Abstract.-Virtual populationand cohort analyses are sometimes
conducted with an age category
known as the "plus group." This
category is used to keep track ofthe
abundance and catches ofolder fishthat cannot be assigned individualages accurately. In this study we
present a procedure for solving thecatch equation backwards in time
when it involves a plus group. The
procedure consists ofan initial analytical approximation, followed bya correction function based on an
empirical analysis, followed by theapplication of Newton's Method.
The results indicate that the procedure works well for a wide rangeof natural and fishing mortality
values. For comparison, we also
applied the same procedure to thebasic catch equation (without aplus group) and found that adequate approximation to the true
fishing mortality value is achieved
even before application of Newton's
Method.

Manuscript accepted 12 September 1994.
Fishery Bulletin 93:308-314 (1995).
308

Approximations for solving the catch
equation when it involvesa Ilpl us group"·
Victor R. Restrepo
Christopher M. legaUlt
Division of Marine Biology and Fisheries
Rosenstiel School of Marine and Atmospheric Science
University of Miami, 4600 Rickenbacker Causeway, Miami, FL 33 J 49

A solution to Fa,t in the catch equation
N

a+l.t+l -F

c a.t Za,ta,t.
(e Z• J -1) ,(1)

where a denotes the age, t denotes
the time period, Ca,t is the catch innumbers, N a+1, t+l is the cohort sizeat the start of the following time
period, F a,t is the fishing mortality
rate and Za,t is the total (fishing +natural) mortality rate (Za,t =Fa,t +Ma,t), is required in many stock assessment problems that track the
exploitation history of a cohort (cohort and virtual population analyses). Here, Equation 1 is expressedin a form known as "backward intime," meaning that a solution to F
for age a during time period t is tobe obtained given known values ofCa,t , N a +1,t+l' and Ma,t.
An analytical solution to Fa,t is
not possible and this prompted several authors to develop useful approximations (see Pope, 1972; Sims,
1982; MacCall, 1986; Allen and
Hearn, 1989). Most current computer
implementations of age-structured
models use numerical algorithms tosolve the catch equation; therefore
approximations are seldom used instock assessment applications. However, approximations provide initial
estimates that may improve the efficiency of the numerical algorithms,
an important consideration when thealgorithm is used repeatedly. More
important, as with analytical solu-

tions, simple approximations enable
scientists to explore relationships
between variables easily.
The objective of this study is topresent a simple procedure for solving a catch equation that involves a"plus group." A plus group lumps
together a number of the oldest age
classes in a population into a single
age category. For example, a plus
group may be used when large (old)fish in the catches cannot be aged
with a desired degree ofaccuracy or
precision as can smaller fish (Restrepo and Powers, 1991). We are notaware of definitive analyses thathave been conducted to evaluate thegeneral merits of data aggregation
into a plus group. Using simulated
data with ageng errors in an agestructured model, Fournier andArchibald (1982) found that using
a plus group gave better results
than did ignoring the ageing errors
and extending the analyses to a last"true" age. Deriso et al. (1989), alsousing simulated data in an agestructured model, found that aggregation ofolder fish into a plus groupproduced comparable estimates tothose obtained from disaggregateddata. However, Hiramatsu (1992)
warned that use of a plus group in
virtual population analyses could
cause large biases under certain cir• Contribution No. 9410501 of the Cooperative Unit of Fisheries Education andResearch, Rosenstiel School of Marine
and Atmospheric Science, University ofMiami, Miami, FL.

309

Restrepo and Legault: Approximations for solving the catch equation

cumstances. It seems that the decision of whether or
not a plus group should be used in an assessment depends on the ageing errors and characteristics of the
population being assessed (Restrepo and Powers, 19911.
The basic plus group dynamics can be presented
as follows. Assume that M is the same for the plus
group and the previous age (we will drop age andyear subscripts for M) and letp denote the plus groupage category. The number of fish alive at the beginning of time period t+ 1 is given by

N p,t+l -- N p,t e- zp .t + N p-l,t e- Zp- 1., ,(.2)

and the corresponding catch equation isN

p.t+l -

CP,t Z p,t
Fp,t (eZ

P.t -Cp-l,t Z p-l,t+1)

Fp-l.t (e

Z

p-l.'

(3)'

-1)

Given known values of Cp,t , C p _1, t , N p , t+l' and M,
no unique joint solution exists for the two unknowns,
Fp,t and F p - 1.t in Equation 3. In this study, we use thesimplifying assumption that the value of a inFa=~purposes, we also use the same approach for catch
equations not involving a plus group.

Range of values examined
In order to evaluate the accuracy of the approximations and to develop empirical corrections, we generated 1,000 uniform random values of each parameter in the following ranges: N p , t [500, 1,500], N p _1, t[500, 1,500], M [0.05, 1.0], Fp_l. t [0.05, 3.0], a [0.5,2.0]. These ranges are arbitrary but we felt that they
represent realistic extremes: the stock size ofthe plus
group and the preceding age can differ by a factor of
3; the range in M is representative of a wide range
in lifespans; the fishing mortality range is extremely
wide because our purpose was to find reasonableapproximations for a wide range of Fs; and the range
in a allows the fishing mortalities of the plus groupand the previous age to differ by a factor of 2.

Initial approximations
Pope's (1972) approximation to Fa. t in Equation 1
can be explained as follows. Consider the equations(from Appendices A and B in Pope, 1972)(4)
N a,t --Na+l,t+l e Z,

Fp - 1. t
is known so that Fp , t in Equation 3 can be replaced
by aFp _ 1, t. Now the solution consists of a single fishing mortality rate, F p _1, t. Theoretically, one or more
a values can be estimated as parameters in an agestructured model. However, estimation of several avalues is difficult, particularly for the last years forwhich catch data are available (Powers and Restrepo,1992). Thus, in many assessment applications, avalues are assumed from a knowledge of the population and the fishery being examined (Powers andRestrepo, 1992). For example, selectivity studies ofthe fishing gear may indicate that fish of ages p-1
and older are equally vulnerable, giving a = 1.

(5)

and
N

a+l,t+l

eM =N

a,t

Z (l-e- Fa .,)a.ta,t F (1- e-za.,)

-C

(6)

a,t

Pope (1972) demonstrated that, over a range offishing and natural mortality values, the function
multiplying C a , t in Equation 6 can be reasonably
approximated by e(M/~). Making use of this approximation and substituting Equation 5 into Equation 6
gives
N a+l,t+l e M -- N a+l,t+l e zo.t _ Ca,t e M /2 ,which can be solved for Fa, t as

Approach
The approach we follow is similar to that used by
Sims (1982). We first provide simple approximations
to F p _ 1, t, similar to those developed by Pope (1972).
On the basis of simulated parameter values, we thenempirically estimate correction factors that can beused to improve upon the initial approximations.
Finally, we use the corrected approximations as starting guesses for Newton's Method (see Sims, 1982),
which can be used to obtain a more accurate numerical solution to the catch equation. For comparative

Fa,t

=In[

Ca,t
eN a + 1•t + 1

M/2 + 1] .

(7)

We followed a similar approach for the purpose ofobtaining an initial analytical approximation to Fp-l, t
in Equation 3. Consider the special case when thefishing mortalities ofthe plus group and the preceding
age are the same (i.e. a=l in Eq. 4, giving Fp, t= Fp-l, t).
Then, the equation analogous to Equation 5 is obtained from Equation 2:

310

Fishery Bulletin 93(2), 1995

and the equation analogous to Equation 6 is-C

eM == N

N
p,t+l

p.t

Fp I
Zp_l,t(1-e- - .,) +N
P,t F
(1
-I')
p-l,t
p-l.t
-e -z p,(9)

Z p- 1,t (1-e- Fp- I . , ) ,_C

p-l.t Fp-l,t(1-e-Zp-I.,)of parameter values; 2) to plot values of the ratio
AF/ TF against (l and M (parameters usually assumed
to be known a priori) and to identify types of functions that can adequately describe the observed relationship, if any; and 3) to estimate the coefficientsof such functions and use them to correct AF so it
becomes closer to TF. This empirical approach is similar to the multiple regression approximations suggested by Allen and Hearn (1989). For comparative
purposes, we also followed the same approach using
the approximation given by Equation 7 for a casewithout a plus group.

which, using the same approximation given by Pope
(1972), becomes

Newtons Method

Substituting Equation 8 into the last equation results in

Sims (1982) suggested the use of Newton's Method
for solving F in the catch equation with a desireddegree of precision. For Equation 1, the functional
equation of interest is

Zp_I' [C+C p-l.t ) e -M/2 ,N p,t+l e M -N
p.t+l e. p.t

C Z«F)a,t a,tNa,t - F (e z•., -1) a+l.t+l,a,twhich can be solved for the fishing mortality rate as
F

p-l.t

== In[(C P.t +Cp-l. t ) e- M / 2 + 1]
N

•

(10)

p,t+l

We found that Equation 10 is often a poor approximation for values of (l # 1 (see Results section). A
much better approximation can be obtained by introducing (l into the equation as

Fp-1,t == In[(Cp't la+Cp_l,t) -M/2
]
N
e
+1 .
p.t+l

and a solution to Fa, t is obtained when {(Fa. t) == O.
Sims (1982) showed that all requirements for convergence were met in order for Newton's Method toconverge to that solution. One iteration of Newton's
Method <denoted by i) changes the estimate of Fa, t
as follows:.
.
(«Fa,t(i»)
Fat(t+l)==Fat(t),.
f' ( Fa•t (i) )

(12)

(11)
with

Empirical correction factors
In many cases, given the widespread availability ofcomputers, the approximation given by Equation 11
can be used as an adequate starting guess for an iterative procedure to get a more accurate solution (e.g.
see the next section). In some cases, however, it isdesirable to improve upon this approximation in order to obtain either a better starting guess, or to obtain as close as possible to an accurate solution because iterative computations are expensive. The latter is the case of solving multiple catch equations
while conducting a cohort analysis on a computer
spreadsheet and is the motivation for this study.
The empirical approach we used was simple. Denote the approximation in Equation 11 byAF and thetrue fishing mortality by TF. Our approach was 1) togenerate a large number of plausible combinations

For the application of Newton's Method to the solution to the catch equation involving a plus group, thefunctional equation of interest is (from Equation 3)
{(F

p-l.t

)==

Cp t (a Fp _ 1 t + M)',

aF
(eaFp-I,,+M -1)
p-l.t+ C p-l,t (Fp-1,t + M) _ N
F

p-l,t

(eFp-I.,+M -1)

p,t+l

and its derivative with respect to F p _ 1, t is

31 J

Restrepo and Legault: Approximations for solving the catch equationf'(Fp-1,f)

tio indicated an overall 3% bias and the largest error was slightly greater than 8%.
Visual inspection of a plot of AF/ TF against M indicated that a linear relationship would improve theapproximation. We fitted the model AF/ TF = a + b M
by minimizing the sum of absolute deviations between the observed ratios and those predicted by themodel. The parameter estimates were=a = 0.9970, and b = 0.0808.

One iteration of Newton's Method proceeds in thesame manner as explained above in Equation 12, if
the empirically corrected estimates of fishing mortality (see previous section) as initial values are used.
Although we did not carry out a rigorous analysis ofconditions for convergence, as Sims (1982) did, we
did not encounter any cases where an iteration did
not result in an improvement.

Results
Table 1 provides some statisticsof the ratioAF/ TF for the 1,000
random combinations ofinputs,
with and without a plus group.
In each case, the first column
provides these statistics for the
initial approximation (AF from
Equation 7 for the case without
a plus group and AF from Equations 10 and 11 for the plus
group approximation). The next
column provides the statistics
for the ratios after an empiricalcorrection function is applied (thecoefficients of these correction
factors are presented in the following subsections). The last twocolumns give the statistics after
one and two iterations of Newton's Method. Implications ofthese results are explained in
more detail below.

emp.A F = init.A F I(a + bM).

(13)

Much improvement in the approximation was obtained by use ofthis simple correction function (Table
1, second column, and Fig. 1, middle panel). The largest observed error was now 3%, which compares favorably with the errors reported by Pope (1972) overa much narrower range of fishing and natural mortality rates. Application of a single iteration of
N ewton's Method resulted in virtual conver-

Table 1
Summary statistics for 1,000 ratios of the approximated fishing mortality rate to
the true fishing mortality rate CAF /TF, see text for a description of how the 1.000
realizations were madel. Case I represents analyses not using a plus group. Case II
represents analyses with a plus group. Each column corresponds to a different stepin the analyses, progressing from an initial approximation (two initial approximations are given for Case II), to an empirically corrected approximation, to the firsttwo iterations of Newton's Method. CV =coefficient of variation.

Case I: DO plus group
Initialapprox.
(Eq.7lMeanMedian
CV
Min.
Max.

1.03501.03180.02171.00071.0826

Empirical
correction(Eq.13)

Newtoniter. 1
CEq. 12)

Newtoniter. 2
CEq. 121

0.99691.00000.00690.97001.00461.00001.00000.00001.00001.00011.00001.00000.00001.00001.0000Case II: plus group
Initialapprox.
IEqs. 10, 11)Case I: without a plus
group
The initial approximation provided by Equation 7 (from Pope,
1972) was reasonable, as expected (See Table 1 and Fig. 1,
top panel), The mean AF/ TF ra-

Thus the empirical correction to the initial approximation to F was

MeanMedian
CV
Min.
Max.

1.11931.14180.18050.63151.67571.0663
1.06600.0876
0.7654
1.3493

Empirical
correction
CEq.14l

Newtoniter. 1
CEq. 12)

Newtoniter. 2
CEq. 12)0.99761.00000.03680.87781.1204

0.99830.99950.00280.98141.00001.00001.00000.00000.99941.0000312

Fishery Bulletin 93(2). J 995

gence to the true F values (Table 1 and Fig. 1, bottom panel).

4 ----,----,----...--------,

3

2

o

3
lI..
"C
Gl

iii
.§
><

2

o

Case II: with a plus group
The initial approximations obtained for the plus
group problem were rather poor compared with thoseof the catch equation without a plus group (Table 1,
Fig. 2, top panell. This was not unexpected, because
the plus group catch equation is not amenable to algebraic manipulations that lead to analytical approximations. However, note that the initial approximation from Equation 11 was much better than thatfrom Equation 10: the observed AF/ TF ratios indicated smaller biases and a tighter approximation
overall (see Table 1). (Note: Subsequent statistics anddata reported in Table 1 and Figure 2 are .based on
the approximation given by Equation 11.)
In order to find empirical correction factors, weplotted the observed AF/ TF ratios against M (for different ex values) and against ex (for different M values) (see Fig. 3). Visual inspection of these figuresindicated that 1) the relationship between AF/ TF andM could be approximated by a linear model; 2) the
relationship between AF/ TF and ex could be approximated by a logarithmic model; and 3) there was aninteraction between M and ex in terms of explaining
variability in AF / TF. Therefore, we fitted the
model

ii
Q.
<

o -jr:-._,--_+--_...--_+-_-.--_--j

3-j----+-----+----~

to the observed ratios, again by minimizing the sumof absolute residuals. The empirical correction factor used was thenemp.A F =.init.A F I(ao + b1ln(a)2 -j----+--------:;,....-----j+~M+bsaM)

(14)

with

o ----1F--,---+--...---+---.------lo2

3True F

Figure 1
Progression in the approximation to the fishing mortality rate for a catch equation not involving a plus
group (400 ofthe 1,000 pairs of approximated and true
F are shown). (Top) Initial approximation (Eq. 7);(Middle) approximation after application of empiricalcorrection function (Eq. 131; (Bottom) approximation
after one iteration of Newton's Method (Eq. 12).

ao =. 0.9951,
b I = 0.2053,
b2 =. 0.0636, andba =. 0.0161.
This empirical correction function provided a substantial improvement in the approximations (Table
1, Fig. 2, middle panel). However, solution errors on
the order of 12% were still obtained after the correction. One iteration ofNewton's Method was sufficient
to reduce the errors to within 2% (Table 1 and Fig. 2,
bottom panel) and the second iteration resulted in
virtual convergence (Table 1).

Restrepo and Legault: Approximations for solving the catch equation

313

4

1.4

-

......
I'l<.

....; ~.- -.
..: ..."io-" ..
I, • •., -. . ..,.

-

•

~:.

3

-

CD
:::I

t=.

l<.

.,..~

..... .. . ....:.

"C
CD{~ ~ ~.:,

2

Oi. '".)If:".. ....

-/""..'-.

a =0.5

';(

e
c.

.

~:r..:,• f~or·

1.0

E

~.

-

a =2.0

1.2

Q.

<

o.s

0.6

0

0.2

I

0.4

O.S

0.6

1.0

M

1.4

l<.

3
l<.
Ql

iii

E

';(
0

,

... ,#I

2

Q.

-

Q.

:::I

t=.

)~'f:
.., .
J1-'f!:.

..

-

"C

Gl

~;

Gl

Oi

E
';c
0

Q.

~.

1.0

c.

<

/

o

M=0.05

"C

./

<

M=1.0
1.2

I,L.

o.s ........L-........_--,-_---,._---._---,_---,
I

I

0.5

1.5

1.0

2.0

Alpha
3

Figure 3

.1//

2

//

-

1/.

o

-/
o

2

3

True F

Figure 2
Progression in the approximation to the fishing mortality rate for a .catch equation involving a phl,s group(400 oftl).e 1.000 ,pairs of approximated and true Fareshown). (Top) Initial approximation CEq. 11); (Middle)
approximation after.application ofempirical ~orrection
function (Eq. 14); (Bottom) approximation after one
iteration of New.ton's Method (Eq. 12).

Summary
The r~sults of ou,r study indicate that the empiricalcorrection in Equation 13 applied to Pope's (1972)

Example of visual analysis to determine the shape of reasonable empirical correction functions. (Top) Values of theratio of initial approximated F to the true F (AF / TF) as afunction of M for two levels of a, with a linear fit; (Bottom) AF/TF values as a function of a for two levels of M,
with a loga·rithmic fit. S~ text for definition of.par;imeters.

approximation CEq. 7) provides an accurate solution
to the catch equation that does not involve ~ plus
group (Table 1, Case I). Over a wide range of plausible fishing and natural mortality values, Equation
7 gives errors of up to 8% whereas the ~mpirical correction giv~s error~ of up ~ 3%. These errors arepractically eliminated after one iteration .ofNewton's
Method following the empirical correction.
For thecateh equation that involves a plus group,
the initial approximations ana1og~)Us to Pope's (1972)
approxix:nation may not be very accurate. For f;l wide
range of plausible mortality values, eITors of up .to
68% and 35% are obtained from the useof;Equactions
10 and n, respectively (Table 1, Case II). The .empirical corre~tion in Eq1,lation 14 applied to the approximation in Equation 11 re<luces the errors towithin 1.2% of the exact solution. One iteration of
Newton's M~thod following the empiric.al correction

314

reduces the errors to under 2%, and a second iteration practically eliminates the errors.
The computational requirements of our approachfor obtaining a solution to the catch equation that
includes a plus group are small. If errors up to 12%
can be tolerated, the initial approximation (Equation 11) and the subsequent empirical correction(Equation 14) can be incorporated into a single formula. One or two additional computations from theapplication ofNewton's Method substantially reduceor eliminate the biases altogether.

Acknowledgments
We are grateful to an anonymous reviewer for comments that helped clarify our presentation. Support
for this study was provided through the Cooperative
Unit of Fisheries Education and Research (CUFER)
by National Oceanic and Atmospheric Administration(NOAA) Cooperative Agreement NA90-RAH-0075.

Literature cited
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1989. Some procedures for use in cohort analysis and otherpopulation simulations. Can. J. Fish. Aquat. Sci. 46:483-488.
Deriso, R. B., P. ft. Neal, and T. J. Quinn II.
1989. Further aspects of catch-age analysis with auxiliary
information. In R. J. Beamish and G. A. McFarlane (eds.J,

Fishery Bulletin 93(2). 1995

Effects of ocean variability on recruitment and an evaluation ofparameters used in stock assessment models, p. 127135. Can. Spec. Publ. Fish. Aquat. Sci. 108.

Fournier, D., and C. P. Archibald.
1982. A general theory for analyzing catch at age data.
Can. J. Fish. Aquat. Sci. 39:1195-1207.
Hiramatsu, K.
1992. Possible biases in the VPA estimates of population
sizes of the plus group. ICCAT (Int. Comm. Conserv. AU.
TunasJ Collect. Vol. Sci. Pap. 39:497-502.
MacCall, A. D.
1986. Virtual population analysis (VPAJ equations fornonhomogeneous populations, and a family of approximations including improvements on Pope's cohort analysis.
Can. J. Fish. Aquat. Sci. 43:2406--2409.
Pope, J. G.
1972. An investigation of the accuracy of virtual population analysis using cohort analysis. ICNAF lInt. Comm.
Northwest AU. Fish.) Res. Bull. 9:65-74.
Powers, J. E., and V. R. Restrepo.
1992. Additional options for age-sequenced analysis.
ICCAT (Int. Comm. Conserv. AU. Tunas) Collect. Vol. Sci.
Pap. 39:540-553.
Restrepo, V. R., and J. E. Powers.
1991. A comparison ofthree methods for handling the Mplus"
group in virtual population analysis in the presence of ageing errors. ICCAT (Int. Comm. Conserv. Atl. Tunas) Collect. Vol. Sci. Pap. 35:346--354.
Sims, S. E.
1982. Algorithms for solving the catch equation forward andbackward in time. Can. J. Fish. Aquat. Sci. 39:197-202.