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

Thank you for your listening

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.