Fisheries Research, 6 (1988) 153-165 153 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

Mean Populat ion Number and the De Lury and Lesl ie Methods

M.J. SANDERS

FAO Fisheries Project ( RAF/79/065 ), P O. Box 487 (Seychelles)

(Accepted for publication December 31, 1986)

ABSTRACT

Sanders, M.J., 1988. Mean population number and the De Lury and Leslie methods. Fish. Res., 6: 153-165.

Further modifications to the De Lury and Leslie methods for estimating the initial population number (N,,) and the catchability coefficient (q) are presented. These are based on the mean population number (/~) during each time interval being

N = ( N J Z T ) [1 -exp ( - Z T ) ]

where Z is the total mortality coefficient and T is the duration of the interval. The more useful of the modifications avoids the requirement, which is implicit in all carlier modifications, that Z remain constant between intervals. The extent of the errors attributable to using approximations of the mean population number are considered. This includes a comparison of the estimates of N,, and q obtained from hypothetical data, for which the correct values are known.

INTRODUCTION

The linear regression models developed by Leslie and Davis (1939) and De Lury (1947) are frequently used to estimate the catchability coefficient (q) and initial population number (No) from catch numbers and fishing efforts. They are applied most often in situations where the changes in population numbers are the consequence of mortalities from fishing. De Lury {1951) in- cludes a modification of the Leslie method for use when the population num- bers are also reduced by natural mortalities. This has subsequently been reworked by Chien and Condrey (1985). A modification of the De Lury method, also for use when the natural mortalities are significant, is given in Ishioka and Inoko (1982).

The premise underlying all of these methods is that the catch per unit effort (c/f) in a particular time interval is proportional to the mean population num- ber (hT) during the interval; that is

0165-7836/88/$03.50 © 1988 Elsevier Science Publishers B.V.

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cff = ql~

where q is the catchability coefficient referred to earlier. The methods use dif- ferent approximations for the mean population number. Subsequent to Braa- ten (1969), the De Lury method uses the number remaining at the middle of the time interval, while for the Leslie method it is when half the removals have occurred. The removals are here taken as the sum of the catch and natural death numbers.

In this paper an alternative definition of the mean population number is used. It is incorporated within further modifications of the De Lury and Leslie methods for estimating No and q. The more useful of these avoids the require- ment that the total mortality coefficient remain constant, which is implicit in all earlier modifications. The extent of the errors attributable to using the various approximations for the mean population number are also considered. This includes a comparison of the estimates of No and q obtained from hypo- thetical data for which the correct values are known.

THEORY

Mean population number and catch per unit effort

The previously mentioned relationship upon which the De Lury and Leslie methods are based is

e / f = q N (1)

where c/f is the catch per unit effort. This can be readily derived from a com- bination of the following equations

F=q f and

F= c/N

where F is the fishing mortality coefficient, c is the catch, and f is the fishing effort. A derivation of both these equations is given in Sanders and Morgan (1976).

Estimation of mean population number

Much of fisheries population dynamics includes the premise that changes in population numbers, as the consequence of mortalities, occur exponentially with time, according to

Aft--No exp ( - Z t ) (2)

where Nt is the number remaining after time t, and Z is the total mortality

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coefficient. When this is correct, it follows that the mean population number is determined from

N = ( N o / Z T ) [ 1 - e x p ( - Z T ) ] (3)

with T being the duration of the time interval; see Gulland (1983).

METHODS

De Lury method

In the De Lury method as described in Ishioka and Inoko (1982), the mean population number (/~') is defined as the number remaining at the middle of the time interval, according to

N' =No exp [ - (qE+M')] (4)

The E is half the fishing effort for the interval, and M' is half the natural mortality coefficient. Combining eqns. (1) and (4) gives

c/f=qNo exp [ - (qE+M')] (5)

which after taking the natural logarithm of both sides and re-arranging gives

In ( c / f )+M'=ln ( q N o ) - q E (6)

In the application of the method the required data are the catches per unit effort and efforts for each of a number of consecutive time intervals, plus a value for the natural mortality coefficient. The fishing efforts and natural mor- tality coefficient must then be accumulated to the middle of each interval, according to

x - - 1

E= ~ f i+ (f~x0.5) and (7) i : l

x - - I

M ' = ~ Mi+ (MxX0.5) (8) i = l

where the subscript x identifies the particular interval. Then according to eqn. (6), the linear regression of In (c/f) +M' against E gives a y-intercept of In (q No) and a negative slope of q. The estimate of No is obtained from the ( antiln y-intercept )/slope.

Further modifications of the De Lury method

When the total mortality coefficient is constant An approach to obtaining estimates of No and q when the mean population

number is as defined by eqn. (3) is to assume that

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N' = w ' / ~ (9)

with w' being the coefficient of proportionality. Combining this with eqn. (1) gives

c/f= ( q/w' ) 1~'

This can be combined with eqn. (4) to obtain the following analogies of eqns. (5) and (6)

c/f= ( q/w' ) No exp [ - (qE + M' ) ]

In ( c / f )+M' = l n [(q/w') N o ] - q E (10)

where E is half the fishing effort for the interval, and M' is half the natural mortality coefficient.

In the application of the method, catch numbers and fishing efforts are re- quired for each interval, and the natural mortality coefficient. Again the fish- ing efforts and natural mortality coefficient must be accumulated to the middle of each time interval, using eqns. (7) and (8) as described previously. Then the linear regression of In (c/f) + M' against E will give a y-intercept of In [ ( q/w' ) No ] and a negative slope of q. No is obtained from the ( antiln y-inter- cept) w'/slope, and requires knowledge of w'.

The value for w' can be obtained from

w'= [ gexp ( - Z / 2 ) ] / [ 1 - e x p ( - Z ) ] (11)

which is derived from N ' / N with/~ ' equal to Nt in eqn. (2) when t=0.5, and /~ as in eqn. (3) with T= 1. The required value of Z is obtained internally using

Z = q f + M (12)

In this case [ is the mean of the fishing efforts, and q is the slope from the above-mentioned linear regression.

When the total mortality coefficient is variable The validity of the above is dependent on w' being constant for all time

intervals. According to eqn. (11) this is only true when Z is constant, and hence the usefulness of the approach is limited. The preferable alternative is to use eqn. (6), and to accumulate f and M to the time t within each interval when the population number remaining is the same as N.

This time can be determined from

t = (In { Z / [ 1 - e x p ( - Z ) ] } ) / Z

which is derived from a combination of eqns. (2) and (3), with Nt = N and T-- 1. A different value of t must be determined for each time interval, which

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in turn requires estimates of Z for each interval. These latter are estimated internally by iteration, as described below.

The starting procedure is to undertake the linear regression of in (c/f) + M' against E, with the f and M values accumulated to the middle of each time interval using eqns. (7) and (8). The slope provides an initial estimate of q. Estimates of Z are then obtained from this q and the f and M values for each interval, using eqn. (12). These values are in turn used to obtain the initial estimates of t.

The linear regression of In (c/f) + M' against E is then repeated, this time with the f and M values accumulated to the time t in each interval. The equa- tions describing the accumulations are

x - - 1

E= ~ fi+(fxxG) i = l

x - - 1

M'= ~ Mi+(M~Xtx) / = 1

The regression parameters give improved estimates of No and q. As previously shown q is then used with the f and M values to obtain improved estimates of Z and then t for each interval.

The procedures of the last paragraph are repeated until no additional im- provement can be obtained in the estimates of No and q. Experience indicates that only one or two repetitions will be required to obtain these final estimates.

Leslie method

In the method of Leslie and Davis (1939), as modified after Braaten (1969), the mean population number (hT") is defined as the number remaining after half the removals have occurred. After including a term, (d), for the numbers dying naturally during the interval, the relationship becomes

- - tp N =No - K (13)

where K=0.5 (c+d) is half the removals. Combining eqns. (1) and (13) gives

c/f=q No - q K (14)

In the application of the method the required data are the catches per unit effort, catch numbers, and natural death numbers for each successive time interval. The removals must be accumulated to include half the catch and nat- ural death numbers for the last interval, according to

X - - 1

K = ~ ( c i + d i ) + 0 . 5 (cx+dx) (15) i = 1

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with the subscript x identifying the particular interval. Using eqn. (14) the linear regression of c/f against K gives a y-intercept of q No and a negative slope of q. An estimate of No is obtained from the (y-intercept)/slope.

Further modifications to the Leslie method

When the total mortality coefficient is constant An approach to obtaining more valid estimates of No and q is to assume that

~" =w"N (16)

which, after insertion into eqn. ( 1 ), gives

c/f= (q/w") N"

when combined with eqn. (13 ) this provides the following analogy of eqn. ( 14 )

elf= (q/w") No - (q/w") K (17)

where K is half the removals for the interval. In the application of this approach the required data are the catches per unit

effort, catch numbers and natural death numbers for each interval. The re- movals must be accumulated to include half the catch and natural death num- bers for the last interval, using eqn. (15) as previously described. Then the linear regression of c/f against K gives a y-intercept of (q/w") No and a nega- tive slope of (q/w"). An estimate of No is obtained from the {y-inter- cept)/slope, while estimating q from the slope requires knowledge of w".

The value for w" can be obtained from

w"=O.5Z [ l+exp ( - Z ) ] / [ 1 - e x p ( - Z ) ]

This is derived from/~"//~, with hT" according to the following variant ofeqn. (13)

N"=O.5No [ l + e x p ( - Z ) ] (18)

and h7 from eqn. (3) with T = 1. The required Z is obtained internally using the following

Z = - l n { [No - (~c + ~d)] /No} /n (19)

where the 2c and ~d are the sum of the catch and natural death numbers, respectively, accumulated over the n time intervals.

When the total mortality coefficient is variable Again the usefulness of the above approach is limited by the requirement

that w" and hence Z are constant. The preferable alternative is to use eqn. (14), and to accumulate the removals so that the population numbers remain- ing in each interval are the same as the values for/~.

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The proportion (p) of the removals consistent with N remaining can be determined for each interval from

p = { 1 - [ 1 - e x p ( - Z ) ] / Z } / [ 1 - e x p ( - Z ) ]

which is derived from a combination of eqn. (3), with T= 1, and the following

N t = N o - p N o [ 1 - e x p ( - Z ) ]

A value for p must be obtained for each interval, which in turn requires esti- mates of Z for each interval, the latter being obtained internally by iteration.

The starting procedure is to undertake the linear regression of cff against K, with the removals accumulated to include half the catch and natural death numbers for the last interval using eqn. (15). The slope provides an initial estimate of No. Estimates of Zx are then obtained using the population number at the start of each interval, (N, ) , and the cx and dx values for each interval, with the following variant of eqn. ( 19 )

Z ~ = - l n { [ N , - (c~ +d~)]/N,}

Starting with the first interval (for which an initial estimate of No has been obtained), the population number at the start of each interval is estimated by subtracting the removals from the population number at the start of the pre- vious interval. The Z values obtained in this way are in turn used to obtain initial estimates ofp.

The linear regression of cff against K is then repeated with the removals accumulated to the proportion p in each interval. This is done using the follow- ing analogy to eqn. (15)

X--1

K-= ~_, (ci+di)+p (c,+dx)

The resulting negative slope provides an improved estimate of q and the (y- intercept ) /slope is an improved estimate of No. The value of No is used as before with the c and d values to obtain further improvements to the estimates of Z and then p for each interval.

The procedures of the last paragraph are repeated until the estimates of No and q have stabilised.

APPLICATION

Each of the modifications described in the previous section were applied to the five hypothetical sets of data presented in Table I. In all of these the catch numbers (cx) and natural death numbers (dx) in each time interval were es- t imated with No= 100 000, M=0.7, and q= 1.0X 10 -4, using

cx= [qfx/(qfx+M)] N1 { 1 - e x p [ - ( q / ~ + M ) ] } and

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TABLE I

Hypothetical catch numbers, and natural death numbers estimated when No = 100 000, M = 0.7, q= 1.0 X 10 -4 and the [, are variable

Time intervals

1 2 3 4 5 6

Set 1

Fishing efforts 3000 3000 3000 3000 3000 3000 Catch numbers 18963.617 6 976.325 2 566.446 944.143 347.331 127.776 Naturaldeath numbers 44 248.439 16 278.091 5 988.375 2 203.000 810.438 298.144

Set 2

Fishing efforts 13 000 13 000 13 000 13 000 13 000 13 000 Catch numbers 56 203.207 7 606.277 1 029.398 139.314 18.854 2.552 Naturaldeath numbers 30 263.265 4 095.687 554.291 75.015 10.152 1.374

Set 3

Fishing efforts 2000 2500 5000 4500 3000 1000 Catch numbers 13 187.341 6 561.381 4 578.260 1 266.392 284.370 37.973 Naturaldeath numbers 46 155.693 18 371.868 6 409.564 1 969.943 663.531 265.809

Set 4

Fishing efforts 5500 6000 10 000 15 000 17 000 24 500 Catch numbers 31393.789 9 619.529 3 753.967 864.799 101.797 10.674 Naturaldeath numbers 39 955.731 11 222.784 2 627.777 403.573 41.917 3.050

Set 5

Fishing efforts 24 500 17 000 15 000 10 000 6000 5500 Catch numbers 74 444.835 2 759.998 235.685 20.709 2.642 0.673 Naturaldeath numbers 21269.953 1 136.470 109.986 14.496 3.082 0.857

d x = [ M / ( q f x + M ) ] N1 { 1 - e x p [ - ( q f ~ + M ) ] } .

The population number (N1) at the beginning of each interval was determined from the number at the beginning of the previous interval, less the catch and natural death numbers for the interval. In the first t ime interval N1 = No. Both M and q were kept constant for all intervals.

The fishing efforts (fx) for each interval were kept constant for Sets 1 and 2 but were varied in respect to Sets 3, 4 and 5. The sum of the fishing efforts over the six intervals are the same for Sets 1 and 3, at •fx = 18 000, and the same for Sets 2, 4 and 5, at 5:fx -- 78 000. Sets 4 and 5 are somewhat extreme.

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TABLE II

Estimates of N,, and q ( × 10 4 ) from using the modifications of the De Lury and Leslie methods with the catch numbers, natural death numbers, and fishing efforts given in Table I

Set 1 Set 2 Set 3 Set 4 Set 5

De Lury method with N . = 104 219 117 518 104 286 109 206 145 990

modification of Ishioka q= 1.00000 0.99998 1.00187 0.94995 1.05009

and Inoko (1982)

De Lury method with N . = 100 000 99 998 100 059 93 863 122 948

author's modification q= 1.00000 0.99998 1.00187 0.94995 1.05009

when Z is constant

De Lury method with PC,,= 100 000 99 998 100 000 100 000 100 004

author's modification q = 1.00000 0.99998 1.00000 1.00000 1.00003

when Z is variable

Leslie method N,,= 100 000 100 000 99 848 99 856 100 101

modified by Braaten q= 0.92423 0.76159 0.93861 0.88891 0.58179

(1969) and with a

natural death term

Leslie method with N,,= 100 000 100 000 99 848 99 856 100 101

author's modification q= 1.00000 1.00000 1.04121 ' 0.64438

when Z is constant

Leslie method with N,,= 100 000 100 000 100 000 100 000 100 000

author's modification q= 1.00000 1.00000 1.00000 1.00000 1.00000

when Z is variable

~No solution as (Xc + ~d) > N,,; see eqn. (19).

In the former, the fishing efforts were allowed to increase progressively but in the latter they decrease progressively.

As M and q were constant for all time intervals, the total mortality coeffi- cients (Zx) are also constant for Sets 1 and 2, and varied for Sets 3, 4 and 5. The sum of the total mortality coefficients over the six intervals are the same for Sets 1 and 3, at ZZx=6.0, and the same for Sets 2, 4 and 5, at ~Zx =12.0. The mean values for the total mortality coefficient are Zx= 1.0 for Sets 1 and 3, and Zx = 2.0 for Sets 2, 4 and 5.

The estimates of No and q from applying the various modifications of the De Lury and Leslie methods to the five data sets are given in Table II. These indicate that the De Lury method as modified by Ishioka and Inoko (1982) provided correct values for q, when the total mortality coefficients were con- stant ( i.e. with Sets 1 and 2) . Even when the Z's were not constant, the esti- mates of q are still reasonably close to the correct values. The extent of error was greatest, at ~ 5% for Sets 4 and 5. This modification did not provide cor- rect values for No, however, even when the total mortality coefficients were

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constant (i.e. with Sets I and 2). The error was found to be highest, at ~ 46%, with Set 5.

The author's modifications to the De Lury method provided correct values, for both No and q, when the total mortality coefficients were constant. When the Z's were not constant, the second modification continued to provide correct estimates for both parameters. Applying the first modification to Sets 3, 4 and 5 gave values of No which were improvements over those from the Ishioka and Inoko (1982) modification. The error was still substantial, at 22%, with Set 5. The first modification provided the same estimates of q as the modification of Ishioka and Inoko (1982).

The Leslie method as modified by Braaten (1969), including a term for the numbers dying naturally, provided correct values for No when the total mor- tality coefficients were constant. When the Z's were not constant, the esti- mates of No were still close to the correct values. The error did not exceed 1% for any of the five data sets. The modification did not provide correct values for q however, even when the total mortality coefficients were constant. The error in q was highest, at 42%, with Set 5.

The author's modifications to the Leslie method provided correct values for both No and q, when the total mortality coefficients were constant. When the Z's were not constant, the second modification continued to provide correct estimates of both parameters. Applying the first modification with Set 5 gave an estimate of q having an error of about 34%. No estimate of q could be ob- tained with Set 4. The estimates of No from this modification, are the same as from the modification of Braaten (1969) with a natural death term.

DISCUSSION

In the consideration of the De Lury and Leslie methods presented here, it has been assumed that the changes in population number within each time interval occur exponentially, according to eqn. (2). In this circumstance the methods as previously modified give estimates of No and q which are approxi- mate. This is because they are based on the mean population number being AT' for modifications of the De Lury method, and N" for modifications of the Les- lie method, instead of the more valid AT. The extent of the differences in the estimates of AT, AT' and/~" for given values of Z can be substantial, as indicated in Table III.

Errors in No and q, associated with using the approximations for the mean population numbers, are somewhat reflected through estimates of w' and w". When Z is constant, ( 1 - w ' ) 100 is the percentage by which A7 is under-esti- mated by AT'. It is also the percentage by which No is over-estimated from the (antiln y-intercept)/slope, when using eqn. (6) instead of eqn. (10). Simi- larly, when Z is constant, (w" - 1 ) 100 is the percentage by which N is over- estimated by AT". It is also the percentage by which q is under-estimated from

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TABLE III

Es t imates of h 7, N ' and AT" (when N o = 100 000, and T = 1 ), and (1 - w') 100 and (w" - 1 ) 100 for given values of Z

Z h7 N ' AT" ( 1 - w ' ) 100 ( w " - l ) 100

0.1 95 163 95 123 95 242 0.042 0.083 0.2 90 635 90 484 90 937 0.167 0.333 0.3 86 394 86 071 87 041 0.374 0.749 0.4 82 420 81 873 83 516 0.664 1.330 0.5 78 694 77 880 80 327 1.034 2.075 0.6 75 198 74 081 77 441 1.484 2.982 0.7 71 916 70 469 74 829 2.013 4.050 0.8 68 834 67 032 72 466 2.618 5.277 0.9 65 937 63 763 70 328 3.297 6.661 1.0 63 212 60 653 68 394 4.048 8.198 1.5 51 791 47 237 61 157 8.794 18.083 2.0 43 233 36 788 56 767 14.908 31.304 2.5 36 717 28 650 54 104 21.969 47.356 3.0 31 674 22 313 52 489 29.554 65.719 3.5 27 709 17 377 51 510 37.285 85.898 4.0 24 542 13 534 50 916 44.856 107.463 4.5 21 975 10 540 50 555 52.038 130.055 5.0 19 865 8 208 50 337 58.679 153.392

No te : /~ , /~ ' and N" were es t imated using eqns. (3) , (2) wi th t=0 .5 , and (18), respectively.

the slope, when using eqn. (14) instead of eqn. (17). Est imates of (1 - w' ) 100 and ( w" - 1 ) 100 for a range of values of Z are given in Table III.

These est imates of error are only indicative and become progressively less useful as the values of Z increase. When Z = 1.0 (as for Set 1 ) the est imated error in No is 4.048%, as indicated by (1 - w') 100 in Table III. This is close to the observed error of 4.219 [ = (104 219 - 100 000) /100 ] % from Table II, when applying the De Lury method as modified by Ishioka and Inoko {1982). When Z=2.0 (as in Set 2), however, the est imated error in No from Table III is 14.908%, compared with the observed error of 17.518%.

Similarly, the est imated error in q of 8.198%, as indicated by ( w " - 1 ) 100 from Table III when Z = I . 0 , is close to the observed error of 7.577 [ = (1 .0-0 .92423) 100] % from Table II, when applying the Leslie method as modified by Braaten (1969) and including a natural death term. When Z = 2.0, however, the est imated error in q is 31.304, compared with the observed error of 23.841%.

The reason for the errors as est imated by ( 1 - w ' ) 100 and ( w " - 1 ) 100 being only approximate, and progressively less useful as the values of Z in- crease, is the consequence of the true relationship b e t w e e n / ~ and N ' , and between N and/V". These relationships are not proportional, as assumed in eqns. (9) and (16), but curvilinear (see Fig. 1 ).

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IOO O O o

!z rF

O

50

N

--!

N

I ~" I I I . . , 1 I I I I I

O 50 IOO

N ('000) Fig. 1. Plots of/~' and/V" against N, as estimated with N , , = 100 000 for a range of values for Z.

It also follows from this, that the author 's first modifications of the De Lury and Leslie methods provide only approximations of No and q, even when the Z's are constant. Notwithstanding, it was not possible to observe this error when applying the modifications with Sets 1 and 2.

The above discussion of errors in the est imates of No and q has little direct relevance when the Z's are not constant. It is interesting to note, however, that for Set 3 the observed error of 4.286% in No, from applying the Ishioka and Inoko (1982) modification of the De Lury method, and 6.139% in q from ap- plying the Leslie method as modified by Braaten (1969) and with a natural death term, are little different than the est imated errors from using ( 1 - w ' ) 100 and ( w" - 1 ) 100. These latter are 4.048 and 8.198, respectively.

It is also interesting to note that when the Z's increase progressively (as for

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Set 4 ) , t he observed e r rors in No f rom apply ing the I sh ioka and Inoko (1982) modif ica t ion , and in q f rom apply ing the modi f i ca t ion of B r a a t e n (1969) wi th a na tu ra l dea th te rm, are less t h a n the e s t i m a t e d e r rors a t Z = 2.0 f rom (1 - w' ) 100 and (w" - 1 ) 100; and are g rea te r when the Z 's decrease progress ive ly ( as for Set 5) .

T h e au thor ' s second modi f ica t ions will p rov ide es t imates of No and q, with- out the er rors m e n t i o n e d above, p rov ided the changes in popu la t ion n u m b er s wi th in each t ime in te rva l are exponen t i a l accord ing to eqn. ( 2 ). T h e addi t iona l v i r tue of the modif ica t ions , is t h a t t hey r ema in valid, even when the to ta l mor- ta l i ty coeff ic ient is no t c o n s t a n t be tween t ime intervals .

It should also be no t ed t h a t these modi f ica t ions can be used even when the na tu ra l mor t a l i t y coeff ic ient var ies be tween intervals . W h e n using the De Lu ry modif ica t ion, it would be necessa ry to have a separa te value of M for each t ime interval . Such deta i led knowledge is unl ikely , however , and so it would nor- mal ly be necessary to assume t h a t M is cons tan t . W h e n using the au tho r ' s second modi f ica t ion of the Leslie method , it would be necessa ry to know the na tu ra l dea th n u m b e r s in each interval . T h e possibi l i ty of this i n fo rma t ion being avai lable is also unl ikely.

REFERENCES

Braaten, D.O., 1969. Robustness of the De Lury population estimator. J. Fish. Res. Bd. Can., 26: 339-354.

Chien, Y.U. and Condrey, R.E., 1985. A modification of the De Lury method for use when natural mortality is not negligible. Fish. Res., 3: 23-28.

De Lury, D.B., 1947. On the estimation of biological populations. Biometrics, 3: 145-167. De Lury, D.B., 1951. On the planning of experiments for the estimation of fish populations. J.

Fish. Res. Bd. Can., 8: 281-307. Gulland, J.A., 1983. Fish Stock Assessment. Wiley and Sons, 223 pp. Ishioka, K. and Inoko, Y., 1982. Estimation of initial stock number based on catch and effort

statistics of a cohort, with an example of blue crab in the Etajima Bay, Hiroshima Prefecture. Bull. Nansei Reg. Fish. Res. Lab. 14: 33-54.

Leslie, P.H. and Davis, D.H.S., 1939. An attempt to determine the absolute number of rats on a given area. J. Anim. Ecol., 8: 94-113.

Sanders, M.J. and Morgan, A.J., 1976. Fishing power, fishing effort, density, fishing intensity and fishing mortality. J. Cons. Int. Explor. Mer., 37: 36-40.