circ., 118: 1-4.

Parr, M. J., Gaskell, T. J., and George, B. J. (1968). Capture-recapture
methods of estimating animal numbers. J. Biol. Educ., 2 : 95-117.
Seber, G. A. F. 1965. A note on the multiple recapture census. Bio¬
metrika, 52 : 249-59.

Southwood, T. R. E. 1966. Ecological Methods. Methuen, Lond.

>; ;

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Date Due