Some Procedures for use in Cohort Analysis and Other Population Simulations

K. Radway Allen

20/8 Waratah Street, Crowarlla NSW 2230, Ausbra!ia

and W. S. Hearn Division of Fisheries Research, CSIWO Marine Laboratories, GPO Box 1538, Hobart Tas 7001, Australia

Allen, K. R., and W. S. Hearn. 1989. Some procedures for use in cohort analysis and other population simulations. Can. j. Fish. Aquat. Sci. 46: 483488.

The catch equation N,, , = must ohen be solved either forwards or backwards. Because F, is an implicit function of catch, population size, and M, the solution must be either iterative or approximate. In this paper an improved approximation to N,, , is developed of the form

N,+, - N,exp(-M) - C, exp (-AM)

for the forward solution and a corresponding equation for backward solution. Appropriate values of A are pre- sented for a range of combinations of M and F; and A = 0.585 gives approximations to N,,, with an error of less than 1 % provided that (F,+ M) is less than abut 1.5. Much closer approximations are obtained if A is calculated for each case as a polynomial in M and either CjNtfor the forward solution or Ct-,lN, for the backward solution. The appropriate coefficients for these polynomials are derived both by truncating Taylor's expansion and by statistical methds, and are presented here. One such polynomial for A gives errors of less than 0.02% where (F+ M) i s less than 4 -1 . Similarly good approximations can also be obtained by interpolating into a table of N,, ,/N, against M and C/N.

t1$quation de la prise N,, , = N,e-'Ftf M' doit souwnt &re extrapol6e ou intrapolee. Parce que F, est une fonction implicite de !a prise, de Itimportance de la population et de MI la solution doit &re iterative ou approximative. Dans cet article, une approximation anaelior& de N,,, est propos6e; elle prend la forme suivante

pour I'extrapolation et il existe une $quation correspondante pour l'intrapolation. Des valeurs appropri&s de A sont prksent&s qui correspondent B diffkrentes combinaisons de M et F ; A = 0,585 donne des approximations de N,,, avec une erreur de plus ou moins 1 % pouwu que (Ft+ MM) soit inferieur % environ 1,5. Des approximations beaucoup plus precises sont obtenues lorsque A est calceel$ dans chaque cas comme un polynorne de M et CjN, pour I'extrapolation ou C,-,N, pour I'intrapslatiola. bes coefficients appropries 3 ces polynomes sont deduits A la fois en tronquant I'expansion de Taylor et par des methodes statistiques; i ls sont presentes dans cet article. Un tel polynorne pour A produit une erreur de moins de 0,02 % quand (F+ M) est inferieur 3 1 , I . De %a m@me facon, on peut obtenir de bonnes approximations en faisant une interpolation suivant une table de N,,,IN, contre M et C/N.

Received December 7 8, 1 987 Accepted October 24, 1986 (~9516)

everal techniques used in the analysis of exploited popu- lations of marine animals involve calculation of the pop- ulation of one year from the population at the beginning

of an adjacent year, knowing in the first year and bowing, or assuming, th ity rate. In cohort analysis (Gulland 1977) the calculation is per-

formed backwards; the population at the beginning of one year is calculated from that at the beginning of the following year. h other cases, the calculations are done forward (Allen 1983; Wmpton and Majkowski 1986). In the following discussion the fmd simulation is treated F i t because i t is rather m e direct, idthough i n practice the backward solution, employed in cohort analysis and. W A i s probably m e widely used.

In the most usud population model, when catching is contin- uous, and Both fishing (0 and natural (RI) mortality rates remain constant, the basic equations are

(1) c, = Ffl, (1 - e-qJc=-(F, 9 M)))I(F, 9 m, and

where N, is the population at the beginning of year 8 and C, is the catch in the same year. Equations (1) and (2) are solved for N,, , and Ft in terns of N, and C, in forward simulation, and for N,-, and Ft- , in terms of N, and C,- , in the backward case.

Equation (1) cannot be solved analytically; an iterative solu- tion is required. I t can be solved, for example, by Newton's

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method, or by the method proposed by MacCdl(1986). such a solution can now be obtained with a computer, or even with some desk cdcu1ators, the number of cdculations required may be substantid. For a typical cohort analysis, for example, this number will be approximately the product of the number of age-gmups md the number of yeas for which there are dab. If higher level iterations are required (e.g. to test h e effect of different vdues of M or of the terminal F), then the number of cdcnlations needed will be increased by a further factor. A substantid mount of computer time may therefore ofen be saved by calculating m appmximation to Nt + , or Nt- directly, using rn algorithm that gives an accephble level sf accmacy. An approximtion frequently used for &is purpose (Ricker 1975) is

which Pope (1972) derived as an approximation to equtions (1) md (2) when combined as

where

MacCdl (1986) discussed the merits of simplifying cohort analysis fomulae and proposed m approxim&ion that is slightly more xcur&e than equation (3). Sims (1982) has dso propsed a~mximations to Ne, ,INt and I&V,- JNe which are much better than those given by Pope (1972).

Two different approaches are here used to develop approxi- mations to B (in tems of M a d C$NT) that estimate Nt + much more accurately than does equatio~ (3). A tbkd approwh dso presented is to obtain m approximation to N,+ ,I& in tems of A4 md C$Nt. These approxbs are dso ~ $ 4 to develop appmx- hations to the ' 'backwmd9 ' solution.

Quation (4) can be rewritten exactly as

Table 1 and Appndix 1 show the values sf A for combina- tions of a rmge of values of F, md M. Pope's (1972) approx- imation is a version 0% equation (61, which sets A = 0.5. It is evident that vdues of A higher than 0.5 will produce better approximations to N, + , .

Gray (1 879) painted out that the expression exp( - ~12.25) is rn approximation to (1 - exp( -x))/x with m error of less than 2%, over the rmge for x of 0 - 1.6. Substituting Ft + M, md F, for x in equation ( 5 ) shows that this is equivalent to giving A in equation (6) a value of 0.556, which is consistent with the values in Appendix 1.

In practicd applications the basic data available are the catch in a given yea, the population at the begiming or end of that year9 a d a value for M for that yea- From these, we have for each year CjNt, or Ct- ,INt and M, md it is therefore usehl to consider obtaining approxim&ions in which A or B is replaced in equations (7) md (5) by expressions in M and C$Nt.

Taylor9 s Approximation

One way of calculating approximation formulae is to use a Taylor's series. As a starting p i n t take an expression for the appmxh&iow for B of the form

md determine QB md b by matching tems in Taylor's expan- sions for B and B* .

From the defiition of B in equation (5), and since

If F, and M are small then from equation (1)

and therefore

Also, since the Taylor's expansion of equation (8) is

matching tems in equations (12) md (13) results in

md substituting into quation (8) gives

where

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TABLE 2. Cmfickllts of regression q u a h n s to cdculate A* from M and C t N t values for both faward (F) and backwad (B) simulation.

Coefficients for A*

No. of hw%& variables b w k w d Consmt M AcP Ct/NtB2 MCtSNg

Over the range 8 < M < 0.3, 0 < F < 1.2 the absolute emor iira B* as an approximation to B is less than 2%. For this mge, the resulting errors in Nt+ , are less than kdf those kom Pop's (1 972) approximation A = 0.5, and also much less than hose given by the improved expression of McCdl(1986).

Multiple Regression Approximations

Mme xcumk appoximations could be obtained by sing further tems in the Taylor" expansion of B, but several tems would be required for a high aecmcy. However, empirical development of approximtions of similar general fom has been found to give very satisfactory results. G d appmxhatioras can be obtained by fitting coefficients

fm the parameters of the formulae by multiple regression on the calculated values of A over a grid of vdues of M and Ft. This has been done for two expressions:

(16) A* = 6, + b,M + b f i Z + b3C{N, + b,(Cj~,)~ 4- tb@C{Nt.

Table 2 presents the coefficients for both expressions for A*, which were obtained by fitting the approximtion to exact vd- uesofAona18 x 10gridwithM = 0.1-1.BBmdF = 0.15 - 1.5.

In backward simulation, as required in cohort analysis, the basic equation comespndimg to quation (6) is:

where A laas the same vdues as in the forward ex&apolation. Approxhations for N t - , can be obtained as before either from the Taylor's expansion or by regression. Since the known q m - tity that has to be used for Ft- , is now Ct-,I&, instead of C j N t , the coefficients age different. They are shown in Tabk 2.

In the third approach mentioned above, the ratio WV,, ,/Nt is tly kom CJN, and M by solving qa t i on (1) for

Ft in t m s of CjNt md M, md substituting in quation (2); an equivalent procedure is followed to calculate N,- ,INt in tems of C,- ,/WV, and M. Tables for these ratios for b t h fornard and backward cdculations are given in Appendix 2.

When M and F are both small (e. g . less aDam 0.2) there is no difficulty in obtaining a gsod approximation to Nt+,/Nt or its baa&wd equivalent. The Pope (1972) approximation is prob- ably adequate for many ses, and that prop& by Sims (1982) is excellent. It is when M and F are higher, particularly

when F + M > 1.0, hat the problems arise, and it is in this situation that the p r w d m s discussed in this paper are superior to the approximations described earlier. Even the less accmate approximations can be used, as S h s (1982) suggests, to give starting values for an iterative p r m d m , but much time can be saved by using approximations which are g o d enough to make this unnecessary-

The procedures discussed in this paper can be u s d to obtain an approximation to the desk4 population size in several dif- ferent ways. Two of these involve dculating Wa, + , by quation (6) or its equivalent for Nt - , , These are: (I) to use an appropriate constant value of A thoughout the

series; (ii) to calculate m individual value of A for each cell for a

yea-class and season from h e initkd population and catch sizes using one of the approximations for A.

It would also be possible to obtain annual values of A by interpolating in a table of this parameter against M and CIN. However, if interpolation is to be used it is more d k t , faster, and little less accurate, (iii) to cdcdate Nt+ , or &-, by interpolating for the appro-

priate ratio of N, , ,IN, or Nt - ,INt in tables of these ratios against M md CJN, or Ct - ,IWTg.

In all these cases an wpmpfiak value of M has to be provided for each estimation.

In simulation or cohort analysis it m y sometimes be conven- ient to use the multiplicative expression.

instead of the exponential d g o r i h s based on equation (6). This equation may have some advantage if the model were to be subjected to further mathemtical treatment, for ex differentiating with respect to M. If it is desired to do this, D may be calculated by using the relationship D = exp(M(0.5 - 4 ) .

(a) Single Value of A

This is the simplest procedure and is appropriate when M cm reasonably k taken as mmtmt, andl when the rate of exploi- tation does not vary greatly over the pepid of the study. It will provide a simple and rapid way of carrying out a preEimi analysis. Tabb 1 would provide a useful guide in selecting the preliminary vdue of A for a given M. In general the values compndirmg to F = 0- l might be appropriate for a lightly exploited stock, or those for F = M if exploitation is fairly heavy. If more specific estimates of F and M are available, appropriate values can be obtained b m Appendix 1. There are analytic advmtages in using a single vdue of A in quatiom

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TABLE 3. Percentage emof in the vdw of He+, or - , using the simple expression with A = 0.55 ; calculated over a grid with M = 8.054.5 md F = 0.054.5.

h m d F

TABLE 4- Mean and rmt mean sq (Rh4S) percentage emrs in predicted values of Nt+ , or Ne- , from four-tern and six-iem formulae for A* for f w d (F) and backward (B) simulations cdculated o w a grid of M = 0.1 - 1.0 m d F = 0.15 - 1.5, md from interpolation into tables of Nt+ ,INt cdculated over 8000 random pints within the same range of F and M.

No. of Foma& Mean e m r RMS emor variables Backwad

(6) a d (171, because these equations will &en be linear in CIN.

If the exploibtion rate has varied extensively or M changes (e. g. as a result of age-depndency), it may be useful to use vdues of A which minimize average emom over a tmge of cob- binations of F and M. Over a rmge of M and F between 0.1 md 1.0, A = 8.585 minimizes the average root mean square (RMS) emr* The RMS e m = in &+, a d Nt- are 1.3 and 0.55% "sr f s w a d md backwad mns, respectively.

If the likely values for M a d F are less than hove, the comspnding values sf A will be smaller. Over a range of M and F up to 0.5 the average RMS e m r is least with A = 0.55 being 0.10% for forwad calculation, and 0.08% for backward

cdculaion. The full emor grid for forward simulation with A = 0.55 is shown in Table 3.

It is important to remember that in backwad ealculatisns the exponent is (1 -A) , not A.

Tests have dso shown that where M is fairly low (e.g. about 0.2) md the m e Ft does not vary from year to year by more thm about zk 20% of the mean, use of the simple equation (6) with A appropriate to the mean Ft will yield approximations a b u t as good as those obtained by using the six-coefficient polynomid in equa~on (16).

(b) Individud Values of A - Cdculated

Use of individual values of A provides g o d estimates of the next population size even when M md the exploitation rate are changing. 'This can be done simply and rapidly by including one of the polynomial formulae, with the approp~ate vdnes for the coefficients, in a computer p r o g m .

arises the emor levels given by these formulae aver a grid sf M = 0.1 - 1.0 a d F = 0.15 - 1.5. As an example, the detailed emor matrix for the six - coefficient for- wad fornula for A is shown in Table 5. The arithmetic mem emor is - 6.008% md the W S emor is 0.12%$ but this is raised by the comparatively high errors for F = 1.5, This table shows a pattern of distribution which results from the nature of the polynomial surface; errors are very smdl when M and F are smdl (less than 0.03% when F is less 0.51, but they beome relatively large, with a complex distribution pattern, towards the lower right comer of the table. Within a grid of M = 0.1 - 1.0, F = 0.1 - 1.0 the same coefficients give an Rh4S emor of 0.050% md a mean emor of 0.0 17% 'Q. For the backwud cdcalatiom the accuracy of the A d g o r i h is close to that for the forward cdcula~ons.

Cm. $. Fish. Atpa. Sci., Val. 46, 1989

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TABLE 5. Percentage e m r in the vdue s f Nt+ , using the six-tern formula in A* with the coefficients given for f8mad simulaeion in Table 2.

F

(c) Direct Interpolation for N, , ,I&

The level of accuracy will depend on the number of digits stmed md the spacing of the pints. The tables in Appendix 2 show N t , ,INt or Nt- to four decimal places over the rages M = 0.1-1.0, C j N t = 0-0.9 md Ct- , /Nt -- 0-3.8. The errors from interpolation in this table are consistently positive so that the average M S error cm be reduced by subtracting a correc- tion factor from the linealy interpolated vdue.

Appropriate correction factors are 0.27% for forward simu- lation and 0.04% for backward simulation. M e r these mmec- tions are applied the M S errors are abu t 0.16 and 0.02395, respectively. There is no advantage in using an interpolation table with a large number of digits.

Using a nodinear inteplation pmedupe will also increase the level of accuracy md remove much of the positive bias so that a correction factor is not necessary. The extent of the improvement will obviously depend on the routine used.

y can dso be increased by reducing the intervals in the interpolation table. If these are halved to 0.65 and 0.15 for M and CIN, respectively, the RMS errors are re- duced to a b u t 6.02 % and 8.0087% for forwad md back sim- ulations, respectively. The bias is dso reduced; for the forward operation a correction factor of 0.046596 is appropriate and has k e n applied to obtain the errors given abve; for the backwd cdculation no correction is necessary to reduce the RMS errors below 0.01%.

W e r e M is constant over a large number of cdculatisns, a f a t md accurate procedure would be to set up initidly a vector of 4, ,INt against C j N r for this vdue of M, md interpolate in this as needed.

The errors for the interpolation p m e d m are included in Table 4 for comparison with those for the four- a d six- coef- ficient algorithms. They are given with less precision thm those for the formula since, for obvious reasons, the means are cd- culakd over a few thousmd random pints, rather than for the fixed grid used for the cdcdated values.

Exact comparison between h e size of the errors produced

the field as a whole and v q only within the individual cells of the table.

Choice of Method

Choice of the method to be used will generally depend on the degree of mdytic simplicity md the level of accuracy desired, the importance- of sped, a d the computer facilities available. The simplest anad fastest method is to use the A dgo- rihm with a single value appropriate to the data. W e n greater accuracy is wamted pameters should k adjusted for each cell and the choice is between calculating A values each time or using interpolation methods. If it is impossible (e.g. on many desk calculators) or otherwise not desirable to store an inter- plation table then cdculation methods must k used.

Were available the interpolation methods will generally be preferable. Not only will they probably be faster, but they dso give almost uniform accuracy over the whole field of parameter vdms to be used, whereas the calculated values are highly accurate over p a t of the field but lose accuracy rapidly as the upper margins are approached. It seems therefore that, where d l procedures are possible, interpolation in a tabk of Nt+ ,INt (or Nt - ,INr), as in Appendix 2, should be the method of choice.

Watever the method used, the errors introduced by my of the approximation procedures described here are likely to be smdl in most cases compared with the uncertainties introduced fiom sampling a d other errors in the original data.

We are grateful to Dr. Geoffrey Evans, Mr. John Hmpbn, Dr. Alex MacCdl, Mr. Dennis Reid, and Dr. Keith Sainsbaary for saaggeseions md comments that have improved h i s manuscript.

ALLEN, K. R. 1983. Development and application of cetacean popamlation models, p. 333487. Pn T. H. Coder [ed.] Advances in applied biology. Academic Press, London.

with the c&ula~ intePlafiOns-has little Gw, D. E 1979. Sane extensions to the least ~ W S approach deriving mortality csefficienns. Invest. ksq. 43: 24 1 - 2 4 .

meaning for two One is that the values are G ~ u I M ) , J. A. Im. m e analysis ofd- dewjopment of p. 67- reproducible, k ing based on a f i e d uniformly distributed grid 95. Pn B. A. Gd1md [ed.] Fish ppdation dynamics. John Wiley and Sons, of points, while the interpolation points are randomized. The hndon.

L - -

are distributed over the field in a mmnifom way depending em bluefin tuna parental biomass, m m i k n t , and catches under the 1982 fishing regime. North Am. J. Fish. Manage. 6: 77-87.

01% the shape of the ~ l ~ n o f i d suda-7 a d having high and lvhe:Ca, A. D. 1986. V i a l population analysis (VPA) equations for non- low error regions, while the interpolated e m s are uniform over homogeneous ppu8ations, marad a family of approximations including

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iiwzprovements on Pope's c$t ma1pis. h. J. Fish. Aqua. Sci. 43: kcm, W. E. 1975. e s m p t a ~ o n and kteqretation s f biological statistics of 2-2W. f ~ h populations. Bull. Fish, Res. Board Can. 191: 382 p.

Porn, J- G . 1972. An inws~gat i~n sf the accm4:y of virtud ppulAsm m d y ~ i ~ SMS, S. B. 1982. Algorithms fm solving the catch equation forward md back- using cohort malysis. I d . C s m . Northwest Atl. RE&. Res. Bull. 9: 65- wad in time. Cam. J. Fish. Aqua. Sci. 39: 197-202. 74.

APPENDIX 1 . Values sf A for M = 0.1-1.0: F = 0.1-1.0.

~ B B N D I X 2. (A) Forward simulation; vdues of iVt, JWP, againsf M and CjMf.

ClN

Can. S. Fish. Aqwt. Sci., Val. 46, 1989

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