ICES Journal of
Marine Science
ICES Journal of Marine Science (2013), 70(6), 1075– 1080. doi:10.1093/icesjms/fst105
Short Communication
Can stock – recruitment points determine which spawning
potential ratio is the best proxy for maximum sustainable
yield reference points?
Christopher M. Legault* and Elizabeth N. Brooks
National Marine Fisheries Service, Northeast Fisheries Science Center, 166 Water Street, Woods Hole, MA 02543, USA
*Corresponding Author: tel: +1 508 495 2025; fax: +1 508 495 2393; e-mail: [email protected]
Legault, C. M., and Brooks, E. N. 2013. Can stock– recruitment points determine which spawning potential ratio is the best proxy for maximum
sustainable yield reference points? – ICES Journal of Marine Science, 70: 1075 – 1080.
Received 17 April 2013; accepted 30 May 2013
The approach of examining scatter plots of stock– recruitment (S– R) estimates to determine appropriate spawning potential ratio (SPR)based proxies for FMSY was investigated through simulation. As originally proposed, the approach assumed that points above a replacement
line indicate year classes that produced a surplus of spawners, while points below that line failed to achieve replacement. In practice, this has
been implemented by determining Fmed, the fishing mortality rate that produces a replacement line with 50% of the points above and 50%
below the line. A new variation on this approach suggests FMSY proxies can be determined by examining the distribution of S – R points that
are above or below replacement lines associated with specific SPRs. Through both analytical calculations and stochastic results, we demonstrate that this approach is fundamentally flawed and that in some cases the inference is diametrically opposed to the method’s intended
purpose. We reject this approach as a tool for determining FMSY proxies. We recommend that the current proxy of F40% be maintained as
appropriate for a typical groundfish life history.
Keywords: maximum sustainable yield, proxy reference points, replacement line, stock – recruitment.
Basis for current FMSY proxy for groundfish
Determination of stock status relies on the ability to estimate current
stock size relative to maximum sustainable yield (MSY)-based reference points. In cases where MSY-based reference points cannot be
estimated directly, proxy reference points are necessary. For many
groundfish stocks in the Northwest Atlantic, the time-series of
data used in stock assessments do not provide sufficient contrast
in spawning stock biomass (SSB) to defensibly estimate a stock–
recruitment (S –R) curve. Given recent work on the difficulty of
fitting S –R functions (Conn et al., 2010; Lee et al., 2012), we
expect that proxy methods are more the norm than the exception.
Proxy reference points based on spawning potential ratio (SPR)
have been shown to be robust to uncertainty in the underlying S–
R function and associated biological parameters (Clark, 1991;
Williams and Shertzer, 2003). SPR expresses the fraction by which
fishing mortality (F) reduces a recruit’s lifetime reproductive
output (Gabriel et al., 1989; Goodyear, 1993). An FMSY proxy reference point of F40%, the fishing mortality rate that reduces a recruit’s
lifetime reproductive output by 40% relative to unexploited conditions, was recently adopted for management of many groundfish
stocks in the Northwest Atlantic after attempts to fit S–R curves
were rejected (NEFSC, 2008). Justification for selecting F40% was
based on the work by Clark (1991, 1993), which explored groundfish
life histories “close to the typical pattern, as exemplified by the listing
of New England stocks in Table 1” (Clark, 1991, pp. 737 –738).
Clark (1991) started with known S –R functions and attempted
to find an F%SPR that would provide a large fraction of the true
MSY. F35% was recommended based on deterministic analyses
(Clark, 1991); however, this was revised to F40% after incorporating
the effects of random or serially correlated recruitment variation
(Clark, 1993). Both F35% and F40% were found to be robust to uncertainty in values of life-history parameters, although there was
Published by Oxford University Press on behalf of International Council for the Exploration of the Sea 2013. This work is written by (a) US
Government employee(s) and is in the public domain in the US.
1076
C. M. Legault and E. N. Brooks
Table 1. Basic life history and fishery parameters used in simulations.
Age
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20 +
M
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
Weight (kg)
0.087
0.459
1.045
1.706
2.344
2.908
3.379
3.759
4.058
4.290
4.467
4.601
4.702
4.778
4.835
4.878
4.909
4.933
4.950
4.963
Maturity
0.1
0.5
0.9
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Selectivity
0
0.1
0.2
0.5
0.8
0.9
1
1
1
1
1
1
1
1
1
1
1
1
1
1
sensitivity to the form of the S –R relationship (Ricker, 1954; or
Beverton and Holt, 1957) and the schedule of maturity at age relative
to fishery selectivity. Subsequent analyses found that long-lived
stocks with low resiliency (e.g. steepness ,0.67) would require a
higher SPR such as F50% –F60% or more (Clark, 2002).
Despite the thorough analyses done by Clark and others, and the
endorsement by an independent panel of experts (NEFSC, 2008),
the basis for using F40% as a proxy for managing groundfish in the
Northwest Atlantic has recently been characterized as arbitrary.
A “new” approach for specifying which percentage SPR is appropriate for an FMSY proxy has been proposed at recent stock assessment
meetings [e.g. winter flounder (Pseudopleuronectes americanus),
yellowtail flounder (Limanda ferruginea), and cod (Gadus
morhua); see e.g. pp. 387–388 in NEFSC, 2012] and has been represented as less arbitrary. This “new” approach derives from
Sissenwine and Shepherd (1987); one simply counts the number
of points which fall below the replacement line associated with a
given %SPR on the S– R plot. If there are few observations below
the replacement line, this is taken as evidence that the fishing mortality rate that produced the replacement line allows replacement
and can be used as a proxy for FMSY. This approach makes the underlying assumption that points above a proposed replacement line indicate sufficient replacement, while points below the proposed line
do not.
the slope of the replacement line to increase (hence, the point of
intersection with the S– R curve shifts to the left). This point of intersection indicates the equilibrium point if (i) there were no variability
in the S –R relationship, (ii) there were no variability in the biological parameters or fishery selectivity, and (iii) the stock was fished at
that F for an extended period. Equation (1) also defines the calculation of SPR, which is given by:
SPR(F ) = SSBPR(F )/SSBPR(F = 0).
(2)
When F ¼ 0 in the numerator, then SPR ¼ 1, indicating no reduction in lifetime reproductive output (i.e. unexploited conditions).
A value of F that produces SPR ¼ 0.6 is referred to as F60%.
In a strictly deterministic context, every point on the S –R curve
reflects equilibrium at a different F through the replacement line (including the unexploited replacement line when F ¼ 0). Each of these
equilibrium points has an associated yield to the fishery; the F that
produces the largest yield is, by definition, FMSY. Consider the replacement line that corresponds to FMSY for a given S – R curve
(i.e. lines connecting the origin and each of the dollar signs in
Figure 1). At equilibrium, every F . FMSY will be associated with
points on the S –R curve above the FMSY replacement line, while
every F , FMSY will be associated with points on the S –R curve
below the FMSY replacement line (Figure 1). Applying the proposed
method of examining whether S –R points are above or below the
replacement line, one would conclude that overfishing (F . FMSY)
always produces replacement, while underfishing (F , FMSY) never
does. The logic underlying this approach is fundamentally flawed.
Stochastic considerations
Of course, no S –R relationship is deterministic; there is often a wide
range of recruitment associated with any given amount of spawning
stock biomass. We simulated a typical groundfish stock with biological and fishery parameters similar to stocks found in the
The deterministic case
A replacement line on an S– R plot is a line starting at the origin with
slope 1/SSBPR(F), where SSBPR (spawning stock biomass per
recruit) is given by:
SSBPR(F ) =
A+
a=1
a−1
wa ma exp −pMa − pFsa
exp(−Mi − Fsi )
i=1
(1)
where wa is weight at age, ma is maturity at age, p is the proportion of
the year elapsed before spawning, Ma is natural mortality at age, and
sa is selectivity at age. As F increases, SSBPR(F) decreases, causing
Figure 1. Four deterministic S – R relationships (solid lines) based on
parameters in Table 1 with replacement lines (dashed) for F30% and
F40%, the equilibrium spawning stock biomass (SSB) and recruitment
(Recruits) associated with maximum sustainable yield (denoted by
the dollar sign), and equilibrium SSB and R associated with F ¼ 0.8 and
F ¼ 0.15, overfishing and underfishing, respectively.
Can S– R points determine which SPR is the best proxy for MSY reference points?
Northwest Atlantic (Table 1). Fishery selectivity is shifted to older
ages relative to maturity, so that simulated fish have the ability to
spawn even under high fishing mortality rates. Spawning is
assumed to occur on 1 January, with recruitment at age 1 occurring
the following year on 1 January. Simulated populations began in an
unexploited state and were then fished at a constant rate for 50 years,
with only recruitment variability causing stochasticity in the simulations. Variability in recruitment (R) was assumed to be lognormal
with:
R = E(R) exp 1 − 0.5s2 ,
(3)
where e N(0, s2). A Beverton –Holt S –R relationship was
assumed in the simulations:
4R0 hSSB
E (R) = SSBPR|F=0 R0 (1 − h) + (5h − 1)SSB
(4)
where R0 is the unexploited recruitment, SSBPR|F¼0 is the spawning
stock biomass per recruit when there is no fishing, and h is the
steepness of the curve (defined as the proportion of R0 when SSB
is one-fifth of the unexploited SSB; Mace and Doonan, 1988).
Given these parameters, the MSY reference points change as a
function of the steepness parameter (Table 2).
We varied the constant fishing mortality, steepness, and s and
then examined the effect on the number of S –R points which fell
below the F30% and F40% replacement lines. Specifically, we examined the most recent 20 S –R points so that transients from unexploited conditions were not considered. Four constant F values
were explored: two were selected from Table 2 to make F30% or
F40% a good proxy for FMSY, and two were substantially different.
The s parameter was varied over a range of plausible recruitment
variability (0.4, 0.8).
The stochastic simulations demonstrate that the position of S –R
points relative to a given replacement line does not always result in a
replacement line that is an appropriate proxy for FMSY (Figure 2).
The top left and bottom right panels of Figure 2 depict situations
Table 2. The maximum sustainable yield (MSY; thousand t)
reference points as a function of steepness in the S – R relationship
given the parameters in Table 1 and R0 ¼ 10 million fish (this results
in SSB0 ¼ 109 thousand t and SSBPR|F¼0 ¼ 10.904 6).
Steepness
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
FMSY
0.03
0.06
0.09
0.12
0.15
0.18
0.21
0.24
0.28
0.31
0.35
0.40
0.44
0.49
0.55
SSBMSY
51.247
48.133
45.738
43.847
42.321
41.065
40.015
39.125
37.341
36.829
35.647
34.127
33.456
32.480
31.357
RMSY
5.417
5.753
6.088
6.420
6.749
7.073
7.392
7.705
7.946
8.264
8.535
8.793
9.094
9.385
9.684
MSY
1.169
2.122
2.925
3.619
4.231
4.777
5.270
5.719
6.133
6.517
6.875
7.211
7.529
7.829
8.116
SPRMSY
0.867
0.767
0.689
0.626
0.575
0.532
0.496
0.466
0.431
0.409
0.383
0.356
0.337
0.317
0.297
FMSY ¼ fishing mortality rate at MSY, SSBMSY ¼ spawning stock biomass at
MSY (thousand t), RMSY ¼ recruitment at MSY (millions), SPRMSY ¼ spawning
potential ratio at MSY.
1077
where the fishing mortality rate was selected to ensure that either
F30% (top left) or F40% (bottom right) are appropriate proxies for
FMSY. In these cases, Fmed is, in fact, a reasonable proxy for FMSY.
However, the other two panels illustrate that conclusions about
under- or overfishing from the Fmed approach would result in conclusions diametrically opposed to the actual situation. In the bottom
left panel, if we were to rely on the positions of the S –R points relative to replacement lines to inform us about proxies, we would conclude that underfishing is occurring relative to both the F30% and
F40% replacement lines, when the actual F (0.8) is far higher than
FMSY (0.31). The opposite situation occurs for the top right panel.
Furthermore, proxies derived from Fmed in these two cases would
lead to “optimal” exploitation rates that are more than double
(bottom left) and less than half (top right) FMSY. Simply determining the proportion of points below a replacement line will not
inform whether the F associated with that replacement line is an appropriate proxy for FMSY. Instead, the points only indicate whether
fishing has generally been above or below the F associated with a
given replacement line, if all else remains constant.
Analytical considerations
This issue can also be examined analytically in terms of the proportion of a recruitment distribution that falls below a replacement line
at different SSB levels. This was done by setting SSB to a fraction
equal to 0.05 –0.9 of unexploited spawning stock biomass (SSB0)
in steps of 0.05 and calculating the proportion of the recruitments
which would fall below the F30%, F40%, and unexploited (F100%) replacement lines based on equation (3) for R. A range of steepness
and s values were considered.
These analytical calculations also demonstrate that the proportion of S – R points below a given replacement line cannot be used
to determine which SPR is an appropriate proxy for FMSY
(Figure 3). The proportion of recruitments for a given spawning
stock biomass that falls below a replacement line depends more
on the uncertainty in the S –R relationship (s) than on the steepness
of the S –R relationship. Note the relatively high proportion of S –R
points that fall below the F ¼ 0 replacement line when s is high, implying that F ¼ 0 is too high a proxy for FMSY under the “new” approach.
Final thoughts
All of these analyses assume a best-case scenario of unchanging
weights at age, natural mortality at age, maturity at age, and
fishery selectivity. In many real populations, at least some of these
parameters vary. This means that there are multiple replacement
lines for any given strategy, e.g. FMSY, F30%, F40%. These changes
over time are difficult to incorporate due to cohorts experiencing
different biological and fishery conditions over their lifespan,
while replacement lines are conditioned on equilibrium values.
Cook (1998) demonstrated how sensitive replacement lines are to
changes in fishing mortality rates, and O’Brien (1999) demonstrated
that uncertainty in replacement lines is difficult to fully capture.
Thus, creating an appropriate replacement line for the “new” approach is not as easy as it first appears, with many hidden caveats.
The S –R points for a given assessment are conditioned on the
data used and modelling assumptions. Relatively small changes in
data or assumptions can cause these points to move both vertically
and horizontally in the S – R plot. Similarly, some assessments
exhibit a retrospective pattern, i.e. there is a consistent directional
bias in estimates each time the model is updated with an additional
year of data. In such assessments, one can expect that recent S –R
1078
C. M. Legault and E. N. Brooks
Figure 2. One realization of stochastic simulations for four combinations of steepness and constant F along with the replacement lines for F30% and
F40%. Off-diagonal plots also have a line for F%SPR corresponding to Fmed. The numeric values for spawning stock biomass (SSB) and recruitment
(Recruits) are not shown on the axes to emphasize the approach of counting S– R points below a given replacement line. Top left and bottom right
plots show fishing at FMSY; the top right plot shows underfishing, while the bottom left shows overfishing (Table 2). Based on Table 2, for steepness of
0.7, F40% is an appropriate FMSY proxy, while for steepness ¼ 0.95, F30% is appropriate.
estimates will also change with additional years of data, if the retrospective pattern continues into the future. For this reason, retrospective adjustments are sometimes made to the terminal year
population abundance and F estimates to determine stock status
or to provide more risk-neutral catch projections (NEFSC, 2008).
However, these retrospective adjustments are not typically made
to any other S –R points. Thus, the location of the individual S –R
points is not well defined and could easily change from below to
above a given replacement line, or vice versa.
One reason the Fmed approach was originally proposed was that
stock assessment S –R points often do not indicate compensation
(e.g. Mace and Sissenwine, 1993). When a stock has been overfished
or underfished for the entire period of available S –R points, there is
often no ability to estimate an S –R curve due to lack of contrast (see
e.g. the lower left and upper right panels of Figure 2). However, the
implications of using Fmed as a proxy for FMSY in these two cases is
quite different and will cause continued overfishing (lower left
panel) or continued underfishing (upper right panel). Thus, the
fitting of S– R curves requires sufficient range in the data to
support estimation, an important caveat to the application of
MSY-based reference points that is often overlooked.
Although F40% has been accepted as a proxy for FMSY in many
Northwest Atlantic fishery stock assessments, as described earlier,
it is not the only possible proxy. Some Canadian stocks use F0.1,
the fishing mortality rate which reduces the slope of the
yield-per-recruit curve to one-tenth that at the origin, as a proxy
for FMSY (DFO, 2012). A number of ICES stocks use biomass
reference points based on Blim, the location on an S –R plot where
lower stock size appears associated with lower recruitment, and
proxies for FMSY such as M, F0.1, and F20 – 40% (Kell et al., 1999;
ICES, 2012). One way to examine the robustness of any FMSY
proxy is through simulation testing. For example, the approach
used by Clark (1993) could be applied on a stock-specific basis
for a range of plausible assumed S –R curves and associated variability. This approach is currently being explored in the Northeast USA
for some of its groundfish stocks. Alternatively, a management
strategy evaluation (MSE) approach could be explored to evaluate
the performance of different FMSY proxy-based control rules
against a range of simulated population conditions (Deroba and
Bence, 2008). Until such work is completed, there is no basis
to change from the current FMSY proxy of F40% for Northwest
Atlantic groundfish.
In conclusion, the answer to the question posed in this paper’s
title is, “No, a scatterplot of stock –recruitment points cannot determine which SPR is the best proxy for MSY reference points.” We
propose that F40% be maintained as the default proxy for FMSY for
Can S– R points determine which SPR is the best proxy for MSY reference points?
1079
Figure 3. Analytical calculations of the proportion of S –R points that would fall below the replacement line for F30% (squares), F40% (triangles), or
F100% (circles) given the spawning stock biomass relative to unexploited conditions (SSB/SSB0) for four combinations of steepness and s. Based on
Table 2, for steepness of 0.7, F40% is an appropriate FMSY proxy, while for steepness ¼ 0.9, F30% is appropriate. The horizontal dashed lines at 0.5
denote where the number of S – R points above and below the replacement lines would be equal.
Northwest Atlantic groundfish exhibiting a “typical groundfish life
history.”
Acknowledgements
We thank Fred Serchuk, Pamela Mace, and Coby Needle for their
reviews, which helped clarify the paper.
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