1
Not to be cited without prior reference to the authors
International Council for the
Exploration of the Sea
ICES C.M. 1991/D:36
Statistics Cttee
Evaluation of stock protection strategies based on areal quotas
•
Lenore Fahrig
Department ofFisheries and Oceans, P.O. Box 5667, St. John's, NF AIC 5XI, Canada*
Abstract.- The one-third rule for harvesting northern cod specifies that one third of the
offshore catch must be taken from each of the NAFO divisions 21, 3K, and 3L, within
which the stock is found. The rationale for this rule has been that historically the relative
biomasses of the stock in the three areas has not differed significantly from equal
proportions in each of the three divisions. In this paper I present results of simulations in
which lexamine the likely long-term effects of this type of management rule relative to
other possibilities. The simulations are conducted using a stochastic model of the stock
dynamics, including movement of the fish between areas and harvesting in the different
areas using different harvesting rules. The simulations resulted in the following
conclusions: (i) given that the fleet has a preference for one area (3K), some rule for
harvest allocation among the three areas must be used to ensure that the substock in 3K is
not overfished, (ii) due to stochastic variability and stock-recruittnent feedback, the static
one-third rule is likely to result in overharvesting in at least one area in the long-run and (iii)
a dynamic harvest allocation rule, in which the proportions allocated to the three areas are
set using the most recent information on the actual relative biomasses in the three areas, will
produce the lowest likelihood of overharvesting in the long-run.
*current address: Department of Biology, Carleton University, Ottawa, ON KIS 5B6,
Canada.
2
Introduction
The northem cod stock is spread over three NAFO divisions: 2J, 3K, and 3L. The spawning
populations in these three areas are thought to be distinct (1. Baird, pers. comm.). For the years in
which there are data, it appears that the biomass of the spawning stock is approximately evenly
split among the three areas, that is, about one third of the stock is present in each of the three areas.
However, the areas are not equally accessible to the otIshore fishing fleet. If given the choice, the
fleet would fish most or all of its quota in 3K because that area is most accessible by ship (1. Baird,
pers. comm.). If the fleet were permitted to do this, there is the fear that the spawning potential of
the sub-stock in division 3K wOuld decline, thus reducing the productivity of the northem cod stock
and perhaps jeopardizing the persistence of this sub-stock. This situation has 100 to the
management rule for northem cod referred to as the 1/3 rule, in which the otIshore fleet is required
to harvest 1/3 of its northem cod quota in each of the three divisions.
Simulations
To examine the likely etIect of this type of management rule I have conducted simulation
experiments using a general model of fish population dynamies. Tbe model is general because it is
not a model of northem cod per se, but is designOO to look at the general problem of the long-term
effects of areal quotas for sub-stocks. 1 examinOO the consequences of four types of fishing
behaviour, two in which areal quotas are not set and two in which they are:
(1) Preference: The fleet is permitted to decide where to fish and the fleet has a preference for a
particular area. For the northem cod case this is equivalent to having no areal quotas and a fleet
with a preference for 3K. It is assumed that each year the fleet attempts first to fish in its preferred
area. If the biomass in the preferred area is reduced to a certain level (termed the "switehing level",
expressed as a percent of the unexploited sub-stock biomass), the fleet moves to another area to
fish. However, the next year it again attempts to fish in its preferred area.
(2) Inertia: Again the fleet is permitted to decide where to fish but in this case the fleet does not
have a preference for any particular area. It chooses an area at random and if it finds fish there it
stays there (perhaps for several years) until the biomass declines to below the switehing level. At
this point the fleet switehes to another area and again stays there until the biomass reaches the
switehing level
(3) Even Split: Areal quotas are set such that the fleet is forced to take equal amounts of fish from
all areas (This is equivalent to the 1/3 rule in the northem cod case).
(4) Proportional: Areal quotas are set such that the total quota is divided among the areas in
proportion to the estimated relative biomasses of the areas.
TheMode/
Tbe model is a stochastic simulation model of a fish stock dividOO into two sub-stocks
(Figure 1). Tune in the model occurs as discrete steps, representing half-year periods (seasons).
Figure 2 illustrates the annual cycle of events in the model. First the biomass in each sub-stock is
assessed and the quota for the year is set If areal quotas are to be applied these are also determinoo
at this time. 1ben recruitment, natural mortality and movement of fish between the areas (optional)
occurs. Fish are then removed from the areas according to the specifiOO type of fishing behaviour
(four types above); the total is limited to the total quota. Furtherrecruitrnent, natural mortality and
between-area movement then occur before the next year's biomass assessment is made.
•
3
"
The stock-recruitment relationship assumed in the model is illustrated in Figure 3.
Recruitrnent is assumed to be highly variable. The curve in Figure 3 gives the maximum possible
recruitrnent, given a cermin sulrstock biomass. However, the actual recruitrnent value used each
time step in the model is chosen at random (uniform distribution) from between 0 and the value of
the curve, for the given sulrstock biomass value in that season. The results of the simulations are
not sensitive to the particular fonn of the recruitment function used, but the function must include
the following two characteristics: (i) recruitrnent must decline with sulrstock size and (ii) there
must be stochastic variability in the recruitrnent level at a given sub-stock size.
Movement rate is the fraction of the biomass present in one area that moves into the other
area. The rate of movement between the two areas is assumed to be the same so that the amount of
movement is density-dependent, but the rate is not Each season (Figure 2) the movement fraction
is chosen at random from a normal distribution with a mean given for that simulation (below), and
coefficient of variation (CY) of 0.2.
Natural mortality rate is the fraction of the biomass dying due to natural (not fishing) causes
each season (Figure 2). It is modelled as a lognormal deviate. The mean survival rate is given by
1
-111 (SF)
eL
, where L is the maximum lifespan of the fish and SF is the fraction of the original recruits
that survive to the maximum age (in the absence of fishing). L is set at 24 seasons (12 years) and
SF is set at 0.01 in the present simulations. The mean seasonal mortality rate is therefore 0.175. The
CV of the lognormal is set at 0.1.
Simulations
1conducted simulations for each combination of the following three parameters: (i) each of
the four fishing behaviours - preference, inertia, even splil, proportionaJ- , where the preferred
area in the preference option is area 1, (ü) each of seven between-area movement rates - 0.0, 0.001,
0.005,0.025,0.125,0.25,0.5 -, and (ili) each offive fleet switehing levels - 0%, 1%,2%,5%,
10% of the unexploited sulrstock biomass. Two sets of the simulations were conducted using
different random number seeds. A total of 280 simulations was therefore conducted. In all cases
the total quota was set such that the exploitation rate was 40% of the total stock biomass, as
determined at the beginning of the year when the biomass counts are made (Figure 2). Each
simulation was run for 250 years; only the final 200 years were used in summarizing the results to
remove the effects of the transient period in the dynamics. For example, any sulrstoek that became
extinct during the 250 years was extinct by the 50th year.
The following values were calculated to summarize the output: mean annual total cateh,
mean total biomass, CV of the mean total biomass, and mean biomass in each area. The two
replicate sets of simulations produced qualitatively identical results. Therefore the results from
only one of the sets is reported here. The overall cateh, biomass, and CV of biomass values for
switehing levels 0%, 2% and 10% are shown in Figures 4 to 6. The biomass in the individual areas
for switehing levels 0% and 10% are shown in Figure 7.
Conclusions and Recommendations
As expected, the results from the preference fishing behaviour indicate that this should be
avoided. The preference behaviour is representative of the fleet behaviour that is expected if the
northem cod quota is not sub-allocated by division; in this case it is expected that the fleet will
show a strong preference to fish in NAFO division 3K. The simulations predict that this would
result in low stock biomass, low catehes and high CV's. The sub-stock in 3K (as in area 1 in the
simulations) would be depressed. The problem is most severe when the between-area movement
4
rate of fish is low. It is therefore cenainly advisable to set areal quotas for the different divisions.
However, the simulations suggest that the 1/3 rule is probably not an appropriate method
for setting the areal quotas. The 1/3 rule is modelled here as the even split fishing behaviour. Tbe
problem with the 1/3 rule is that, due to variability in recruitrnent between areas, one area will
inevitably have lower biomass than the others at some point If the 1/3 rule is applied, the stock in
the reduced area will suffer a higher than intended exploitation rate, reducing its biomass further,
and possibly reducing its recru.itrnent rate. As the 1/3 rule continues to be applied, the reduced area
eventually declines to 0 biomass.
In effect, under the even split rule equal biomasses in the three divisions represents an
unstable equilibrium point for the system. As long as the biomass in each of the three areas remains
exactly even, the 1/3 rule is optimal. However, a small perturbation from this situation is
exacerbated by the even spUt rule, driving the system to a new equilibrium in which the biomass
in at least one area is inevitably reduced to zero. Note that this problem will arise for any areal
allocation rule that does not allow for response to fluctuations in the relative biomasses of the
different areas.
The simulations also show that this scenario for the 1/3 (or even spUt) rule is particularly
important when the movement rate of fish between the areas is low (less than 0.1). For higher
movement rates the incoming fish reseue an area that has been reduced temporarily, so the negative
effect of the 1/3 rule is no longer evident In evaluating the likely long-term impact of the 1/3 rule
it is therefore very important to have some understanding of the rate of interchange of fish among
the sub-stocks within the northem cod stock. Since the current feeling appears to be that this rate
of interchange is low (1. Baird, pers. comm.), it is clear that the 1/3 rule should be abandoned.
Tbe question remaining is then: what should replace the 1/3 rule? As discussed above, areal
quotas must be set to avoid the consequences of area preference and inertia of the fleet The
simulations suggest that application of proportional areal quotas would be preferable to the 1/3
rule. Application of this strategy depends on the availability of some annual measure of the relative
biomasses in the three divisions. 1 suggest using the autumn survey relative biomass estimates for
this purpose. Although the survey occurs several months before the peak of the offshore fishery,
this is accounted for in the simulations because in the simulations there is a delay between the
biomass counts and the fishing, during which recruitrnent, natural monality, and inter-area
movement occurs. Furthermore, there is some recent evidence that the fall survey biomass
estimates are correlated with the relative biomasses later in the season (I Baird, pers. comm.).
Acknowledgments
1 am grateful to lim Baird, lohn Hoenig and Bill Nuttle for helpful discussions.
Programming assistance was provided by Noel Cadigan. Peter Shelton provided comments on the
manuscripl
•
•
5
Area 1
Fish
Area2
Moving
Fishing
Figure 1. Set-up of the model stock.
Biomass Counts
in the Two Areas,
Set Total Quota, ,
Set Areal Quotas
Recruinnent,
Recruinnent,
Natural Mortaliry,
Between-Area Movement
Natural Mortaliry,
Between-Area Movement
Fishing
Figure 2. Annual sequence in the simulation model.
Figure 3. Stock-Recruitment Relationship
for Sub-Stocks in the Model
6E04 r - - - - - - - - - - - - - - - - - - - - - - - - - - - ,
5E04
(/)
(/)
co AE04
E
0
(()
~
c
Q)
3E04
E
........
:::l
L-
u
Q)
2E04
a:
1E04
OL.L-~--L-_L_JL___"_..J_.<'__L._.L.I._~_L._..L_._"l.:..._L.L_....L...._....'___.L_____L_..4._.L___L_~~L...L.._.'___.L_--c--.L_.J
o
2.5E04
5E04
7.5E04
1E05
Sub-Stock Biomass
1.25E05
1.5E05
1.BE04
A
7
1.6E04
..c.
~
co
1.4E04
Ü
--- ----. - -- -- --- ---- - -- ---- - ------- - --- --
\
~
c:
~~
:'--.:-:'
-"..-.
.--._._.-.-.-
lti
1.2E04
<
,- . ...-.~.--
c:
co
Q)
lE04
~
~
.........
...
./
.... ./.
../ ./
Preference
.... /.
8000
~
o
o
I08rtia
Even Splil
.
\I
Proportional
6000
Q)
>
I
I
o
.1
!
.3
.2
Q)
!
.4
.5
Cl
.S
Between-Area Movement Rate
.I::.
.B
e
.~
CI)
Preference
3.5E04
8
JE'"
Vl
<ri
.....
Inerlla
l\ .
~
.....
C
Even Splil
Proportional
mm..m.. m.mm
'mi
Q)
E
Q)
;J)
o>
E
CO
E
o 2.5E04
ä5
.x:
_._._._._._._.-
u
o
(jj
c:
2E04
------
CO
Q)
~
I.5E04 ~/
lE04
e
--- - ---~.;.=:--
--. -- -- --
cn
>
.I::.
-------------------------
.....
ro
(.)
(.)
ro
:::3
C
C
I
I
o
.1
.2
!
.3
I
.4
.5
Between-Area Movement Rate
ro
c
ro
Q)
~
cn
.:::
.6
:::3
cn
C
Q)
Preference
Even SpÜl
'"'"
ro
I -~l
E
0
Propomonal
0
.4
\"'-
ü5
--.- -'-'- -._._._._.-.-._.-.-.-
Ö
>
I
Ü
.-
I
o
!
.1
.2
.3
Between-Area Movement Rate
.4
.5
.0
CI)
"'C
~
'0
Q)
Q)
.....
a.
'- x
:::l
0>
u...
.3
2
:::3
E
,,'
ä5
.x:
u
cn
c cn
o ro
'';:: E
ro 0
lnenia
.5
'-
Q)
c:
~.
8
1.8E04
A
1.6E04
.s::.
~
ca
1.4E04
Ü
------ .... _---::"-.:': -:~-.
,-
'"iii
~
c:
<:
c:
ca
Q)
~
1.2E04
~. __
- ~~,;:: -:--- .-'-
_.-
-----------_.
...
1'" --
~
o
C\J
Prelerenal
1E04
1ner1ia
11
Even Spil
r:-
8000
Q)
>
Propor1lOnal
Q)
I
6000
I
.1
o
.2
!
!
.3
0)
I
"~
.5
.4
.I::.
U
.-
8etween-Area Movement Rate
"§
e
Cf)
ci
.~
.c
Preference
3.5E04
B
lnertia
Even Spül
Proponlonal
Q)
(J)
(J)
~
3E04
.
..
E
Q)
..
o>
E
ca
E
o
äi
t3o
2.5E04
._._.-._._._._._.-
ü5
c:
ca
Cf)
>
L:-
U
.CO
.-- -- -- -
Q)
:iE
_.-.-
2E04
,----I
1.5E04
o
u
CO
-----------------.1
::J
-------
!
I
.2
.3
c:
c:
CO
c:
.5
.4
e
CO
8etween-Area Movement Rate
Q)
~
Cf)
.::J
Cf)
Preference
.6
c
Q)
~
Inertia
c:
o
"';::
Even Spil
Proponlonal
.5
'"
'"ro
----------
E
Cf)
1'.-
E
o
äi
--D
o
U5
ö
ctS
::J
Ln
.4
Cl>
~
:::J
0)
>
u
r
-._.-._
u..
_._.-._._._._._.-._._~
.3
)
o
.1
.2
.3
Between-Area Movement Rate
t
t
.4
5
CI)
CI)
CO
E
o
:c
"0
Q)
.-
o
0X
Q)
c:
;:)
1.BE04
..
9
1.6E04 t-
A
~ ~ ::..:-~ __~..::-: ~ 7".'::: :-:".-:" ~::-: ...~.:. =-= ~-=--'':-:- ~ ~-~ _.. -- _. -..c
u
Cti
1.4E04
ü
rn
::l
c
c
1.2E04
<:
c
<tl
(1)
:E
lE04
Preference
~
0
Inertia
[.i
0
Even Spül
Booo
-r-
Proportional
11
Q5
6000
.2
.1
0
.3
.5
.4
Between-Area Movement Rate
>
Cl)
0)
c
..s::::.
e
()
"§
Cf)
35E04 [
I_. I---_.
8
3.25E04
'"'"<tl
E
Preference
,
Inema
Even Splil
II - -
Proportional
I
CO
I
i
i
~
C
Cl)
.
E
Cl)
>
0
E
3E04
0
CD
~
u
._._._._._._._._._._._._._._._._._.-
2.75E04
0
U5
CI)
c
<tl
(1)
Cl)
>
-
2.5E04
..s::::.
:E
u
CO
()
225E04
------------~----------------
2E04
e
0
.1
.2
.3
CO
.5
.4
:::J
C
C
CO
Between-Area Movement Rate
c
CO
Cl)
~
..
CI)
==:::J
.6
c
CI)
Cl)
~
.5
~
'"
E
I: _. -
Preference
I'
Inema
- -
- _.
Ll _ _____
ci5
1
o
2
E
:::J
:c
.
CO
Cl)
~
'. . .:.---------!
.1
CO
E
Proporuonal
I
2--
0';::
Cf)
I-
.=-
CI)
CI)
0
I · · · · · · · · · · · · Eyen SpUl
r -'
c
.3
Between-Area Movement Rate
.4
.5
:::J
C>
u::
CO
0
"'C
Cl)
"0
Q.
X
Cl)
C
:::J
Figure 7. Simulation results: Biomass in each of the two areas vs. movement rate. A:
Switching level = 0% unexploited biomass. B: Switching revel = 10% unexploited biomass.
4E04 -
4E04 -
A
Preference
3E04
C\l
C'lI
Preference
C'lI
QJ
c
.......
111
~ 2E04
Even Srl;1
4:
Even Spfil
c
Proportional
111
_
- _.- - - _.- .-'- -
..
k.--:....-.._----~::_:::_:==~-_.--r'"'"'~.--.----
c
c
.- - .-.- .-.-
...-' ...-'
in
in
- - Proportional
I
~ 2E04
E
o
.
E
o
C'lI
co
QJ
~
QJ
~
3E04
QJ
Inerlla
4:
- - - - - Inerlia
lE04
-----------------------------
o L _ _~
o
.1
.-.-
_.-'-
.-.-......L__---=-'=.t:=..:.=---1:__----'.
.2
.3
~--------7-------
OL-L-_ _...L-
o
.5
.4
lE04
.1
-IL-----...L----.-J
-L
.2
5
.3
Between-Area Movemenl Aale
Between-Area Movement Aale
4E04 •
4E04
- .-
B
Prelerence
- - - -' fnerlia
C\l
co
3E04
co
Prelerallce
Ql
~
.E
IIIar1la
4:
EVUII Splil
.E
111
co 2E04 .
E
o
gj 2E04 E
o
- - -'-'-'-'-'-
c
co
c
co
lE04
-'-
-
-
-'-'- -'-'-
_
..
in
iD
~
Even Splil
- - Proportional
Ul
Proportlonal
(/)
3E04 -
QJ
------------------------------------------------
_.-.-.-._.-.-._.-.-.-.-
------------------------------
QJ
~
----------~----------
lE04
-'-'-'-'-'-'-'-"
t-'
o
0L-
o
...L-
.1
-L
.2
--lL-----'-----'
.3
_
Belween-Area Movemenl Aalt>
.4
.5
OL-L-_ _- L
o
e
1
-L
.2
L------'-----'
.3
Between-Area MovelOenl Rale
.4
.5