SIMULATIONS OF MANAGEMENT OF ANCHOVY IN THE BAY OF BISCAY (ICES
DIVISION VIII)
Commission Staff Non-Paper (2007)
Introduction
This note outlines the technical basis for a request to STECF to investigate yield/risk
tradeoffs in managing anchovy in the Bay of Biscay, and provides a detailed
specification of the assumptions to be made about management actions, and the
scenarios to be calculated. This definition is not meant to be proscriptive but rather to
inform STECF about the Commission's view of a suitable technical basis for the risk
calculation. The definition could be modified or augmented if the Commission and
STECF experts agree that it is necessary to do so.
ICES Advice on management
ICES advice from ACFM in 1999 implies a management approach in which the
fishery should be closed if stock size is projected to fall below Bpa, until there is
evidence of a good recruitment. The precautionary stock size (Bpa) is defined such
that if a stock at spawning time is at Bpa in year y and the incoming recruitment is of
lowest previously-observed abundance, then the stock in the subsequent year will still
be above Blim.
This approach to management is intended by ICES to be "consistent with the
precautionary approach" in that it seeks to achieve a low probability of falling below
the Blim reference point, in accordance with international agreements on the
precautionary approach to fisheries.
STECF Advice on Management
STECF in its November 1999 report endorsed the ICES advice, but proposed two
options:
1. Closure of the fishery, followed by re-opening if a good recruitment were
detected.
2. Reduction of TAC to a low level, followed by an increase if surveys in
April-May show a strong 1999 yearclass still surviving until spawning
time, or continued closure if the spawning biomass is confirmed to be low.
In principle the abundance of the 2000 year-class could be forecast from
upwelling indices around May, and this index could be used in taking a
decision on the level of the TAC for the remainder of the year.
Both options imply a change in the fishery régime in the second part of the year.
Yield/Risk tradeoff
Although seeking to maintain stock size above historical low levels appears to be a
desirable objective, it is not currently known to what extent decreases in catches result
in a reduction in biological risk expressed in such terms.
For example, closing the fishery from 1 January 2000 implies a reduction in risk
(expressed as probability that SSB will be below Blim in 2001) from about 23% in the
case of F-status-quo fishing to about 16% if the fishery is closed (See Appendix for
technical detail).
The following seeks to specify some basic assumptions for a modelling approach
calculating the yield/risk tradeoff. The proposed model components are:
1. Population dynamics: conventional catch equation model
2. Recruitment : assume stock-independent above a particular critical value,
with a decline in recruitment below this value down to the origin. Test
sensitivity to the critical value assumed.
3. Fleet dynamics: As neither a constant-catch nor a constant-F régime seems
appropriate for this stock, a fleet model with intermediate characteriscs is
proposed (catch increases with stock biomass but F decreases with
increasing stock size).
4. Management model: Assume that if stock size falls below a particular
value (viz. Bpa ) restrictive measures are imposed to reduce catch
possibilities. Test the sensitivity to (a) the Bpa value used (b) the severity
of the catch restriction.
5. Diagnostics: For a range of plausible scenarios, the parameters of interest
are (a) how often the restrictive measures will need to be imposed (b) the
average yield (c) the risk of stock depletion below Blim (d) how often
restrictive measures may need to be imposed for several years.
Technical detail on proposed model formulation is provided in the Appendix. It is
proposed that evaluation of the tradeoff between yield and risk should be made under
plausible assumptions of measurement error in the surveys, but also for a range of
plausible options for
- Stock -recruit relationships
- Fleet dynamics
- Different levels of Bpa (at which fishery restrictions are imposed)
- Various levels of the severity of the restrictions
The technical basis for such options, and the values to calculate under these
assumptions are also provided in the Appendix.
APPENDIX : TECHNICAL BASIS FOR SIMULATIONS OF MANAGEMENT OF ANCHOVY
IN THE BAY OF BISCAY (ICES DIVISION VIII)
Ad hoc risk calculation
A simple calculation of risk can be made as follows: ICES advice indicates a risk of
about 90% of being below Bpa in 2000 for F status quo (catch = 15 000t ), from which
the risk of being below Blim in 2001 is P(B2000 <Bpa)*P(R2000< 5 billion), or about
90% * 3/13 = 21% if recruitment is i.i.d. random. This is because Bpa is defined as
that level of stock size which will leave the stock at Blim the following year in the
event of a poor recruitment.
Also, ICES advice indicates a risk of about 70% of being below Bpa in 2000.
Implicitly the risk of being below Blim in 2001 is P(B2000 <Bpa)*P(R2000< 5 billion),
or about 70% * 3/13 = 16% if recruitment is i.i.d. random. This does not however
include a quantification of the risk that concentration effects at reduced stock sizes
may make the stock more vulnerable to the fishery.
The technical basis described below is intended to provide a more technically sound
basis for such inferences.
Exploration of Management Options by Simulation
Structural Model
The structural model is structured by year y, age a and semester s. The population
model is the conventional exponential decay model with an assumption of stable
selection. Semesters run from 1 January until 31 May (s=1) and from 1 June until 31
December (s=2), representing 5/12 and 7/12 of a year respectively.
Changes in population abundance are governed by the usual structural equations :
Ny,a,s = Ry
if s=1 and a = 0
(1)
Ny,a,s = 0
if a >=4
(2)
Ny+1,a,1 = Ny,a-1,2. exp(- My,a,2 - Fy,a,2)
(3)
Ny,a,2 = Ny,a,1 . exp(-My,a,1 - Fy,a,1)
(4)
The relationship beween catch in number, fishing mortality and abundance is
modelled by
Cy,a,s = Fy,a,s. Ny,a,s (1- exp ( -My,a,s -F y,a,s )) / (Fy,a,s + My,a,s) (5)
Landings in weight are given by
L
y ,s
 a C a , y , sW a , y , s
(6)
where mean weights at age by semester for all years should be taken as mean values
for the international fishery from the first half of the year and the second half of the
year respectively, in the period 1987 to 1998, as given in Table 10.3.2.2 of Anon.
(2000).
Constraints on Fishing Mortality
As a starting point, the following constraints on selection should apply, restricting the
simulations to the selection pattern estimated in the most recent assessment:
Fy,0,s
= 0.0071 . Fy,2,s
Fy,1,s = 0.3886. Fy,2,s
Fy,3,s =
0.6574. Fy,2,s
Fy,4+,s = 0
(7)
However, exploration of the consequences of alternative selection patterns, e.g. that
obtaining in the early 1980s could also be warranted.
Recruitment Ry:
Recruitment is defined as population abundance calculated on 1 January, ie N0,y,1 . It
is indicated to assume that recruitment is independent of SSB when SSB is above a
critical level (B1), but that when stock size is below B1 a linear decline in recruitment
occurs. Consequences of alternative choices of B1 including values of 18 000t and 55
000t should be explored.
- SSB Above Bl : Stationary, nonparametric bootstrap from observed time
series from 1987 to 1997 from most recent analytic
assessment and 1998 value from shrunk prediction
including upwelling and average value Rav= 4774 . 10
^6.
- SSB Below Bl : R = (Rav - (Rav . (SSBy-B1)/B1 )). exp (e), where e is drawn
by nonparametric bootstrap on a logarithmic scale from
recruitments as described above, and Rav is the average
historical recruitment over the period 1987-1998.
Fleet Dynamics Model
Neither TAC constraints nor F-constraints can be used to make in-year forecasts for
this stock, as F is extremely variable and the catch is not usually TAC constrained.
Instead it is proposed to assume that the dynamics of the fleet can be described in a
simple way, based on the Cobb-Douglas harvest function with constant capacity, as
Landings = Q. SSB K (Figure 1).
Yield/Biomass Ratio
Catch
Yield/Biomass ratio
Reported Catch (t)
50000
40000
30000
20000
10000
0
0
50000
100000
150000
Estimated SSB ( t)
1
0,9
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0
0
50000
100000
150000
Estimated SSB (t)
Figure 1. Left panel, historic catch and biomass estimates, and fitted line
corresponding to catch = 14.28 SSB 0.692 . Right panel, corresponding yield/biomass
ratios from the history of the fishery and from the same catch model. See Appendix
for further detail.
To represent catches by semester and by year, define : Ly,1+Ly,2= f(SSB) as follows to
represent annual catches taken when the TAC is not restrictive. This function
includes a representation of stochastic processes such as variability in catchability and
market demand: The model is :
f(SSBy) = Q. SSByK. exp(ε).exp(-σ2/2); ε ~N(0, σ2);
(8)
When K=0 this model represents a constant-catch situation and when K=1 the model
represents exploitation with constant fishing mortality. Estimating Q and K from the
stock history leads to Q = 14.28 and K= 0.692. However, because the value of K is
estimated imprecisely yet is also important for the dynamic behaviour of the fishery,
yield/risk tradeoffs should also be calculated for plausible alternative values of K,
such as K = 0.3, and also to represent the situation where catches are taken of a
similar level as the TAC (=33 000t) but with the same variability as observed
historically (cv=0.42). Plausible parameterisations to be tested should include :
(a.) Q = 14.28; K=0.692; σ =0.277.
(b.) Q = 988.5; K=0.3, σ = 0.33
(c.) Q= 33000; K=0, σ=0.42
For numerical stability in the simulations, assume catch = SSB/3 if SSB < 10t, and F
by semester at age 2 never greater than 5. Based on the foregoing, catches by semester
can be predicted from
Ly,1 = P. f(SSB'y)
(9)
Ly,2 = (1-P)f(SSB'y)
(10)
where

P

y 1998, s 1
y 1987, s 1
y 1998, s  2
y 1987, s 1
L
L
y ,s
y ,s
(11)
Values of Ly,s for y= 1987 to 1998 are to be taken from Table 10.3.1.3. of Anon.
2000.
and
a 3
 
SSB y   N y ,a,1 exp 0.96  F y ,a,1  M y ,a,1
a 1
SW
y ,a
(12)
where SWy,a represents weight at age at spawning time and = 15.92, 28.70, 34.18g at
ages 1-3 respectively.
Natural Mortality : Assumed fixed and = 1.2 annually, ie My,a,1 = 1.2. (5/12); My,a,2
= 1.2. (7/12)
Measurement (Assessment) error : For a simple representation, use
N'age= Nage. exp(φage).exp(-ωage2/2); φage ~N(0, ωage 2);
(13)
initially for ω 0-3 = 0.52, 0.44,0.18, 0.22 (Values from 1999 assessment), where
N' represents the perceived population. For simplicity, treat the measurement error as
if measurements were made on 1 January.
Ideally, a full management simulation would be structured to represent the
population and the process of sampling from it and assessing it.
Starting Conditions
In order to begin the simulations in year y=1, population abundances at ages 0, 1, 2
and 3 should be simulated according to the values estimated for 1 January 1998 in
Anon. 2000, with corresponding log standard error:
Age 0 : 4394 Million, log s.e. = 0.5206
Age 1 : 1434 Million; log s.e. = 0.4418
Age 2 : 2710 Million, log s.e. = 0.18
Age 3 : 286 Million, log s.e. = 0.22
Management Actions to Model
The following scenarios should be simulated, representing the effects of a
management action taken in response to an estimate of stock size at spawing time. We
define two estimators of this quantity. An estimate of SSB at spawning time from
survey information in year y is :
a 3
 
SSB'y   N 'y ,a,1 exp 0.96  F y ,a,1  M y ,a,1
a 1
SW
y ,a
(14)
However, where decisions one year ahead are to be made, this is done on the basis of
a one-year ahead forecast of stock size based on the most recent survey, denoted
SSB*:
*
SSB

y 1
 a 0 N 'y ,a ,1. SW y 1,a
a 1
exp  F 'y ,a ,1  M y ,a ,1  F 'y ,a , 2  M y ,a , 2 0.96F 'y 1,a 1,1 0.96M y 1,a 1,1

(15)
where F'y,a,1 , F'y,a,2, F'y+1, a+1, 1 are found by nonlinear solution of 1-16 above and A1A6, but with σ taken as 0 for forecasting purposes (in which case L' replaces L) .
Note that it is necessary to model populations and fishing mortalities separately for :
(1) The underlying, true populations and realised catches, F,N and L;
(2) The forecasts corresponding to perceived stock size and forecast catches,
F', N' and L'.
The variables SSB*y+1 and SSB'y+1 are used to define a simple harvest control law.
This describes management as switching between essentially unrestricted catches (as
at present) to some more restrictive condition such as a closure in the case that a
forecast stock size should fall below a precautionary biomass. Here define CMin =
lowest possible catches that are taken in a semester, which could be in the range zero
to 8 000 t.
ICES/STECF Approach
Closure if stocks forecast is below Bpa in next year (SSB*y+1<Bpa) and re-open in midyear if the stock size after survey, found to be above Bpa (SSB'y+1>Bpa) .
if SSB*y+1< Bpa
Ly+1,1= CMin
If SSB'y+1 > Bpa
Ly+1 = f(SSBy+1)-CMin [Note: Iterative solution]
Otherwise
Ly+1,2 = CMin
Otherwise
TACy+1 = 33Kt
Ly+1 = f(SSBy+1)
[Note: Iterative solution]
Required Results of the Simulations
For the case described above the simulations should be used to estimate, over a
twenty-year period
(1) The frequency of imposing a restrictive TAC = CMin.
(2) The average yield from the stock.
(3) The proportion of simulations in which B falls less than Blim at least once.
(4) The frequency of occurrence of a restrictive TAC = CMin which extends
for more than one year.
These should be calculated for
- various levels of assumed Bpa, including 18000t, 25000t and 36000 t)
- various levels of CMin including 0, 4000 and 8000 t):
- various levels of B1 including 18 000t and 55 000t
- Catch models with (a) Q=14.28, K=0.692, σ=0.277
(b) Q= 988.5, K = 0.3, σ =0.33
(c) Q=33000, K=0, σ=0.42
This implies provision of four diagnostics from at least (3x3x2x3)= 54 simulation
scenarios. These simulations would be helpful in informing management about about
the relative risks and benefits of alternative actions in terms of choice of B pa and CMin.
Reference
Anonymous, 2000. Report of the Working Group on the assessment of Mackerel,
Horse Mackerel, Sardine and Anchovy. ICES C.M. 2000/ACFM: 5