Can. J. Fish. Aquat. Sci. Downloaded from cdnsciencepub.com by HOKKAIDO DAIGAKU on 09/30/21
For personal use only.

ation Ana
Equations for Nonhomogeneous
ions, and a Fa
f Approximations
Improvements on Pope's Cohort Ana

Southwest Fisheries Center, National Marine Fisheries Service, P.O. Box 279, La lolda, CA 92838, USA

MacCaBI, A. D. 1986. Virtual population analysis (VPA) equations for nonhomogeneous populations, and afamily of approximations including improvements on Pope's cohort analysis. Can. ). Fish. Aquat. Sci.
43: 2406-2409.
A set of "backward" virtual population analysis (VPA) equations relates catch (C,) from continuous fishing
between times t and t
1 to population size (N,, N, + 1 ) when a portion of the stock is unavailable to fishing.
The usual VPA equations become a special case where the entire stock is available (i.e. the stock is homogeneous). A close approximation to the VPA equations is N, = N,, explM)
CtM/(l - exp(-M)), which hasproperties similar to Pope's "cohort analysis" and is somewhat more accurate in the case of a continuous fishery,especially if the natural mortality rate ( M )is large. Much closer simple approximations are possible if the seasonalpattern of catches is known.

++biwe serie d%quations d'analyse de population virtuellle (APV) a rebours lie les prises (C,) tirees de la p@che
continue entre les temps t et t
1 2 la taille de la population (N,, N, + ,) quand une partie du stock n'esa pas
disponible pour la peche. bes equations habituelles dfAPV deviennent un cas special la oh le stock en entier est
disponible (c.4-d. que le stock est homogene). Une approximation etroite des equations d'APV est N, =N,, expfM) C,M/(1 - exp(-M)) qui a des gsroprietes sernblables 21 l'anafyse par cohortes de Pope et est un
peu plus precise dans le cas d'une p k h e continue, surtout si le taux de mortalit6 natureile ( M )est important. Des
approximations simples beaucoup plus etroites sont possibles s i I'on csnnait la tendance saisonni$re des prises.

+

((

))

+

Received April 30, 1985
Accepted luBy 25, 7 986(J8224)

irtual population analysis (VBA), also known generically as cohort analysis, has come to be the preeminent
method of fish stmk assessment since its appearance
20 yr ago (Gulland 1965; Murphy 1965). The method
usually is used in the form of a "backwards solution" where
historical abundances of a cohort are estimated on the basis ofsubsequent catches from a presumed homogeneous stock: allfish of a specified age are assumed to be equally susceptible tocapture. The equations unfortunately do not allow convenient
separation of the unknown quantities from the known and/or
assumed quantities, so the solution must be obtained by numerical methods, e.g. Sims (19821, or by approximation, e.g.
Pope's (1972) popular "cohort analysis." Computers havemade numerical solution very easy, but there remain applications of approximation methods. For example, Pope's approximation has proved useful for statistical and mathematical analysis of the behavior of VPA, whereas the exact equations are
intractable. In practice, good approximations and efficient algorithms remain useful, as computers may not conveniently beavailable in working group meetings, in field locations, and insome developing countries.
The following discussion provides two extensions to existing
VPA methods. The first is a set of equations which include
possible assumptions concerning homogeneity of cohorts, i. e.

part of the cohort may not be available or vulnerable to fishing.
These equations also provide the basis for an easy and rapid
iterative backward solution of the VBA equations. The second
is description of a family of approximation formulas which aresimilar in concept to Pope's cohort analysis.

VPA Equations for Nonhomogeneous Populations
Assume that the cohort consists of two segments, one fishedand one unfished (cf. Widrig 1954). This division could arise
under a variety of circumstances. For example, some membersof the cohort may reside in habitats or regions which cannot be
fished. Another possibility is differential vulnerability to thegearls,depending on fish size or other attribute. The present
treatment corresponds to the camanran model of "knife-edge"
gear selection. Importantly, there must be no interchange between the two segments during the time interval. k t P be theage-specific fraction sf the cohort which is subject to fishing at
the beginning of a time interval. The remaining notation follows that of Wicker (1975), except that F refers to the instantaneous coefficient of fishing mortality on the fishable segmentof the cohort. As usual, F and M are assumed to be independent, and M has the same value in both segments. Both of theseassumptions are open to question (e.g. Lawless 1982, p. 491;
Wicker 1976), but there seems to be no practical alternative
Can. J . Fish. Aquac. Sci., 8/01. 43, 1986

Can. J. Fish. Aquat. Sci. Downloaded from cdnsciencepub.com by HOKKAIDO DAIGAKU on 09/30/21
For personal use only.

given the usual state of knowledge. All parameters are assumed
to be constant within the time period, but may vary among ages
or time periods. A full notation is useful for fishery analysis,
but requires extensive use of subscripts denoting age and time;
this notation can be developed by the reader for particularapplications (Schnute 198%provides a useful structure for thispurpose). The present notation is simplified for ease of exposition, and considers a single cohort and a unit time period. The
time subscript r is shown only where desirable for clarity.
Under continuous fishing, the catch (C,) for a unit time period
beginning at time t , and ending at time t -I- I, is given by
where % = F + M ,and A = I - exp(-Z). The abundance atthe beginning of the next time period is

-4

0.0

where the two additive terms on the right side represent contributions from the fished and the unfished segments, respectively. This expression may be rearranged to giveand equation (I) allows substitution of C,Z/FA = N,B, giving(4)

N, + I = (N, - C,mZ/FA)exp(-M)

where sn = 1 - exp ( - F ) . The backward solution therefore is

Equations (4) and (5) also can be derived from the equations
describing the usually assumed homogeneous case. However,
because P appears in equations (1) and (2), but not in (4) or ( 5 ) ,this derivation demonstrates that they have the somewhat surprising property of being true whether or not the homogeneity
assumption ( P = 1) is met, provided that F is dqfined appropria t e [ ~These.
equations also demonstrate that for a given catch
md natural mortality rate, the four variables, N,, N, + , H, andF , possess only two degrees of freedom among them. Thismeans that specification of any two values should determine
both remaining values, but this property may be of limited Baseas the basis for direct parameter estimation (e.g. for a given
catch, F and P covary strongly, and estimation errors could
easily lead to misinterpretation). In prxtice, one value is given;for example, in the case of a backwards solution, N, + I may
have been provided by m initial guess or by VPA of older aged
catches. The second value is imposed by a simultaneous equation which may explicitly or implicitiy fix P (see equations (6)and (7), respectively). Equation (5) is aesthetically attractive inthat it clearly expresses the sense of "virtual population" as amodified sum of subsequent catches.
There are many possible equations which give F as a function of P, N,, A!, + [ , and M ,depending on the assumed behavior
and distribution of the stock and fishery. Differences among
these equations become especially important when they areapplied sequentially to link successive ages, as is the usual casein VPA. For example, equation (2) can be rearranged
algebraically to give

,

2.8

0.5
1.0
1.5
FISHING MORTALITY RATE (Fl-FIG. 8 . Relative emor (see text for definition) of Pope's and the new
approximationfor a homogeneous stock (P I) and two values of M .
For comparison, the relative emor of the result of the first iteration of
the VPA algorithm is shown for M = I , where relative emor is definedas ( ( A ~ / F ~ ) / ( ~ Z / F-AI .) )time period, so that the individuals in the fished and unfishedsegments have no differentid tendency to remain in their previous states. ABternatively, if P can be estimated independently
for individual ages and times, perhaps from surveys of habitat
distribution or of size composition, equation 46) allows exploitation rates to be separated into component sources of variation: those associated with changes in B and those associatedwith changes in F on the available segment. This form of"separable VPA" (sensu Doubleday 1976) is being developedby Daniel Kimura (NMFS-NWAFC, BIN C1570-0, 7600
Sandpint Way N.E., Seattle, WA 98 115, USA, manuscript inrevision). Of course, in the case of homogeneity (B = I),equation (6) simplifies to
as was suggested by Pope (1972).
Equations (5) and (7) also provide a convenient rapid algorithm for solving the usual VPA equations for a homogeneousstock. The procedure is as follows: (I) given G,,N, + , and M ,use Pope's approximation or the improved approximation described below to obtain a first estimate of N,; (2) use this valueof N, in quation (7) to obtain an estimate of F ; (3) use thisvdue of F in equation (5) to get an improved estimate of N,;(4) repeat steps 2 and 3 until the desired level of precision isreached, judging by the change from the estimate in the previous iteration; and (5) repeat the procedure on the next
younger age of the cohort, letting the previous N, now be
N , + and go to step 1 . The rate of convergence is very high(Fig. I), approaching that of Newton9smethod as described by
Sims (1982); two iterations often are sufficient. Also, thismethod does not converge to extraneous solutions.

,,

A Family sf Approximaticbms
The independence of C , in equation (5) indicates that there
exists a family of VBA approximations of the form

However, if B is treated as an independent population parameter, sequential use of this equation must be based on anassumed redistribution of all of the survivors at the end of eachCan. I . Fish. Aquat. Sci., Vol. 43, 1984

(8)

N, = N, + ,exp(M) + C,V(F.M, . . . )where fit is an approximation of N, and V(F, M , . . . ) is aCan. J. Fish. Aquat. Sci. Downloaded from cdnsciencepub.com by HOKKAIDO DAIGAKU on 09/30/21
For personal use only.

function approximating the vral~aeof mZ/FA. These approximations are applicable to nonhomogeneous fisheries ( P f 1) as
well as the more commonly assumed homogeneous fisheries.
In the case sf nonhomogeneous fisheries, these equations weed
to be coupled with an equation such as (6).
Pope's (1972) cohod analysis is a member of this family of
approximatisns, where V = exp(M/2):
Another approximation is provided by a first-order Taylor
series apgroximatisw of the exponentials in the left side of theexpression
where n = 1-exp(-M).

Rearrangement of (18) gives

which may be substituted into equation 68) to giveMONTH

The relative accuracy of approximations (9) and (12) can be
expressed as the relative error in V , the coefficient of C , ; theremaining terms of the equations are identical and tend to
reduce clarity of the comparisons. Also, because equations (9)and (12) are linear in C : errors in the coefficients propagate ina straightforwar4 manner. Let the "relatjve error" ( r , ) of an
approximation, 5, be defined as r , = (% - X)/%, or equivalently, r, = (%/X)-I, where % is the true value. In thepresent case, the approximation is the coefficient applied to C ,in equations (9) and (l2), and the true value is the coefficientr n Z / F A in equation (5)). Thus, the relative emor for Pope's
approximation ( r , ,,)) is

FIG. 2. Campxison of monthly catch distribution under the assumption sf constant %; and that from a seasonal fishery (the California catchof mackerel in 1941 -42, Anonymous 1944).

in the new approximation is equal to N ~ / Nif there is only
natural mogtaIity. Paradoxically, this means that the new approximation assumes the fishery's catch ts be a continuous
sampling of natural deaths in the absence of a fishery. This
somewhat strange interpretation is consistent with the fact thatthe new approximation is identical to the expression which
results from taking the limit of equation (5) for a given catch
as 8; -+ 0.

VPA Approximation for Seasonal Fisheries
where exp(M/2) is Pope's cohort analysis coefficient of C , .
Similarly, the relative error of the new approximation in equation (12) is

Relative errors for these two approximations are compared inFig. 1; generally, the differences are not large. The relativeemor of the new approximatism in equation (12) is slightly but
consistently smaller than that of Pope's approximation. The
new approxirnation becomes considerably better than Pope's as
M becomes large, thus extending the range of mortality rates
under which approximation is useful. Taylor series expansionsof the relative errors, equations ( 13) and (141, clarify the nature
If we disof the difference between the two approximatio~~~.
card terms higher thaw quadratic, the relative error for Pope's
approximation is (PFM + M"/24, whereas that of the new
approximation is (2FM)/24. The constant factor of M2/2%in
Pope's relative error explains both its nonzero intercept at
F = 0 and the parallel appewance of the plots in Fig. 1 .
Pope's approximation is equivalent to assuming that the entire catch is taken instantaneously at midyear. The new approximation does not have a corresponding capnvenient interpretation. A tentative interpretation may be derived from therelationship N = Non/M, which describes average abundance(p)over a unit time interval versus initial abundance ( N o ) , if
only natural mortality is acting. Thus, the quantity M/n used

The usual assumption in VPA is that F is constant over thetime interval under consideration. This assumption requiresthat the catch decline exponentially over that interval. Many, if
not most, fisheries are seasonal, so the pattern of catches over
the year or season does not conform to this assumption (Fig. 2).
A standard practice in these cases is to divide the fishing year
into shorter time periods such as quarters or months, within
which F is more nearly constant.
If the seasonality of the fishery is predictable, a useful agproximation can be developed as follows. If the exact time (ti)of each catch (Ci) were known, VPA equation (8) would reduce towhere to is the beginning of the fishing year. Equivalently, if
the proportion (pi) of the total catch (C,) taken at time ti is
known, we have

Equation ( 16) has the same form as equation (8), demonstratingthat seasonal fisheries can be treated by the same family of
approximations. Moreover, equation ( 16) requires infomation
only on M and the temporal distribution of catches, making
function V,,,,, and hence the VPA, independent of estimated F .
In practice, the value of V,,,, for a seasonal fishery can be
Can. J . Fish. Plquaf. Sci., Vol. 4.3, 1986

obtained from sequential VPA sf the typical monthly proportions of the total annual catch (see example below). By setting
N , + to zero, V,, is given directly by the virtual populationof the p , values at time to, this being equivalent to the summation in equation (84).

,Can. J. Fish. Aquat. Sci. Downloaded from cdnsciencepub.com by HOKKAIDO DAIGAKU on 09/30/21
For personal use only.

Application sf Approximations
Panish and MacCall (1978) gave VBA estimates for mackerel (Scsmber ~ ~ ~ O P Z ~ Cbased
U S ) ,on annual catch-at-age data,
M = 0.5, and assumed constant F within the fishing season.
Estimated F for fish aged 4 and older in the 1941-42 fishing
I15, giving a V,,
of m%/FA
year (May through April) was I .
= 1.2 152. The monthly proportions of the total unaged annual
catch age based on Anonymous (4944) and are shown in
Fig. 2. These monthly catches are in weight rather than numbers, causing some error due to individual growth during theyear. A value of V,,, can be approximated by letting VmOnth=(M/ 12)/( H -exp( - M /12))- The annual value of V based on
the seasonal catch distribution is approximately
1L(17) V , = C p,Vmthexp((i - 1)M/12)
i -1

giving $I,, = 4.285 1. Thus, the relative enor of Barrish and
MacCal19sannually based VPA is approximately -5.4% forthis year, indicating a moderate underestimate of abundance,
given the assumptions. Interestingly, the new VPA approximation in equation (12) gives V,,, = I .2707, and Pope's approximation gives a nearly exact value of Vp,
1.2840.
Pope's approximation does well because in this case the actual
catch was indeed taken very near midyear. Of course, otherseasond distributions may produce values of V,,, ranging from
B .O (all the catch taken at yea's end) to exp(M) if all of thecatch is taken at the beginning of the year.
Differences in VPA estimates of abundance due to various
homogeneity assumptions and/or approximations often fallwithin the variability due to imprecision in the vdue of thenatural mortality rate, M. For example, given M = 0.5 andconstant F = 0.5, Pope's approximation using M = 0.48 would
have the same relative error as the new approximation using
M = 0.5. The difference becomes larger for large M: given
M = I .O and constant F = 1.0, Pope's approximation would
require M = 0.92 to produce a relative error equal to that of the
new apprsximatisn. Violations of the assumption of constant F
over the time period can produce somewhat larger errors. The
relative error of Pan-ish and MacCall's annual %"PAin thepreceding example is roughly equal to the relative error which
would have occurred if they had used a seasonally corrected-

Can. J . Fish. Aqrtat. Sci., Vol. 43, 1986

VBA with M = 0.39, which is 22% smaller than assumed M.
It can be argued that the choice of approximations is inconsequential became imprecision associated with alternative
VPA approximations and assumptions is smaller than that associated with imprecision in assumed M . A counterargument isthat e m r s due to approximations or assumptions are systematic
biases and should be recognized as such. If a value sf M isstated,VPA using that M should as nearly as possible reflectthat value, rather than some systematically different effective
value arising from the properties of an unnecessary assumption
or approximation.

This work was supported in part by the Ocean Assessments Divisionof the National Ocean Service, U.S. National Oceanic and Atmospheric Administration. 1 wish to thank Daniel Kirnura and William
Lenarz (U.S. National Marine Fisheries Service) for allowing me toread prepublication copies of their manuscripts on VPA of nonhomogeneous stocks, which started me thinking a b u t the implications sf equations ( I ) and (2), and for their insightful comments onan early draft of this paper. Also, Doug Butteworth md Patricia
Symington (University of Cape Town) provided useful ideas on how
to compare approximation formulas, both mathematically and graphically.

References
ANONYMOUS.
1944. The commercial fish catch of California for the years 8 94 1
and 4942., Calif. Div. Fish Game Fish Bull. 59: 71 p.
DOUBLEDAY,
W. G. 1974. A least squares approach to analyzing catch at age
data. Res. Bull. lnt. Comm. Northeast Atl. Fish. 12: 68-8 1 .
GULLAND,
J . A. 1965. Estimation of mortality rates. Annex to Arctic Fisheries
Working Group Rep., ICES CM 1965, Doc. No. 3: 9 p. (Mimeo.)
LAWLESS,J . F. 1982. Statistical models and methods for lifetime data. John
Wiley & Sons, New York, NY. 580 p.
MIJRPHY,G. 1. 1965. A solution of the catch equation. J. Fish. Wes. Board Can.
22: 191-202.
PARRISH,R. H., AND A. D. MACCALL.1978. Climatic variation and exploitation in the Pacific mackerel fishery. Calif. Dep. Fish Game Fish
Bull. 167: 1led p.
POPE,J. G. 1972. An investigation of the accuracy of virtual population
analysis using cohort analysis. Wss. Bull. lnt. Comm. Northwest Atl.
Fish. 9: 65-74.
RICKER,W. E. 1975. Computation and interpretation of biological statistics sf
fish populations. Bull. Fish. Wes. Board Can. 191: 382 p.
1976. Review sf the rate of growth and mortality of Pacific salmon
in salt water, and noncatek mortality caused by fishing, J . Fish. Wes.
Board Can. 33: 8483- 1524.
S C H N ~9.
E 1985.
,
A general theory for analysis of catch and effort data. Can.
J. Fish. Aquat. Sci. 42: 414-429.
Srlws, %. E. 1982. Algorithms for solving the catch equation forward andbackward. Can. J. Fish. Aquat. Sci. 39: 897-202.
Wn~wrc,%. 1954. Method of estimating fish populations, with application to
Pacific sardine. Fish. Bull. U.S. 54: 141- 166.