Sheng-Ping Wang1,2, Mark Maunder2, and
Alexandre Aires-Da-Silva2
1. National Taiwan Ocean University
2. Inter-American Tropical Tuna Commission


For many situations, catch and data are only
available for assessment especially for nontarget species, small scale fisheries...
The Schaefer surplus production model is
commonly used in fisheries stock
assessment.
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
2

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.
3

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.
MSY
m  f ( B M SY / B 0 )
B0
4

Surplus production models represent
population dynamics as a function of a single
aggregated measure of biomass.
B t 1  f ( B t ,  )
 ~  B0 , r , m
e.g. carrying capacity (K or B0), productivity rate (r),
and shape parameter (m)
5

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
6

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
7

Age-structured model can incorporate
biological processes and selectivity for
considerations.B t   N t , a w a
a
N t 1  N t   M t  C t    G t  R t 
Rt  f ( S t )
St 
N
t .a
m a wa
a
C t  f ( Bt , F )
g
Bt 
g
N
t .a
sa wa
a
I t  q Bt
g
g
g
8


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.
9

First we use simulation analysis to illustrate
the impact of selectivity and biological
parameters on the production function based
on equilibrium age-structured model.
10

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.
11

Sensitivity of MSY and related management
quantities to biological parameters and
selectivity is analyzed based on an agestructured 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
12

The analysis is repeated for a variety of
values for the steepness of the stockrecruitment 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
13

Estimates of shape parameter (BMSY/B0)
◦ Age at first capture is fixed at 4 yrs.
M = 0.1
K = 0.1
K = 0.2
K = 0.3
M = 0.2
K = 0.1
K = 0.2
K = 0.3
M = 0.3
K = 0.1
K = 0.2
K = 0.3
h = 0.5
h = 0.75
h = 1.0
0.39
0.38
0.36
0.33
0.31
0.29
0.27
0.23
0.19
0.39
0.38
0.37
0.32
0.30
0.29
0.26
0.22
0.17
0.38
0.37
0.36
0.31
0.30
0.28
0.23
0.16
0.15
Estimates of productivity parameter (r or
MSY/BMSY)
M = 0.1
K = 0.1
K = 0.2
K = 0.3
M = 0.2
K = 0.1
K = 0.2
K = 0.3
M = 0.3
K = 0.1
K = 0.2
K = 0.3
h = 0.5
h = 0.75
h = 1.0
0.04
0.06
0.07
0.07
0.10
0.13
0.10
0.17
0.25
0.07
0.10
0.12
0.12
0.18
0.24
0.18
0.32
0.52
0.10
0.14
0.18
0.18
0.27
0.36
0.31
0.63
1.00
Yield

Biomass
Two types of curves are used to exam the
impacts of selectivity on the production
function and MSY based quantities.
0.8
asd=1
asd=5
asd=10
0.4

Knife-edged selectivity
Double dome-shaped selectivity (only change the shape
of curve on the right hand side)
0.0

Selectivity

5
10
Age
15
20
16
K=0.2
K=0.3
shape
shape
shape
ac
ac
ac
shape
shape
shape
ac
h=0.75
ac
h=1
ac
0.4
0.8 0.0
0.4
shape
h=0.75
h=1
Biomass
0.0
shape
SMSY S0
ac
Yield
0.8
0.0
ac
shape
ac
shape
0.2
0.4 0.0
M=0.1
M=0.2
M=0.3
0.2
shape
shape
BMSY B0
0.4
K=0.1
2
4
6
ac
8
10
2
4
6
8
10
ac capture
Age at first
2
4
6
ac
8
10
0.0
0.8
ac
ac
2
4
6
ac
Yratio
0.4
Yield
0.0
8
Yratio
0.4
ac
ac
10
2
4
6
Yratio
ac
Yratio
0.0
0.8
Yratio MSY
ac
Yratio
0.8
SMSY Yratio
0.0
Yratio
Yratio
0.4
BMSY Yratio
0.8
M=0.1
M=0.2
M=0.3
Yratio
0.4
Yratio MSY
K=0.1
K=0.2
K=0.3
8
h=0.75
ac
10
ac capture
Age at first
2
4
6
ac
h=1
ac
h=0.75
ac
h=1
Biomass
8
10
shape
h=0.75
Yield
shape
K=0.3
ac
shape
ac
K=0.2
shape
0.4 0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6
ac
h=1
ac
shape
ac
ac
shape
ac
shape
Biomass
shape
0.2
0.4 0.0
M=0.1
M=0.2
M=0.3
ac
6
8
ac
6
8
h=0.75
h=1
0.0
0.2
shape
shape
SMSY S0
shape
BMSY B0
shape
K=0.1
2
4
10
2
4
10
2
4
6
8 10
ac deviation of age ac
ac
Standard
for dome-shaped selectivity
1.0
0.8
0.6
0.4
0.2
ac
h=1
0.0
ac
Yratio
Yratio
0.8
0.4
ac
Selectivity
Yratio
h=0.75
0.0
0.4
K=0.3
0.0
Yratio MSY
BMSY Yratio
0.8
M=0.1
M=0.2
M=0.3
K=0.2
Yratio
K=0.1
5
10
15
ac
Yratio
Yratio
ac
h=0.75
ac
Yratio
0.4
ac
Yratio
0.8
Yield
0.0
0.4
ac
ac
h=1
0.0
Yratio MSY
SMSY Yratio
0.8
Age
Biomass
2
4
6
8
10
2
4
6
8
10
2
4
6
8
10
ac deviation of age ac
ac
Standard
for dome-shaped selectivity
20

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


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.
Selectivity
◦ Selectivity of longline (SLL) is assumed to be logistic
curve
◦ Selectivity of purse-seine (SPS) is assumed to be
descending right hand limb.
4
6
Age
8
10
0.0 0.2 0.4
Selectivity
0.0 0.2 0.4
2
0.6 0.8 1.0
Purse-seine
Double dome-shaped curve
0.6 0.8 1.0
Longline
Logistic curve
Selectivity

2
4
6
Age
8
10
Selectivity
◦ The age-specific fishing mortality in 2010 is used
to calculate the longline and purse-seine combined
selectivity (SLL+PS).
0.0 0.2
0.4
0.6
0.8 1.0
Combined selectivity
Based on fishing mortality in 2010
Selectivity

2
4
6
Age
8
10


The gear-specific catch is calculated without
error
Gear-specific catch rate (index of relative
abundance) is calculated incorporating a
multiplicative lognormal observation error.

Pre-specific biological and fishery parameters
Category of parameters
von Bertalanffy growth function
L∞
K
t0
Length-Weigth relationship
a
b
Age at maturity
am
Virgin recruitment
R0
Steepness for spawning biomass recruitment relationship
h
Natural mortality
M
Value
1
0.2
0
1
3
4
100
0.75
0.4

Pre-specific biological and fishery parameters
Category of parameters
Catchability
q for longline
q for purse seine
Standard deviation of random residuals
for Recruitment
for CPUE
Value
0.0175
0.35
0.6
0.2
3.0
Longline
Purse seine
0.0
(B)
0.4
B B0
Longline
Purse seine
1980:2010
0.8
1.2 0.0
1.0
0.8
2.0
(A)
1950
1970
1990
0.4
Year
Longline
Purse seine
0.0
Simulated yield level
E[29:59, 1]
Simulated Effort level

1980
1990
2000
Year
2010
2010

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.
B t 1  f ( B t , B 0 , r , m )
B t 1
 B tm

 Bt 
 m 1  B t   C t
 1
  B0

  1
m

r

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

The shape and productivity parameters are
based on the



vulnerable biomass
spawning biomass
Longline selectivity
Purse-seine selectivity
Gear-combined selectivity
500 simulation runs were carried out for each
scenario.
Estimation category
Fixed r & m
3.0 0.0
0.2
0.4
0.0
0.3
0.6
SLL
Fixed r
Fixed m
1.5
BMSY B0
SPS
Estimate all
Age-structured
Fixed r & m
Fixed r
Fixed m
Estimate all
Age-structured
Fixed r & m
Fixed r
Fixed m
Estimate all
Age-structured
0.0
Productivity
MSY BMSY
Shape
parameter
MSY
SLL
PS
32
Fixed r & m
4
6
8 0
2
4
6
SLL
Fixed r
Fixed m
2
Bcur BMSY
SPS
Estimate all
Age-structured
Fixed r & m
Fixed r
Fixed m
Estimate all
Age-structured
Fixed r & m
Fixed r
Fixed m
Estimate all
Age-structured
0
Ucur UMSY
SLL
PS
Estimation category
33

Shape and productivity parameters estimated based on prespecific values obtained from various selectivity assumptions
and measurement of biomass
34

Current biomass ratio (Bcur/BMSY) estimated based on prespecific values obtained from various selectivity assumptions
and measurement of biomass
35
SLL
PS
0.1
0.2
0.3
0.4
0.5
SPS
BMSY B0
SMSY S0
0.0
Relative biomass
SLL
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
Steepness (h)
36
Residual sum of squares for estimation
models with time-varied r and m.
Fixed m
Fixed r & m
PSS
SLL
SPS
SLL
Time-varied
PSS
SLL
SPS
SLL
Time-varied
PSS
SLL
SPS
SLL
Time-varied
0
5
10
20
Fixed r
Residual
sum of squares

Estimation category
37


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.
38
Estimating shape parameter of PellaTomlinson production model would be
problematic.
◦ The estimations are biased and imprecise.
◦ Lead to the problematic estimates.
.
Proportion of convergence (%)

100
98
96
94
92
90
SLL
SPS
PLL+PS
39

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.
40


Production function is substantially
influenced by biological process and
selectivity assumptions.
Schaefer model might not be appropriate for
most scenarios.
41


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.
42
Thank you for your listening
43

0
Na
 R0
 0 M
  N a  1 e a 1
 0 M
a 1
 N a 1 e
 1  e  M a
S0 

0
N a w a m a ra
a
B0 

a
0
N a wa sa
for a  1
for 1  a  
for a  
• 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).
45

R
(F M
)

N a   N a  1 e a 1 a 1

 ( Fa 1  M a 1 )
N
e
 a 1
 1  e  ( Fa  M a )
for a  1
for 1  a  
for a  
Fa  F  s a
S 
N
a
w a m a ra
a
wa sa
a
B 
N
Where F is the fishing mortality for
full-recruitment, and sa are the
selectivity of fish at age a.
a
46
◦ The Beverton and Holt stock-recruitment
relationship which is re-parameterized in terms of
the "steepness" of the stock-recruitment
relationship.
R 
 
X 
X
S 0 (1  h )
X is the spawning stock biomass per
recruit:
X 
5h  1
4 h R0
Xa
a
4 h R0
 

Xa
 w a m a ra

a 1
  Fa  M a

  w a m a ra e j 1

a 1

  Fa  M a

e j 1
 w a m a ra
(F M )
1 e a a

for a  1
for 1  a  
for a  
◦ Knife-edged selectivity
0
sa  
1
for a  a
c
for a  a
c
◦ Double dome-shaped selectivity
1
2 a
right
sd
 (a  a )2
exp 
2
right
 2  a sd 
sa ,g 
for a  a 



for a  a 
asd=1
asd=5
asd=10
0.8
left



0.4
2  a sd
 (a  a )2
exp 
2
left
2
a


sd

0.0
1
Selectivity
s a , g







s a , g
m ax( s a , g )
5
10
Age
15
20
Y 

a

Fa
Fa  M a

Na 1 e
 ( Fa  M a )
w
a
The parameters of production function and
MSY-related quantities can be obtained by
maximizing the yield equation.

N t ,a
 Rt

(F
M
)
  N t 1, a 1 e t 1 ,a 1 a 1

 ( Ft 1 , a 1  M a 1 )
 ( Ft 1 , a  M a )
N
e

N
e
t  1, a
 t 1, a 1
Ft , a 
F
t,g
 sa ,g
t,g
 q g  sa ,g
g

E
g
for a  1
for 1  a  
for a  
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.
Rt 
4 hR 0 S t
(1  h ) S 0  (5 h  1) S t
 t  / 2
2
e
where ε is normally distributed process error, and
σ2 is variance of process error in recruitment.

Selectivity
sa ,g 
s a , g
m ax( s a , g )
◦ Selectivity of longline (SLL) is assumed to be logistic
curve
s a , g

a  a 50

1  exp   ln 19
a 95  a 50





1
◦ Selectivity of purse-seine (SPS) is assumed to be
double dome-shaped curve.

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.
st , a 
Ft , a
m ax( Ft , a )
 Gear-combined selectivity (SLL+PS) in 2010 is used to
make comparison with assumptions of LL and PS
selectivity.

Yield
Yt , g 

a

Ft , g  s a , g
Ft , a  M a

N t ,a 1  e
 ( Ft , a  M a )
w
a
Catch rate (index of relative abundance)
I t , g  q g Bt , g e
Bt , g 
N
a
t ,a
 t  / 2
2
wa sa , g
K=0.2
h=0.75
ac
msy
ac
msy
ac
K=0.3
msy
msy
0.0 0.4 0.8 1.2 0.0 0.4 0.8 1.2
msy
FMSY
msy
K=0.1
h=1
M=0.1
M=0.2
M=0.3
2
4
6
ac
8
10
2
4
6
8
10
Age at first
ac capture
2
4
6
8
10
ac
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.
K=0.2
K=0.3
msy
h=0.75
ac
ac
msy
1.0
ac
msy
2.0
3.0
0.0
1.0
msy
2.0
M=0.1
M=0.2
M=0.3
h=1
0.0
msy
FMSY
msy
3.0
K=0.1
2
4
6
8
10
2
4
6
8
10
2
4
6
8
10
Standard
ac
deviation of age ac
for dome-shaped selectivity
ac
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.