F I S H and F I S H E R I E S

A simple method for estimating MSY from catch andresilience
Steven Martell1 & Rainer Froese2
1

University of British Columbia, 2202 Main Mall, Vancouver, BC, V6T 1Z4, USA and 2Helmholtz Centre for Ocean

Research, GEOMAR, Düsternbroker Weg 20, 24105, Kiel, Germany

Abstract
The Law of the Sea requires that fish stocks are maintained at levels that can produce the maximum sustainable yield (MSY). However, for most fish stocks, no estimates of MSY are currently available. Here, we present a new method for
estimating MSY from catch data, resilience of the respective species, and simple
assumptions about relative stock sizes at the first and final year of the catch datatime series. We compare our results with 146 MSY estimates derived from fullstock assessments and find excellent agreement. We present principles for fisheries
management of data-poor stocks, based only on information about catches andMSY.
Keywords Carrying capacity, data-poor stocks, harvest control rules, intrinsic rateof population increase, maximum sustainable yield

Introduction

2

The need for simple methods

2

Outline of the Catch-MSY method

2

Material and Methods

2

Model and assumptions

2

Data sources

2

Results and Discussion

3

Applying the Catch-MSY method to Greenland halibut

3

Applying the Catch-MSY method to other stocks with available MSY estimates4

How good are the estimates of r and k?

5

Applying the Catch-MSY method to the ‘data-poor’ Strait of Georgia lingcod

7

Caveats of the Catch-MSY approach

8

Other methods for estimating MSY from catch data

8

Principles of catch & MSY management

8

Acknowledgements

9References

9

Appendix10© 2012 Blackwell Publishing Ltd

DOI: 10.1111/j.1467-2979.2012.00485.x

Correspondence:
Rainer Froese,
GEOMAR
Düsternbroker Weg
2024105 Kiel
Germany
Tel.: +49 431 600
4579
Fax: +49 431 600
1699
E-mail: rfroese@geomar.deReceived 7 Feb 2012
Accepted 16 May
2012

1

Catch-MSY S Martell and R Froese

Introduction
The need for simple methods
In the Law of the Sea of 1982 (UNCLOS 1982),
which entered into force in 1994, the nations of
the World have agreed to maintain exploited populations of marine organisms at levels that can
produce the maximum sustainable yield MSY.
Respective management systems have been introduced by Australia (DAFF 2007), New Zealand(MFNZ 2008), and the USA (MSA 2006), and Europe plans an implementation by 2013 (EC 2011).
However, for the vast majority of exploited populations or stocks, no estimates of MSY are available.
Thus, there is a need for simple methods thatallow inclusion of such stocks in MSY management schemes.
Outline of the Catch-MSY method
The simplest model-based methods for estimating
MSY are production models such as the Schaefer(1954). At a minimum, these models requiretime series data of abundance and removals toestimate two model parameters: the carrying
capacity k and the maximum rate of population
increase r for a given stock in a given ecosystem.
While estimates of removals (defined here ascatch plus dead discards) are available for moststocks, abundance estimates are difficult andcostly to obtain and are mostly missing. However, given only a time series of removals, a surprisingly narrow range of r-k combinations isable to maintain the population such that it neither collapses nor exceeds the assumed carrying
capacity. This set of viable r-k combinations canbe used to approximate MSY. Here, we present a
simple method that uses catch data plus readily
available additional information to approximate
MSY with error margins. We demonstrate theapplication for two stocks, Greenland halibut(Reinhardtius hippoglossoides, Pleuronectidae) and
Strait of Georgia lingcod (Ophiodon elongatus, Hexagrammidae). We apply the method to 48 stocksof the Northeast Atlantic for which independent
MSY estimates were available from a previous
study and compare the results. We also apply themethod to 98 global stocks with MSY estimates.
Finally, after pointing out some caveats, we propose simple management rules based on catchand MSY.

2

Material and methods
Model and assumptions
The Catch-MSY method as proposed here wasinspired by the stock reduction analysis of Kimura
and Tagart (1982) and Kimura et al. (1984). As
input data, it requires a time series of removals,
prior ranges of r and k, and possible ranges of relative stock sizes in the first and final years of thetime series. It then uses the Schaefer production
model to calculate annual biomasses for a given
set of r and k parameters. As no prior distributionsof r and k are available for most fish stocks, we
randomly draw r-k pairs from a uniform prior distribution and then use a Bernoulli distribution as
the likelihood function for accepting each r-k pairthat has never collapsed the stock or exceeded carrying capacity and that results in a final relative
biomass estimate that falls within the assumed
range of depletion. Additional process errors canalso be added to the model if desired. Absent process errors, as in our examples, are equivalent toassuming an observation error only model that isdeterministic. A detailed description of the parameters and equations is given in the Appendix. The
R-code for batch processing of the 146 stocks andthe catch data are available from http://www.fishbase.de/rfroese/ with file names of CatchMSY_2.r,
RAM_MSY.csv, and ICESct2.csv concatenated to
the URL, respectively.
Data sources
We used assessment data for 48 stocks of 19 species of the Northeast Atlantic, as available in the
ICES Stock Summary Database downloaded fromwww.ices.dk/datacentre/StdGraphDB.asp in September 2011. We extracted estimates of F0.1 fromICES advice documents for 2011, as available fromwww.ices.dk. We also used the estimates of Fmsy,
MSY, and carrying capacity k for these stocks from
Froese and Proelss (2010). For each species, we
got a resilience classification from FishBase (Froeseand Pauly 2011). These stocks spanned a wide
range of sizes and exploitation rates, ranging in
spawning stock biomass from 1000 tonnes to 12
million tonnes, with exploitation rates F/Fmsy of
0.5 to 5.8. The advantage of this data set was theapplication of the same standard methods acrossall stocks, and the provision of MSY with 95%
confidence limits by Froese and Proelss (2010).
© 2012 Blackwell Publishing Ltd, F I S H and F I S H E R I E S

Catch-MSY S Martell and R Froese

The disadvantage of this data set was that, with
one exception, it only contained species with medium resilience. We, therefore, also used working
group assessments of MSY for 98 stocks from the
RAM legacy database (Ricard et al. 2011). For thebatch analysis of these stocks, we derived default
ranges of relative biomass in the first and final
year of the time series, based on respective catches
relative to the maximum catch (Froese et al.
2012), see Table 1.
Random samples of the carrying capacity
parameter (k) were drawn from a uniform distribution where the lower and upper limits were givenby the maximum catch in the time series and 100
times maximum catch, respectively. Note thatsuch upper bound for k means that catches neverexceeded 1% of the carrying capacity. If this wereindeed the case, catches would contain very little
information about the productivity of the stockand the Catch-MSY method should not be applied.
Given their nearly unexploited status, such stocks
are not in immediate need of management.
We used resilience estimates from FishBase,which are based on Musick (1999) as modified by
Froese et al. (2000), to assign default values to theallowed range for the random samples of the maximum intrinsic rate of population increase r(Table 2). Note that we do not propose applicationof the Catch-MSY method with the default values

in Tables 1 and 2 for serious stock assessment.
Rather, we would expect that the best available
knowledge about the respective stocks is used.
As most probable values from the resulting density distributions, we used the geometric means ofr, k, and MSY, where MSY was calculated fromthe r-k pairs (see Appendix). We chose geometricmean instead of mean, median, or mode because
it was the only estimate where the central MSY
value derived after the calculation of MSY for eachr-k pair was about the same as the one derived byusing the respective central values of r and k. For
example, for Western Baltic cod, median MSY calculated from r-k pairs was 38 335 tonnes,
whereas MSY calculated from median r and k was
38 997 tonnes, a difference of 662 tonnes. For
the geometric mean, the respective values were
38 975 and 38 906, a difference of only 69 tonnes. Thus, the geometric mean seemed to better
capture the distributions of r, k, and MSY.
As a measure of uncertainty, we used two times
the standard deviation of the logarithmic mean.
This implies that, with a roughly lognormal distribution, about 95% of the MSY estimates would fallwithin this range.

Table 1 Default values for initial and final biomass,used for batch processing of stocks, where B is biomass,
and k is carrying capacity.

Fig. 1 shows the graphical output of the CatchMSY method as applied to the Greenland halibut,a species with low resilience (Froese and Pauly
2011; see Table 1). Panel A shows the time seriesof catches with overlaid estimate of MSY =24 900 tonnes and the limits (19 800–31 400)that contain about 95% of the MSY estimatesderived from the r-k pairs. This is not significantly
different from an independent estimate for thisstock of MSY = 31 023 tonnes with 95% confidence limits of 19 171–53 950 tonnes (Froeseand Proelss 2010). Panel B spans the prioruniform distribution of r = 0.05 0.5 andk = 89 484 8 948 400 tonnes. The r-k combinations (1st iterations) that are compatible with thetime series of catches occupy only a small cornerof that space, showing the typical decline of viable
r-k pairs with increasing r. Panel C is a magnification of the r-k pairs (after 2nd iterations with newupper limit for k) in log space, with overlaid lines
indicating the r-k combinations that would result

First year
Final year

Catch/max catch

B/k

<0.5
 0.5
>0.5
 0.5

0.5–0.9
0.3–0.6
0.3–0.7
0.01–0.4

Table 2 Default values used for batch processing ofstocks, based on resilience assignments in FishBase,
where r is the maximum intrinsic rate of population
increase.
ResilienceHigh

MediumLow

Very lowr (year 1)

0.6–1.5

0.2–1

0.05–0.5

0.015–0.1© 2012 Blackwell Publishing Ltd, F I S H and F I S H E R I E SResults and discussion
Applying the Catch-MSY method to Greenland
halibut

3

(c)

k

ln(k)

6000

80
60

2000

40

0

0

2010

0.1

0.2

Year

0.20

0.5

–2.5

0.30

–2.0

–1.5

ln(r)

(f)

0.00

Density0.0010 0.0020

Density0.10

0.000015
100Density

(e)

0.00

0.4

r

5

(d)

0.3

0.15

1990

0.10

1970

0.05

Catch (1000 t)

(b)

20

(a)

6.2 6.4 6.6 6.8 7.0 7.2

Catch-MSY S Martell and R Froese

0

r

500

1000
k

1500

0

10

20

30

40

MSY

Figure 1 Graphic output from the Catch-MSY method for Greenland halibut (Reinhardtius hippoglossoides). (A) shows
the time series of catches with overlaid estimate of MSY (bold) and the limits (broken) that contain about 95% of theestimates. (B) frames the prior uniform distribution of r and k; the gray dots show the r-k combinations that arecompatible with the time series of catches. (C) is a magnification of the viable r-k pairs in log space, with the geometricmean MSY estimate (bold) ± 2 standard deviations (broken lines) overlaid. (D-F) show the posterior densities of r, k, andMSY, respectively. In Panel F, geometric mean MSY (bold) ± 2 standard deviations (broken lines) are indicated.

in geometric mean MSY ± 2 standard deviations.
Panels D to F show the posterior densities of r, k,
and MSY, respectively.
Applying the Catch-MSY method to other stocks
with available MSY estimates
The key question obviously is how well the MSY
estimates derived with the Catch-MSY method
compare with a wide range of MSY estimates fromfull stock assessments. For this comparison, weused 48 stocks from the Northeast Atlantic and
98 stocks from all over the world, analyzed withthe default assumptions as described above. Thesedefault settings found r-k pairs for all Northeast
Atlantic stocks and for most of the global stocks.
Fig. 2 shows a comparison of the respective MSY
estimates for the 48 Northeast Atlantic stocks. A
log–log linear regression accounted for 98.6% of
the variability of Catch-MSY estimates relative tofull assessment estimates of MSY, with an intercept not significantly different from the origin(n = 48, log intercept = 0.05, 95% CL = 0.118
4–0.018) and a slope not significantly different from1 (slope = 1.003, CL = 0.967–1.039, r2 = 0.986).
The 95% confidence limits of MSY provided by
Froese and Proelss (2010) overlapped in 42 of the
48 stocks with the double standard deviation used
as an error margin by the Catch-MSY method,suggesting that these MSY estimates were not significantly different.
For the global stocks, the default settings did not
result in suitable r-k combinations for about 10 of
the 98 stocks, mostly because these stocks hadintermediate resilience (between very low and low
or between low and medium, see Table 2), or
because they were very lightly exploited, withmaximum catches of 2–30% of the MSY estimateof the respective working groups, see outliers in
Fig. 3. As pointed out previously, in very lightly
fished stocks, the time series of catches does notcontain sufficient information about productivity,
and the Catch-MSY method should not be applied.
But overall, most of the Catch-MSY estimates forthe global stocks fell within a range of 0.5–1.5 of
the independent estimate, see respective lines in© 2012 Blackwell Publishing Ltd, F I S H and F I S H E R I E S

Catch-MSY S Martell and R Froese

Figure 2 Plot of MSY estimated by
the Catch-MSY method vs. full stock
assessments for 48 stocks from theNortheast Atlantic. The broken line
indicates the 1:1 relation, while thedotted lines indicate ratios of 0.5
and 1.5, respectively.

Figure 3 Plot of MSY estimated by
the Catch-MSY method vs. full stock
assessments for 98 global stocks.
The broken line indicates the 1:1
relation, while the dotted lines
indicate ratios of 0.5 and 1.5,
respectively. The six outliers in thelower-right section of the graph arevery lightly exploited stocks.

Fig. 3. Thus, the Catch-MSY method appears well
suited to provide preliminary approximations of
MSY in cases where abundance data are lacking.
How good are the estimates of r and k?
A stock that was able to produce the cumulative
historical catch must have had a certain productivity, for which the maximum sustainable yield is
an appropriate measure. MSY is a function of r
and k, which in the Schaefer model have a log linear negative correlation with a slope of
1(Fig. 1c). In other words, the observed catches
may have been produced by a small population
with high r or a large population with small r, ora very lightly exploited population with any combination of r and k. In the last case, catch data
are insufficient to estimate population properties,
error margins for r, k, and MSY will be very wide,and MSY will be underestimated (outliers in© 2012 Blackwell Publishing Ltd, F I S H and F I S H E R I E S

Fig. 3). High values of r cause strong fluctuations
in stock size, with the associated risks of overshooting carrying capacity or going extinct (May
1974, 1976). Also, the larger k is relative tocatches, the wider is the range of r values thatallow the population to sustain those catches.
These two effects may explain the typical triangular shape of viable r-k pairs in log r-k space(Fig. 1c), with few pairs at high and most pairs at
lower r values. This triangle only expands by its
short side, when the range for prior r is reduced
and for prior k increased.
The triangular shape of viable r-k space does
not affect much the estimation of a representativevalue for MSY, as the line of r-k pairs giving thesame MSY is anchored near the center of the triangle (Fig. 1c). However, the estimates of mostprobable central values for r and k are strongly
dependent on the lower limit chosen for r and theupper limit chosen for k. While the lower limit for5

Catch-MSY S Martell and R Froeser is assumed to represent the best available priorknowledge, the upper limit for k was chosen arbitrarily as 100-fold the maximum catch in the timeseries. We used the following method for finding
an upper limit of k better corresponding with thelower limit of r, based on the knowledge gained ina first analysis of the data: We selected as upper
limit of k the smallest viable k value at the lower
limit of r. This provided a clear new upper cut off
for k determined by the prior lower limit of r
(Fig. 1c).
In Figs 4–6, we compare the r and k estimatesof the Catch-MSY method with related fisheriesreference points. Figure 4 shows a plot of k over
unexploited total biomass from Froese and Proelss(2010). The points scatter around the 1:1 line,
but with an upward bias of about 10%, that is,
the Catch-MSY method overestimated carrying
capacity and related biomass reference points byabout that amount. Similarly, in Fig. 5, most estimates of r fall below the 1:2 line of the relationship between r and the fishing mortality Fmsy thatwould result in the biomass that can produce
maximum sustainable yield (data from Froese andProelss 2010). A better match is obtained inFig. 6, where r is plotted over the conservative
fishing mortality F0.1 derived by ICES working
groups from yield per recruit analysis (Cadima
2003). The rectangular distribution of the data
points in Figs 5 and 6 stems from the fact thatmost species were of medium resilience and thus
had the same default lower prior limit of r = 0.2.
In summary, while MSY estimates of the CatchMSY method are fairly robust with regard to initial assumptions and in very good agreement withestimates derived with more demanding methods,
the r and k estimates strongly depend on the lower
prior limit for r which thus must be carefully set.

Figure 4 Plot of carrying capacity
k estimated by the Catch-MSY
method over estimates of
unexploited total biomass from
Froese and Proelss (2010). Notethat the majority of points fallabove the 1:1 line (dotted), that is,
the Catch-MSY method tends tooverestimate k and reference points
derived from it.

Figure 5 Plot of maximum
intrinsic rate of population increaser over estimates of fishing pressure
associated with maximum
sustainable yields Fmsy, as obtained
from Froese and Proelss (2010).
Note that most estimates fall below
the expected 1:2 line (dotted), thatis, the Catch-MSY method tends tounderestimate r and derived
reference points such as Fmsy.
6© 2012 Blackwell Publishing Ltd, F I S H and F I S H E R I E S

Catch-MSY S Martell and R Froese

Figure 6 Plot of maximum intrinsic
rate of population increase r overestimates of fishing pressure
associated with F0.1 from yield per
recruit analysis for 40 Northeast
Atlantic stocks with available data.
Note that r estimates of the CatchMSY method are roughly evenly
distributed around the dotted 1:2
line, that is, F0.1 ~ ½ r.

From a management point of view, the bias of r
and k is precautionary, because it suggests higher
thresholds for biomass and lower thresholds forfishing mortality.
Applying the Catch-MSY method to the‘data-poor’ Strait of Georgia lingcod
To demonstrate this simple method for estimating
MSY from catch time series data, we used the historical landings from the Strait of Georgia lingcod
fishery, for which no abundance data were available (Fig. 7) (King 2001). Landings of lingcod
from this region date back to 1889. The commercial fishery largely consisted of a handline fishery;
however, lingcod were also taken in trawl fisheries, starting in the 1940s. The stock was considered depleted and the commercial fishery wasclosed in 1990. The remaining recreational fishery
was closed in 2002 (Logan et al. 2005).

The lingcod fishery began around 1860 (King2001), and thus, we assumed that the biomass atthe start of the time series in 1889 was alreadybelow carrying capacity. As early catches weremodest, we assumed a start biomass of 0.8 k. For
the prior densities of r, we took the uniform distribution for species with very low resilience withr = 0.015 0.1 (Table 1). Given the known
depleted state of the stock, we assumed that relative biomass in 2002 was between 1% and 25% of
the carrying capacity. Inserting the estimated MSY
in Fig. 7 facilitated the interpretation: Landings
reached the lower range of MSY estimates in the
1900s, exceeded MSY in the 1910s and the upper
range of MSY estimates between 1920 and 1960,
and declined thereafter. Despite the closure of thecommercial fishery in 1990, the stock showed no
consistent signs of recovery in 2000 (Logan et al.
2005). A full assessment is still not available forthis stock (DFO 2012), but Fig. 7 suggests that

Figure 7 Landings data for theStrait of Georgia lingcod (Ophiodon
elongatus) fishery, with indication of
MSY and the (dotted) limits that
contain 95% of the estimates, basedon Catch-MSY analysis. The stock
was severely overfished in the
1940s and has not recovered,despite closure of the commercial
fishery in 1990.
© 2012 Blackwell Publishing Ltd, F I S H and F I S H E R I E S

7

Catch-MSY S Martell and R Froesethe massive and prolonged overshooting of MSY
was the cause of depletion in 1990.
Caveats of the Catch-MSY approach
A key assumption in the Catch-MSY approach aslaid out here is the ability to define a reasonableprior range for the parameters of the Schaefer
model. In our case studies, we have arbitrarily
chosen 100 times the maximum catch as theupper bound for k. In a developing fishery, or afishery that has a continuous increase in catch, itwill be more difficult to define the upper bound of
k because the maximum potential has yet to berealized. Another key assumption is the statedrange of depletion for which to accept or reject thesets of r-k pairs of parameters. The lower depletion
limit defines the lower boundary of the resulting
MSY distribution, and the upper depletion limit
and the range of values for k determine the upperbound of MSY. To be clear, these depletion levels
are assumptions about the current state of thestock. Finally, the Catch-MSY approach also
assumes a stationary production function or inthis case, no change in the parameters of the
Schaefer model over time.
Other methods for estimating MSY from catch
data
MacCall (2009) provides estimates of depletioncorrected average catch (DCAC) based on catch
data and estimates of natural mortality or of thedepletion in stock size caused by fishing. As MacCall (2009) points out, this method does not give
estimates of MSY but suggests rather a ‘…moderately high yield that is likely to be sustainable,
while having a low probability that the estimatedyield level exceeds MSY…’ Dick and MacCall(2011) present a depletion-corrected stock reduction analysis (DB-SRA) based on a time series ofannual catches, the rate of natural mortality M,
the Fmsy/M ratio, the age at maturity, the Bmsy/kratio, and an estimate of relative biomass near theend of the time series. The method then applies a
production function and accepts estimates of kfrom biomass trajectories that never became negative. Outputs of the model are MSY and unexploited biomass k. The authors tested their method
on 31 northeast Pacific groundfish stocks off the
US West coast and conclude that ‘[f]or most stocks
we evaluated, median estimates of MSY and K […]8tend to be between one-half and double the assessment value’.
The DB-SRA method is similar to the CatchMSY method we present here. However, the application of the DB-SRA method requires more
knowledge about the respective stock, such as agood estimate of productivity. In comparison, theCatch-MSY method requires as input only fairly
wide ranges of potential productivity, which may
be derived from resilience estimates, and fairly
wide ranges of initial and final relative abundance,
which may be derived from initial and final
catches relative to the maximum catch in the timeseries (Froese et al. 2012).
Wetzel and Punt (2011) evaluate the performance of the DCAC and DB-SRA methods andconclude that both are reasonably robust withregard to errors in natural mortality (used for estimating productivity) but sensitive to overly optimistic depletion ratios.

Principles of MSY-based management
We have shown above that with catch data andsimple assumptions about resilience and the statusof the stock, reasonable estimates of MSY witherror margin can be obtained. But what are harvest control rules for management based only oncatch and MSY? An obvious first rule is thatcatches shall never exceed MSY and that the lower
margin of error of MSY should be used as target
for total allowable catch if stock size can beassumed to be above 0.5 k. Note that this lower
margin will be further decreased if process error is
included in the model. Stock size is likely to be
below 0.5 k if any of the following observations istrue:
1. Catches in the past have exceeded MSY;
2. Catch per unit of effort (CPUE) is stable ordecreasing, instead of slightly increasing, as
can be expected as result of ‘effort creep’ (Marchal et al. 2007);
3. Mean length in the catch has declined andthe proportion of large fish is less than 30-40%
of that known from the beginning of the fishery(Froese 2004).
If an overfished status of the stock cannot be
excluded, catches should be reduced strongly until
increases in CPUE and increases of maximum
length in the catch indicate a recovery. Catches
can then be increased slowly, that is, slower than
© 2012 Blackwell Publishing Ltd, F I S H and F I S H E R I E S

Catch-MSY S Martell and R Froese

the expected increase in biomass, until the lower
error margin of MSY is reached.
We hope that this simple method and the proposed principles will prove useful in the management of data-poor stocks.
Acknowledgement
We thank Daniel Pauly for constructive criticism
and for reminding us not to destroy ‘the beautiful
simplicity of the thing’ by too many additional
assumptions. We also thank Selina Heppell for
inviting us to a workshop that inspired this idea.
Steven Martell acknowledges financial support fromNSERC Discovery Grant. Rainer Froese acknowledges support by the Future Ocean Excellence Cluster 80, funded by the German Research Foundation
on behalf of the German Federal State and State
Governments. He also acknowledges support fromthe European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreement
no. 244706/ECOKNOWS project. However, thepaper does not necessarily reflect the views of theEuropean Commission (EC) and in no way anticipates the Commission’s future policy in the area.
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Appendix 1
The method outline here for approximating MSY
is based on a very simple Schaefer production
model, and it should be noted here that othermodels with alternative assumptions about theform of the stock productivity relationship could
be substituted with the additional structural
assumptions. The primary objectives of thismethod are (i) to devise a very simple methodthat can be applied to any catch time series, (ii)the method must be easy to understand andimplement so that it can be used by many people
involved in fisheries science and management,and (iii) the method requires few additional
assumptions.
The minimum data requirement is a catch timeseries from a specific area that is normally defined
as a unit stock where the population is closed toimmigration and emigration (Table A1, equation1). In addition to the catch data, a range of initial and current depletion levels (i.e., the current
stock size relative to the unfished carrying capacity) must also be specified; these are denoted by
k01 and k02 for the initial stock size and by k1
and k2 for the final lower and upper limits,
respectively. The last remaining assumption is tospecify the standard deviation in the processerrors rm; process errors are assumed lognormal,
independent, and identically distributed (10). If
rm = 0 , this is equivalent to assuming a deterministic model. The model parameters (4) ofinterests are the carrying capacity k and the maximum intrinsic rate of population growth r. Starting with an assumed relative biomass of B1 = k01
k in the first year, biomass in subsequent years is
calculated based on (6), where the observed catchis subtracted from the start of the year biomass.
This assumes the catch is measured without
error, unless rm > 0. This is repeated for additional initial relative biomasses, in steps of 0.05
between k01 and k02.
10

un.org/Depts/los/convention_agreements/texts/unclos/
unclos_e.pdf (accessed on 4th January 2010).
Wetzel, C.R. and Punt, A.E. (2011) Model performancefor the determination of appropriate harvest levels in
the case of data-poor stocks. Fisheries Research 110,
342–355.

A very simple importance sampling procedure isthen used to map the joint distribution of model
parameters (in this case, r and k of the Schaefer
production model) that lead to current depletionlevels between k1 and k2. In cases where combinations of (r, k) lead to the population going extinct
or overshooting k before the end of the time series,
we simply assign 0 for that parameter combination. For combinations of (r, k) that result in final
stock sizes between k1 and k2, we assign a valueof 1 (equation 7). Then, for each parameter combination that results in a viable population at theend of the time series, estimates of MSY can becalculated from the population parameters (11).
The basic algorithm is implemented as follows:
1. Specify the initial status of the stock (k01
and k02) and lower (k1) and upper (k2) limits ofthe final status of the stock (e.g., values of
k01 = 0.5 and k02 = 0.9 imply that the stock
was between half and 90% of carrying capacity
at the beginning of the time series, and k1 = 0
and k2 = 1 imply that the stock is somewhere
between completely depleted and at its carrying
capacity at the end). Also, specify rm to a value
greater than 0 if you wish to include a stochastic component.
2. Draw a trial parameter set Θi from therespective prior distributions (e.g., equations 8,9, and 10).
3. Initialize the population model at the trial
value of ki (5).
4. Update the biomass next year using the
Schaefer production model (6).
5. Calculate the likelihood of the parameter
vector Θi using (7).
6. Repeat steps 2–5 many times (e.g.,
100 000) and store the 0 or 1 likelihood foreach trial.
7. Plot distributions of management quantities(11) only for cases in which the likelihood is 1.

© 2012 Blackwell Publishing Ltd, F I S H and F I S H E R I E S

Catch-MSY S Martell and R Froese

Table A1 A simple Schaefer production model and the corresponding management parameters.
Data
ct observed catch from t = 1 to t = n years
k01, k02 lower and upper bounds for relative biomass in year 1
k1, k2 lower and upper bounds for depletion levelrm process error standard deviation

(1)
(2)
(3)

ParametersΘ = {k, r}(4)

Initial states t = 1
Bt = k0 k exp(m t )(5)

Dynamic states t > 1
Bn+1 = [Bt + r Bt (1 Bt/k )Likelihoodl(Θ|ct) = 1=0ct] exp(m t )k1  Bn+1/k  k2
k1 > Bn+1/k > k2

Prior densities
p(log(k)) ~ uniform(log(l k), log(u k))p(log(r)) ~ uniform(log(l r), log(u r))p(m t) ~ normal(0, rm)
Management quantities
MSY = ¼ r k
Bmsy = ½ kFmsy = ½ r© 2012 Blackwell Publishing Ltd, F I S H and F I S H E R I E S

(6)
(7)

(8)
(9)
(10)
(11)

11