selectivity’s distortion of the production function and its influence on management advice
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Selectivity’s distortion of the production function and its
influence on management advice
Sheng-Ping Wang1,2, Mark Maunder2, and Alexandre Aires-Da-Silva2
1. National Taiwan Ocean University2. Inter-American Tropical Tuna Commission
2
For many situations, catch and data are only available for assessment especially for non-target species, small scale fisheries...
The Schaefer surplus production model is commonly used in fisheries stock assessment.
Introduction
It has a symmetrical relationship between equilibrium yield and biomass where maximum sustainable yield occurs when the population is at 50% of the unexploited level
3
The Schaefer model has been criticized because contemporary stock assessment models, which explicitly model the individual population processes, suggest that MSY is obtained at biomass levels substantially less than 50% of the unexploited level for many species.
Introduction
4
Pella and Tomlinson (1969) developed a more general surplus production model with an additional (shape) parameter that allows MSY to occur at any biomass level.
Introduction
MSY
B0
0( / )MSYm f B B
5
Surplus production models represent population dynamics as a function of a single aggregated measure of biomass.
Introduction
1
0
( , )
~ , ,t tB f B
B r m
e.g. carrying capacity (K or B0), productivity rate (r), and shape parameter (m)
6
It is well known that the production function of a stock is highly dependent on biological processes◦ e.g. growth, natural mortality, and recruitment and
density dependence (e.g. the stock-recruitment relationship).
This has led to questioning of the use of traditional surplus production models for the assessment of fish stocks.◦ Estimation of the production function from catch
and an index of relative abundance (or catch and effort data is problematic
Introduction
7
The production function is also dependent on the size (or age) of fish caught by the fishery.◦ In general, fisheries that catch small fish produce a
lower MSY compared to fisheries that catch large fish.
Fisheries that catch small fish also generally produce a lower BMSY/B0.
Therefore, the age/size of the fish caught in a fishery needs to be taken into consideration when estimating the impact of a fishery on the stock
Introduction
8
Age-structured model can incorporate biological processes and selectivity for considerations.
Introduction
,
1
.
.
( )
( , )
t t a aa
t t t t t t
t t
t t a a aa
gt t
gt t a a a
a
g g gt t
B N w
N N M C G R
R f S
S N m w
C f B F
B N s w
I q B
9
Typically, the selectivity increases smoothly with the size or age of the individual and either asymptotes or perhaps reducing at larger sizes.
The shape of the selectivity at size/age should be explicitly taken into consideration when evaluating equilibrium yield or the shape of the production function.
Introduction
10
First we use simulation analysis to illustrate the impact of selectivity and biological parameters on the production function based on equilibrium age-structured model.
Objectives of this study
11
Then we evaluate how changes in selectivity over time influence parameter estimates and management advice from production models. ◦ The simulation analysis is roughly based on the
bigeye tuna stock in the eastern Pacific Ocean. ◦ The fishery has changed from mainly a longline
fishery, which captures large bigeye, to a mix of longline and purse seine, which captures small bigeye.
Objectives of this study
12
Sensitivity of MSY and related management quantities to biological parameters and selectivity is analyzed based on an age-structured model developed to model the population dynamics under equilibrium conditions.◦ Beverton and Holt stock-recruitment relationship◦ Separate fishing mortality◦ von Bertalanffy growth function◦ knife-edged maturity◦ Constant natural mortality
Equilibrium analysis
13
The analysis is repeated for a variety of values for the steepness of the stock-recruitment relationship (h), the von Bertalanffy growth rate parameter (K), natural mortality (M), and the parameters of selectivity.◦ h = 0.5, 0.75, and 1◦K = 0.1, 0.2, and 0.3◦M = 0.1, 0.2, and 0.3
Equilibrium analysis
Estimates of shape parameter (BMSY/B0)◦ Age at first capture is fixed at 4 yrs.
Knife-edged selectivity assumption
h = 0.5 h = 0.75 h = 1.0
M = 0.1K = 0.1 0.39 0.33 0.27K = 0.2 0.38 0.31 0.23K = 0.3 0.36 0.29 0.19
M = 0.2K = 0.1 0.39 0.32 0.26K = 0.2 0.38 0.30 0.22K = 0.3 0.37 0.29 0.17
M = 0.3K = 0.1 0.38 0.31 0.23K = 0.2 0.37 0.30 0.16K = 0.3 0.36 0.28 0.15
Estimates of productivity parameter (r or MSY/BMSY)
Knife-edged selectivity assumption
h = 0.5 h = 0.75 h = 1.0
M = 0.1K = 0.1 0.04 0.07 0.10K = 0.2 0.06 0.10 0.17K = 0.3 0.07 0.13 0.25
M = 0.2K = 0.1 0.07 0.12 0.18K = 0.2 0.10 0.18 0.32K = 0.3 0.12 0.24 0.52
M = 0.3K = 0.1 0.10 0.18 0.31K = 0.2 0.14 0.27 0.63K = 0.3 0.18 0.36 1.00
Yiel
d
Biomass
16
Two types of curves are used to exam the impacts of selectivity on the production function and MSY based quantities. Knife-edged selectivity Double dome-shaped selectivity (only change the
shape of curve on the right hand side)
Equilibrium analysis
5 10 15 20
0.0
0.4
0.8
Age
Se
lectivity
asd=1asd=5asd=10
Knife-edged selectivity assumption
ac
shap
e
0.0
0.2
0.4
0.0
0.2
0.4
0.0
0.2
0.4
M=0.1M=0.2M=0.3
ac
shap
e
acsh
ape
h=0.75
ac
shap
e
0.0
0.2
0.4
0.0
0.2
0.4
0.0
0.2
0.4
ac
shap
e
ac
shap
e
h=1
BM
SY
B0
K=0.1 K=0.2 K=0.3
ac
shap
e
0.0
0.4
0.8
0.0
0.4
0.8
0.0
0.4
0.8
ac
shap
e
ac
shap
e
h=0.75
ac
shap
e
0.0
0.4
0.8
2 4 6 8 10
0.0
0.4
0.8
2 4 6 8 10
0.0
0.4
0.8
2 4 6 8 10
ac
shap
e
2 4 6 8 102 4 6 8 102 4 6 8 10
ac
shap
e
2 4 6 8 102 4 6 8 102 4 6 8 10
h=1
SM
SY
S0
Age at first capture
Yiel
dBiomass
Knife-edged selectivity assumption
ac
Yra
tio
0.0
0.4
0.8
0.0
0.4
0.8
0.0
0.4
0.8
M=0.1M=0.2M=0.3
ac
Yra
tio
ac
Yra
tio h=0.75
ac
Yra
tio
0.0
0.4
0.8
0.0
0.4
0.8
0.0
0.4
0.8
ac
Yra
tio
ac
Yra
tio
h=1
MS
YB
MS
Y
K=0.1 K=0.2 K=0.3
ac
Yra
tio
0.0
0.4
0.8
0.0
0.4
0.8
0.0
0.4
0.8
ac
Yra
tio
ac
Yra
tio
h=0.75
ac
Yra
tio
0.0
0.4
0.8
2 4 6 8 10
0.0
0.4
0.8
2 4 6 8 10
0.0
0.4
0.8
2 4 6 8 10
ac
Yra
tio
2 4 6 8 102 4 6 8 102 4 6 8 10
ac
Yra
tio
2 4 6 8 102 4 6 8 102 4 6 8 10
h=1
MS
YS
MS
Y
Age at first capture
Yiel
dBiomass
Double dome-shaped selectivity assumption
ac
shap
e
0.0
0.2
0.4
0.6
0.0
0.2
0.4
0.6
0.0
0.2
0.4
0.6
M=0.1M=0.2M=0.3
ac
shap
e
ac
shap
e
h=0.75
ac
shap
e
0.0
0.2
0.4
0.6
0.0
0.2
0.4
0.6
0.0
0.2
0.4
0.6
ac
shap
e
ac
shap
e
h=1
BM
SY
B0
K=0.1 K=0.2 K=0.3
ac
shap
e
0.0
0.2
0.4
0.0
0.2
0.4
0.0
0.2
0.4
ac
shap
e
ac
shap
e
h=0.75
ac
shap
e
0.0
0.2
0.4
2 4 6 8 10
0.0
0.2
0.4
2 4 6 8 10
0.0
0.2
0.4
2 4 6 8 10
ac
shap
e
2 4 6 8 102 4 6 8 102 4 6 8 10
ac
shap
e
2 4 6 8 102 4 6 8 102 4 6 8 10
h=1
SM
SY
S0
Standard deviation of age for dome-shaped selectivity
Yiel
d
Biomass
Double dome-shaped selectivity assumption
ac
Yra
tio
0.0
0.4
0.8
0.0
0.4
0.8
0.0
0.4
0.8
M=0.1M=0.2M=0.3
ac
Yra
tio
ac
Yra
tio h=0.75
ac
Yra
tio
0.0
0.4
0.8
0.0
0.4
0.8
0.0
0.4
0.8
ac
Yra
tio
ac
Yra
tio
h=1
MS
YB
MS
Y
K=0.1 K=0.2 K=0.3
ac
Yra
tio
0.0
0.4
0.8
0.0
0.4
0.8
0.0
0.4
0.8
ac
Yra
tio
ac
Yra
tio
h=0.75
ac
Yra
tio
0.0
0.4
0.8
2 4 6 8 10
0.0
0.4
0.8
2 4 6 8 10
0.0
0.4
0.8
2 4 6 8 10
ac
Yra
tio
2 4 6 8 102 4 6 8 102 4 6 8 10
ac
Yra
tio
2 4 6 8 102 4 6 8 102 4 6 8 10
h=1
MS
YS
MS
Y
Standard deviation of age for dome-shaped selectivity
Yiel
dBiomass
5 10 15 20
0.0
0.2
0.4
0.6
0.8
1.0
Age
Sel
ectiv
ity
The dynamic age-structured model is used to simulate a age-specific biomass, fishing mortality, catch series and index of relative abundance for BET in the EPO.◦ Beverton and Holt stock-recruitment relationship
recruitment is modeled using multiplicative lognormal process variation
◦ Gear-specific separate fishing mortality◦ von Bertalanffy growth function◦ Knife-edged maturity◦ Constant natural mortality
Application on bigeye tuna stock in the EPO
The BET stock in the EPO has two main fisheries, purse seine setting on floating objects and longline.
Thus the dynamic age-structured model is developed for incorporating gear-specific selectivities. ◦ Gear-specific fishing mortality is the product of
gear-specific effort, catchability and selectivity.
Application on bigeye tuna stock in the EPO
Selectivity ◦ Selectivity of longline (SLL) is assumed to be
logistic curve◦ Selectivity of purse-seine (SPS) is assumed to be
descending right hand limb.
Application on bigeye tuna stock in the EPO
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
Age
Se
lect
ivity
LonglineLogistic curve
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
Age
Se
lect
ivity
Purse-seineDouble dome-shaped curve
Selectivity ◦ The age-specific fishing mortality in 2010 is used
to calculate the longline and purse-seine combined selectivity (SLL+PS).
Application on bigeye tuna stock in the EPO
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
Age
Se
lect
ivity
Combined selectivityBased on fishing mortality in 2010
The gear-specific catch is calculated without error
Gear-specific catch rate (index of relative abundance) is calculated incorporating a multiplicative lognormal observation error.
Application on bigeye tuna stock in the EPO
Pre-specific biological and fishery parameters
Application on bigeye tuna stock in the EPO
Category of parameters Valuevon Bertalanffy growth function
L∞ 1K 0.2t0 0
Length-Weigth relationshipa 1b 3
Age at maturityam 4
Virgin recruitmentR0 100
Steepness for spawning biomass -recruitment relationship
h 0.75Natural mortality
M 0.4
Pre-specific biological and fishery parameters
Application on bigeye tuna stock in the EPO
Category of parameters ValueCatchability
q for longline 0.0175q for purse seine 0.35
Standard deviation of random residuals for Recruitment 0.6 for CPUE 0.2
Application on bigeye tuna stock in the EPO
1980:2010
E[2
9:59
, 1]
LonglinePurse seine
(A)
0.0
1.0
2.0
3.0
Sim
ulat
ed E
ffort
leve
l
LonglinePurse seine
(B)
1980 1990 2000 2010
0.0
0.4
0.8
1.2
Sim
ulat
ed y
ield
leve
l
Year
1950 1970 1990 2010
0.0
0.4
0.8
YearB
B0
LonglinePurse seine
Gilbert’s version of the Pella-Tomlinson model is fit to the simulated data with the shape (m) and productivity (r) parameters either fixed based on the pre-specific values from the age-structured model or estimated.
Application on bigeye tuna stock in the EPO
1 0
1 10
( , , , )
11
t t
mt
t t t tm
B f B B r m
BrB B B C
Bm
Pre-specific values of the shape (m) and productivity (r) parameters are obtained by equilibrium age-structured model with various selectivity assumptions.◦ Selectivity assumed to be SLL
◦ Selectivity assumed to be SPS
◦ Selectivity assumed to be SLL+PS
Application on bigeye tuna stock in the EPO
The shape and productivity parameters are based on the
vulnerable biomass spawning biomass
500 simulation runs were carried out for each scenario.
Application on bigeye tuna stock in the EPO
Longline selectivityPurse-seine selectivityGear-combined selectivity
32
Results by fitting to total catch and the LL catch rate
Shape parameter
Productivity
SLL
0.0
0.3
0.6
BM
SY
B0
SPS SLLPS
MS
YB
MS
Y
0.0
0.2
0.4
Age
-str
uctu
red
Est
imat
e al
l
Fix
ed m
Fix
ed r
Fix
ed r
& m
MS
Y
0.0
1.5
3.0
Age
-str
uctu
red
Est
imat
e al
l
Fix
ed m
Fix
ed r
Fix
ed r
& m
Age
-str
uctu
red
Est
imat
e al
l
Fix
ed m
Fix
ed r
Fix
ed r
& m
Estimation category
33
Results by fitting to total catch and the LL catch rate
SLL
Bcu
rB
MS
Y
02
46
SPS SLLPS
Age
-str
uctu
red
Est
imat
e al
l
Fix
ed m
Fix
ed r
Fix
ed r
& m
Ucu
rU
MS
Y
02
46
8
Age
-str
uctu
red
Est
imat
e al
l
Fix
ed m
Fix
ed r
Fix
ed r
& m
Age
-str
uctu
red
Est
imat
e al
l
Fix
ed m
Fix
ed r
Fix
ed r
& m
Estimation category
34
Comparison between vulnerable and spawning biomass
Shape and productivity parameters estimated based on pre-specific values obtained from various selectivity assumptions and measurement of biomass
35
Comparison between vulnerable and spawning biomass
Current biomass ratio (Bcur/BMSY) estimated based on pre-specific values obtained from various selectivity assumptions and measurement of biomass
36
Comparison between vulnerable and spawning biomass
0.2 0.4 0.6 0.8 1.0
SLL
0.0
0.1
0.2
0.3
0.4
0.5
Rel
ativ
e bi
omas
s
BMSY B0
SMSY S0
0.2 0.4 0.6 0.8 1.0
SPS
0.2 0.4 0.6 0.8 1.0
SLLPS
Steepness (h)
37
Using time-varied parameters
Residual sum of squares for estimation models with time-varied r and m.
Tim
e-va
ried
SLL
SP
S
SLL
PS
S
05
1020
Fixed r
Tim
e-va
ried
SLL
SP
S
SLL
PS
S
Fixed m
Tim
e-va
ried
SLL
SP
S
SLL
PS
S
Fixed r & m
Estimation category
Res
idua
lsu
m o
f squ
ares
38
Discussion The results of this study indicate that the
selectivity and biological processes can substantially impact the production function.
Vulnerable biomass and spawning biomass are calculated based on different equations basis. However, production model only estimates biomass based on vulnerable pattern and thus we cannot know which measurement is appropriate to be used for comparison.
39
Discussion Estimating shape parameter of Pella-
Tomlinson production model would be problematic.◦ The estimations are biased and imprecise.◦ Lead to the problematic estimates..
SLLSPS
PLL+PS9092949698
100
Prop
ortio
n of
con
verg
ence
(%)
40
Discussion Since historical catch and catch rate were
mainly contributed by LL, time-varied parameters of production calculated based on gear-combined selectivity cannot significantly improve fits of production model.◦ Assuming the parameters of production based on
LL selectivity would improve the fits of model.
41
Conclusions Production function is substantially
influenced by biological process and selectivity assumptions.
Schaefer model might not be appropriate for most scenarios.
42
Conclusions Although Pella-Tomlinson model is much
flexible, estimating shape parameters leads to problematic estimations for all selectivity assumptions.
The estimations of production model are distinct from the those of age-structured model (“true values”) since population dynamics is actually related to age-specific selectivity.
43
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Euqations-Equilibrium analysis
45
Initial conditions
1
1
00 0
1
01
for 1
for 1
1 for
a
a
a
Ma a
Ma
M
R aN N e a
N e
e a
00
00
a a a aa
a a aa
S N w m r
B N w s
• Where wa, ma and ra are the weight, maturity, and sex-ratio (proportion of females) of fish at age a.
• wa is calculated based on von Bertalanffy growth function and length-weight relationship.
• Maturity is assumed to be knife-edged with age-at-maturity (am).
46
Population dynamics
1 1
1 1
( )1
( )1
( )
for 1
for 1
1 for
a a
a a
a a
F Ma a
F Ma
F M
aRN N e a
N e
e a
a aF F s
a a a aa
a a aa
S N w m r
B N w s
Where F is the fishing mortality for full-recruitment, and sa are the selectivity of fish at age a.
◦ The Beverton and Holt stock-recruitment relationship which is re-parameterized in terms of the "steepness" of the stock-recruitment relationship.
Recruitment
0
0
0
(1 )
4
5 1
4
XR
X
S h
hR
h
hR
1
1
1
1
( )
for 1
for 1
for 1
a
a aj
a
a aj
a a
aa
a a a
F M
a a a a
F M
a a a F M
X X
w m r a
X w m r e a
ew m r a
e
X is the spawning stock biomass per recruit:
◦ Knife-edged selectivity
◦ Double dome-shaped selectivity
Selectivity
0 for
1 for
c
a c
a as
a a
2
2
, 2
2
( )1exp for
2 2
( )1exp for
2 2
left leftsd sd
a g
right rightsd sd
a aa a
a as
a aa a
a a
5 10 15 20
0.0
0.4
0.8
Age
Se
lect
ivity
asd=1asd=5asd=10
,,
,max( )a g
a ga g
ss
s
The parameters of production function and MSY-related quantities can be obtained by maximizing the yield equation.
Yield
( )1 a aF Maa a
a a a
FY N e w
F M
Equations-Application on bigeye tuna stock in the EPO
Dynamic age-structured model
1, 1 1
1, 1 1 1,
( ), 1, 1
( ) ( )1, 1 1,
for 1
for 1
for
t a a
t a a t a a
t
F Mt a t a
F M F Mt a t a
R a
N N e a
N e N e a
, , ,
, ,
t a t g a gg
t g g a gg
F F s
E q s
where Ft,g is the fishing mortality for fully-selected fish derived by fishery g in year t, Et,g is the fishing effort of fishery g in year t, qg is the catchability of fishery g, and Sa,g is the fishing gear selectivity of fish at age a derived by fishery g.
Recruitment◦ The Beverton and Holt stock-recruitment
relationship which is re-parameterized in terms of the "steepness" of the stock-recruitment relationship.
Dynamic age-structured model
2 /20
0
4
(1 ) (5 1)tt
tt
hR SR e
h S h S
where ε is normally distributed process error, and σ2 is variance of process error in recruitment.
Selectivity
◦ Selectivity of longline (SLL) is assumed to be logistic curve
◦ Selectivity of purse-seine (SPS) is assumed to be double dome-shaped curve.
Dynamic age-structured model
1
50,
95 50
1 exp ln19a g
a as
a a
,,
,max( )a g
a ga g
ss
s
Selectivity ◦ The total age-specific fishing mortality scaled to a
maximum of one is used to represent longline and purse-seine combined selectivity in the equilibrium model to estimate MSY based quantities.
Gear-combined selectivity (SLL+PS) in 2010 is used to make comparison with assumptions of LL and PS selectivity.
Dynamic age-structured model
,,
,max( )t a
t at a
Fs
F
Yield
Catch rate (index of relative abundance)
Dynamic age-structured model
,( ), ,, ,
,
1 t a aF Mt g a gt g t a a
a t a a
F sY N e w
F M
, , ,t g t a a a ga
B N w s
2 /2, ,
tt g g t gI q B e
Knife-edged selectivity assumption
ac
msy
0.0
0.4
0.8
1.2
0.0
0.4
0.8
1.2
0.0
0.4
0.8
1.2
ac
msy
ac
msy
h=0.75
ac
msy
0.0
0.4
0.8
1.2
2 4 6 8 10
0.0
0.4
0.8
1.2
2 4 6 8 10
0.0
0.4
0.8
1.2
2 4 6 8 10
ac
msy
2 4 6 8 102 4 6 8 102 4 6 8 10
acm
sy
2 4 6 8 102 4 6 8 102 4 6 8 10
M=0.1M=0.2M=0.3
h=1
Age at first capture
FM
SY
K=0.1 K=0.2 K=0.3
When age at first capture is increased to a specific level (retains large amount of small fish), MSY will occur at a very high value of fishing mortality.
Double dome-shaped selectivity assumption
ac
msy
0.0
1.0
2.0
3.0
0.0
1.0
2.0
3.0
0.0
1.0
2.0
3.0
M=0.1M=0.2M=0.3
ac
msy
ac
msy
h=0.75
ac
msy
0.0
1.0
2.0
3.0
2 4 6 8 10
0.0
1.0
2.0
3.0
2 4 6 8 10
0.0
1.0
2.0
3.0
2 4 6 8 10
ac
msy
2 4 6 8 102 4 6 8 102 4 6 8 10
acm
sy
2 4 6 8 102 4 6 8 102 4 6 8 10
h=1
Standard deviation of age for dome-shaped selectivity
FM
SY
K=0.1 K=0.2 K=0.3
When SD of age is smaller than a specific level (fishes are caught at a narrow age/size range), MSY will occur at a very high value of fishing mortality.
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