FTP
Surplus-Production Models
Overview
2

Purpose of slides:




Introduction to the production model (revision)
Overview of different methods of estimating the parameters
Go over some critique of the method
Source:


Haddon 2001, Chapter 10
Hilborn and Walters 1992, Chapter 8
Introduction
3

Nomenclature:



Simplest analytical models available
Pool recruitment, mortality and growth into a single
production function


surplus production models = biomass dynamic models
The stock is thus an undifferentiated mass
Minimum data requirements:

Time series of index of relative abundance



Survey index
Catch per unit of effort from the commercial fisheries
Time series of catch
Revision
General form of surplus production
4
Revision
Are related directly to Russels mass-balance formulation:
Bt+1 = Bt + Rt + Gt - Mt – Ct
Bt+1 = Bt + Pt – Ct
Bt+1 = Bt + f(Bt) – Ct



Bt+1: Biomass in the beginning of year t+1 (or end of t)
Bt: Biomass in the beginning of year t
Pt: Surplus production



the difference between production (recruitment + growth) and
natural mortality
f(Bt): Surplus production as a function of biomass in the
start of the year t
Ct: Biomass (yield) caught during year t
Functional forms of surplus productions
5

Classic Schaefer (logistic) form:
 B
t
f
B

r
B
1

t
t
 K


The more general Pella & Tomlinson form:
p


r
B


t
f
B

B
1



t
t  
p 
K



Note: when p=1 the two functional forms are the same
Revision
Revision
Population trajectory according to Schaefer model
Stock biomass Bt
6
K: Asymptotic carrying capacity
100
90
80
70
60
50
40
30
20
10
0
Unharvested curve
B

B

r
B
1

B
K


t

1
t
t
t
0
5
10
15
Time t
20
25
7
Surplus production as a function of stock size
Surplus production
 B
t
rB
1

t
14
K


Revision
Maximum production
12
10
8
6
K
4
K /2
2
0
0
10
20
30
40
50
60
Stock biomass Bt
70
80
90
100
Revision
The reference points from a production model
• Biomass giving maximum sustainable yield:
BMSY = K/2
• Fishing mortality rate leading to BMSY:
FMSY = r/2
• Effort leading to BMSY:
EMSY = FMSY/q = (r/2)/q = r/(2q)
• Maximum sustainable yield
MSY = FMSY * BMSY = r/2 * K/2 = rK/4
Add fishing to the system
Bt 1  Bt  rBt 1  Bt K   Ct
• Recall that
Ct  Ft Bt  qEt Bt
• r, B and q are all parameters we need to estimates
• We also need an index of abundance
C
P
U
E
f
B

t
t
C
t
• most often assume: C
P
U
E


q
B
t
t
E
t
A side step – what does the model do
0.35
Ft
Fmsy
0.30
Parameters
r
K
p
B1
0.4
180
1
144
Reference points
Bmsy
90
Fmsy
0.2
MSY
18
0.25
0.20
0.15
0.10
0.05
0.00
0
20
40
60
80
200
25
20
0.50
180
Catch
MSY
Biomass
Bmsy
K
CPUE
160
140
120
15
10
5
0
20
40
60
80
100
0.45
0.40
0.35
0.30
100
0.25
80
0.20
60
0.15
40
0.10
20
0.05
0
0
100
0
20
40
60
80
0.00
100
FTP
Estimating r, K and q
12
Estimation procedures
Three types of models have been used over time:

Equilibrium methods



Regression (linear transformation) methods



The oldest method
Never use equilibrium methods!
Computationally quick
Often make odd assumption about error structure
Time series fitting

Currently considered to be best method available
Ct = Production
The equilibrium model 1
13

The model:

The equilibrium assumption:
C
t
C
P
U
E


q
B
t
t
B

B

r
B
1

B
K

C
E


t

1
t t
t
t
t
Ct  rBt 1 Bt / K 

Each years catch and effort data represent an equilibrium
situation where the catch is equal to the surplus production at
the level of effort in that year. This also means:
Bt 1  Bt


If the fishing regime is changed (effort) the stock is assumed to
move instantaneously to a different stable equilibrium biomass
with associated surplus production. The time series nature of the
system is thus ignored.
THIS IS PATENTLY WRONG, SO NEVER FIT THE DATA USING
THE EQUILIBRIUM ASSUMPTION
Ct = Production
The equilibrium model 2
14
Given this assumption:
It can be shown that:


C

r
B
1
B


t
t
tK
C
t
B

B

r
B
1

B
K

Y
P
U
E


q
B


t

1
t
t
t
tC
t
t
E
t
can be simplified to
Surplus production
14
C

a
E
b
E
12
2
10
8
6
4
2
0
0
10
20
30
40
50
60
70
80
90
C
abE
E
100
Effort

Then one obtains the reference values from:
EMSY = a/(2b)
MSY = (a/2)2/b
15
The equilibrium model 3
Ct = Production
Equilibrium model 4
If effort increases as the fisheries develops we may expect to get data like the blue trajectory
Yield
16
Ct = Production
20
18
16
14
12
10
8
6
4
2
0
Observations
and the fit
The true
equilibrium
0
10
20
30
40
50
Effort
60
70
80
90
100
17
The regression model 1
Next biomass = this biomass + surplus production - catch
Y
F
B
q
E
B
t
tt
tt
1
2
B
t
B

B

r
B
(
1)

q
E
B
t

1 t
t
tt
K
C
P
U
E
U
t
t
B


t
q
q
• Substituting 2 in
U t 1 U t
Ut

r
q
q
q
1
gives:
 Ut 
1 
  EtU t
 qK 
….and multiplying by
q
Ut
U
r
U
U
r
t

1
t
t

1

1

r


E
q


1

r

U

q
E
t
t
t
U
q
K
U
q
K
t
t
18
The regression model 2
U
r
t

1

1

r
U
q
E
t
t
U
q
K
t
…this means:
 density 

 rateof   instrinsic 

 
  dependent   fishing 

 
 change    growth   

 in biomass  rate   reductionin   mortality

 
  growth rate


U t 1
 1 on Ut and Et which is a multiple regression of the form:
Regress:
Ut
Y

b

b
X

b
Xw
h
e
r
eX

U
n
dX

E
0
1
1
22
1
t a
2
t

r

b
0
brb

,

,
b


q
K

0
1
2
q
K
b

b
1
2
Time series fitting 1
19

The population model:
B
t
BB

r
B
1


Y
t

1
t
t
t
K


The observation model (the link model):


i
ˆ
ˆ
U

q
B

o
r
U
q
B
e
t
t i
t
t

The statistical model
2
 
t

t
2

ˆ
ˆ
S
S

U

U
o
r
S
S

l
n
U

l
n
U


m
i
n
t t
m
i
n
t
t
t

0
t

0
Time series fitting 2
20

The population model:


The observation model (“the link model”):


Given some initial guesses of the parameters r and K and an
initial starting value B1, and given the observed yield (Yt) a series
of expected Bt’s, can be produced.
The expected value of Bt’s is used to produce a predicted series
of Ût (CPUEt) by multiplying Bt with a catchability coefficient (q)
The statistical model:

The predicted series of Ût are compared with those observed (Ut)
and the best set of parameter r,K,B0 and q obtained by
minimizing the sums of squares
21
Example: N-Australian tiger prawn fishery 1
40000
Effort
0.30
35000
0.25
30000
0.20
25000
20000
0.15
15000
0.10
10000
0.05
5000
0
1970
7000
CPUE
1975
1980
1985
1990
1995
0.00
1970
1975
1980
1985
1990
1995
History of fishery:
1970-76: Effort and catch low, cpue high
1976-83: Effort increased 5-7fold and catch
5fold, cpue declined by 3/4
1985-: Effort declining, catch intermediate,
cpue gradually increasing.
Catch
6000
5000
4000
3000
Objective: Fit a stock production model to
the data to determine state of stock and
fisheries in relation to reference values.
2000
1000
0
1970
1975
1980
1985
1990
1995
Haddon 2001
Example: N-Australian tiger prawn fishery 2
22
0.30
CPUE
Bt/Bmsy
0.25
0.20
0.15
0.10
0.05
0.00
1970
1.2
1975
1980
1985
1990
1995
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
1970
1975
1980
1985
1990
1995
The observe catch rates (red) and model fit
(blue).
Parameters: r= 0.32, K=49275, Bo=37050
Reference points:
MSY: 3900
Emsy: 32700
Bcurrent/Bmsy: 1.4
Et/Emsy
1.0
0.8
0.6
0.4
0.2
0.0
1970
1975
1980
1985
1990
1995
The analysis indicates that the current status
of the stock is above Bmsy and that effort is
below Emsy.
Question is how informative are the data and
how sensitive is the fit.
Haddon 2001
Adding auxiliary information
23

In addition to yield and biomass indices data may have
information on:

Stock biomass may have been un-fished at the beginning of the
time series. I.e. B1 = K. Thus possible to assume that

U1 = qB1=qK



However data most often not available at the beginning of the fisheries
Absolute biomass estimates from direct counts, acoustic or trawl
surveys for one or more years. Those years may be used to
obtain estimates of q.
Prior estimates of r, K or q


q: tagging studies
r: basic biology or other similar populations


In the latter case make sure that the estimates are sound
K: area or habitat available basis
Greenland halibut
Bootstrap uncertainty – history and predictions
FTP
Some word of caution
One way trip
27

“One way trip”

Increase in effort and decline in CPUE with time

A lot of catch and effort series fall under this category.
Effort
CPUE
Time
Time
Catch
Time
One way trip 2
28
Fit # r
R
1.24
T1 0.18
T2 0.15
T3 0.13

K
3 105
2 106
4 106
8 106
q
10 10-5
10 10-7
5 10-7
2 10-7
MSY
100,000
100,000
150,000
250,000
Emsy
0.6 106
1.9 106
4.5 106
9.1 106
Identical fit to the CPUE series, however:

K doubles from fit 1 to 2 to 3


Thus no information about K from this data set
Thus estimates of MSY and Emsy is very unreliable
SS
---3.82
3.83
3.83
The importance of perturbation histories 1
29


“Principle: You can not understand how a stock will respond to
exploitation unless the stock has been exploited”. (Walters and
Hilborn 1992).
Ideally, to get a good fit we need three types of situations:
Stock size
low
high (K)
high/low

Effort
low
low
high
get parameter
r
K (given we know q)
q (given we know r)
Due to time series nature of stock and fishery development it is
virtually impossible to get three such divergent & informative
situations

steel JK slide from the first lecture on the development
One more word of caution
31

Equilibrium model: Most abused stock-assessment technique –
hardly used


Published applications based on the assumption of equilibrium should be
ignored
Problem is that equilibrium based models almost always produce
workable management advice. Non-equilibrium model fitting may
however reveal that there is no information in the data.


Latter not useful for management, but it is scientific
Efficiency is likely to increase with time, thus may have:
q  f (t )

e.g
q t  q1  qadd t
Commercial CPUE indices may not be proportional to abundance.
I.e.
C
P
U
E
q
B
t
t


Why do we be default assume that the relationship is proportional?
And then K may not have been constant over the time series of the
data
CPUE = f(B): What is the true relationship??
commercial CPUE
32
Why do we by
default assume
that the
relationship is
proportional?
10
9
8
7
6
5
4
3
2
1
0
Hyperstability
Proportional
Hyperdepletion
0
5
Biomass
10
33
Accuracy = Precision + Bias
Not accurate and not precise
Precise but not accurate
(Precisely wrong)
Accurate but not precise
(Vaguely right)
Accurate and precise
It is better to be vaguely right than precisely wrong!