Biological Models Used to Evaluate Medium-Term Impacts

9-January-2004

Stock Assessment

• Objectives of an assessment:– Stock status– Uncertainties– Projections

Assessment Input Data• Basic Biology:

• lifespan • growth • movements

• Fishery Information:• maturity rate• historical development (areas, gears)• past and current regulations (size limits, gear restrictions).• catch (landings, discards, age/size distribution)• effort (catch rates)

• Surveys• distribution• relative abundance and biomass over time• age/size structure• life history (growth, maturity)

Types of Assessments• A nonlinear progression from “data-poor” to “data-rich” situations.

– Index Methods (n = 7):• Descriptive assessment of catch and survey data.

– Biomass Dynamics Methods (n = 1):• Combined analysis of catch and survey data with a simple

biomass-based population model.– Age-Structured Methods (n = 11, 8 of 11 include discards):

• Virtual Population Analysis: Back-calculate stock numbers at age using age distribution of the catch, calibrated with surveyindices to minimize measurement error.

• Statistical Catch-at-Age Analysis: Forward-projection of stock numbers at age using age distribution of the catch, calibrated with survey indices or other auxiliary information in a likelihood-based framework.

Age-Based Methods

• Age distribution of the catch.– From census of total

catch biomass and port samples.

– SNE yellowtail example:

• 1987 yearclass dominated the catch in the late 80s early 90s.

Southern New England Yellowtail Catch

-2003

-1998

-1993

-1988

-1983

-1978

-1973

0 1 2 3 4 5 6 7 8

Age

Year

VPA• Reconstruction of all

yearclasses gives a total population estimate.

• Input Data:– catch at age– estimate of natural

mortality– initial guess about

abundance of survivors at the oldest age.

Southern New England Yellowtail Abundance

-2003

-1998

-1993

-1988

-1983

-1978

-1973

0 1 2 3 4 5 6 7 8

Age

Year

VPA Calibration• Initial guesses are replaced

with estimates:– oldest age of historical

yearclasses estimated by assuming that age-7 fish have the same vulnerability to the fishery as ages 4-6:

– yearclasses that are alive now require more information.

• need an independent index of relative abundance over time.

Southern New England Yellowtail Abundance

-2003

-1998

-1993

-1988

-1983

-1978

-1973

0 1 2 3 4 5 6 7 8

Age

Year

( )tZt

ttt eF

ZCN −−=

1

VPA Calibration• Abundance of

living yearclasses in 2002 are estimated using a predictive relationship between historical VPA abundance and survey indices.

0

5

10

15

20

25

0 10 20 30 40 50 60 70

Age-3 Abundance from VPA (millions)

Age

-3 S

prin

g Su

rvey

Inde

x (#

/tow

)

observedpredicted

2002

1990(1987 yearclass)

VPA Estimates• Informative

Assessment:– Example: SNE

yellowtail– Estimates of stock size

and F,– But also age

distribution, recruitment, mature biomass, etc.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

1973

1975

1977

1979

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

Fish

ing

Mor

talit

y (4

-6)

F40%

0

5

10

15

20

25

1972

1974

1976

1978

1980

1982

1984

1986

1988

1990

1992

1994

1996

1998

2000

SSB Year; Recruitment Yearclass

Spaw

ning

Bio

mas

s ('0

00s

mt)

0

20

40

60

80

100

120

140

Age

-1 A

bund

ance

(mill

ions

)

recruitmentSSB

Groundfish Stock Status - 1996

Biomass 1996 / B-MSY0.0 0.3 0.5 0.8 1.0

F 19

96 /

F-M

SY

0.00.51.01.52.02.53.03.54.04.55.05.56.0

GM Cod

GB Cod

GM HadGB Had

CC YT

GB YT

SNE YTWitch

Plaice

GM WinGB Win

SNE Win

W Hake

Pol

RedfishPout

N Wind

S Wind

overfishingnot overfished

no overfishingoverfished

no overfishingnot overfished

overfishingoverfished

F-MSY

1/2 B-MSY

Groundfish Stock Status - 2002

Biomass 2002 / B-MSY0.0 0.3 0.5 0.8 1.0 1.6 1.8 2.0

F 20

02 /

F-M

SY

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

GM Cod

GB Cod

GM Had

GB Had

CC YT

GB YT

SNE YT

Witch

Plaice

GB Win

SNE Win

W Hake

Pol

RedfishPout N Wind

S Wind

overfishingnot overfished

no overfishingoverfished

no overfishingnot overfished

overfishingoverfished

F-MSY

1/2 B-MSY

VPA Uncertainty

• Similar to production model: survey measurement errors are reshuffled many times to estimate precision. (“bootstrapping”).

• The estimate of 2001 SSB is 1850mt, with a 80% confidence limit of 1500 to 2500mt.

SNE Yellowtail

0

10

20

30

40

50

60

70

1200

1300

1400

1500

1600

1700

1800

1900

2000

2100

2200

2300

2400

2500

2600

2700

2800

2900

3000

3100

3200

3300

2001 SSB (mt)

Freq

uenc

y

Projections

"It is far better to foresee even without certainty than not to foresee at all" Poincare, The Foundations of Science

Short-Term Projection

• Fluke (SAW35):– Landings in 2003

would need to be 10,580 mt (23.3 million lbs) to meet the target F rate of Fmax = 0.26 with 50% probability.

Long-Term Projection

• Georges Bank Cod– the stock is

expected to have approximately a 50% chance of rebuilding to SSBMSY by 2026 if fished at an F of 0.18.

Year

2002 2006 2010 2014 2018 2022 2026

Spaw

ning

Bio

mas

s (0

00s

mt)

0

50

100

150

200

250

300

75th percentile Median Spawning Biomass25th percentileBMSY

Age-Structured Model

• Population Numbers, Survival, Spawning Biomass

• Catch, Landings, and Discards

• Population Harvest

Population Numbers at Age

( )

( )( )( )

( )

N t

N tN tN t

N t

R

R

R

A

=

+

+

1

2

M

Survival by Age Class

N t N t e

for a R to A

a aM t F ta( ) ( ) ( ) ( )= ⋅

= + −

−− − − −−

11 11

1 1

Survival of Plus Group

N t N t eN t e

A AM t F t

AM t F t

A

A

( ) ( )( )

( ) ( )

( ) ( )

= − ⋅

+ − ⋅

− − − −

−− − − −−

11

1 1

11 11

Spawning Biomass

[ ]SSB t W FM N t eS a a aZ t M t F t

a R

APROJ a( ) ( ),

( ) ( ) ( )= ⋅ ⋅ ⋅ − ⋅ +

=∑

Catch Numbers at Age

[ ]C tF t

M t F te N ta

a

a

M t F ta

a( )( )

( ) ( )( )( ) ( )=

+− ⋅− −1

Landings

[ ]L t C t DF t Waa

A

a L a( ) ( ) ( ) ,= ⋅ − ⋅=∑

11

Discards

D t C t DF t Wa a D aa

A

( ) ( ) ( ) ,= ⋅ ⋅=∑

1

Population Harvest

• Input fully-recruited fishing mortality F(t)

• Input partial recruitment vector PR(t) and ZPROJ(t)

• Input discard fraction at age vector DF(t) if applicable

• Input landings quota Q(t)

• Input partial recruitment vector PR(t) and ZPROJ(t)

• Input discard fraction at age vector DF(t) if applicable

• Solve for F(t)

Fishing Mortality at Age

F t F t PR ta a( ) ( ) ( )= ⋅

Catch Numbers at Age as a Function of Fishing Mortality

[ ]C FPR t F

M t PR t Fe N ta

a

a

M t PR t Fa

a( )( )

( ) ( )( )( ) ( )=

⋅+ ⋅

− ⋅− − ⋅1

Landings as a Function of F

[ ]L F C F DF t Waa

A

a L a( ) ( ) ( ) ,= ⋅ − ⋅=∑

11

Solve for Fishing Mortality to Harvest Landings Quota

Q L F− =( ) 0

Age-Structured Model

• Stock-Recruitment Relationship

• Initial Population Abundance

• Abundance and Fishing Mortality Thresholds

Stock-Recruitment Relationship• Deterministic component

• Stochastic component

( ) ( )N t f SSB t R tR ( ) ( ) , ,= − ⋅θ ε ϖ

Recruitment Models

• Dependent on spawning biomass (n = 10)

• Independent of spawning biomass (n = 5)

• Uncorrelated stochastic component (n = 10)

• Correlated stochastic component (n = 5)

Beverton-Holt Curve Lognormal Error

( )

n ta ssb t Rb ssb t R

e

where w N

Rw

w

( )( )( )

~ ,

=⋅ −+ −

⋅

0 2σ

Empirical Cumulative Distribution Function

( )

N t T R R UST

R

where S U T

R S S S( ) ( )

( )

= − − −−−

+

= + ⋅ −

+111

1 1

1

Georges Bank yellowtail flounder recruitment CDF

Recruitment (000,000s age-1 fish)

0 20 40 60 80 100

Freq

uenc

y R

< R

ecru

itmen

t

0

5

10

15

20

25

Population Abundance and Fishing Mortality Thresholds

• Abundance– Spawning biomass– Mean biomass of USER-selected age range– Total biomass

• Fishing mortality– Fully-recruited fishing mortality– Fishing mortality weighted by biomass

Probability of Achieving Threshold

( )Pr ( )( )

( )SSB t SSB

K tK tTHRESHOLDTHRESHOLD

TOTAL≥ =

Initial Population Abundance

• No uncertainty for estimate of N(1)

• Uncertainty for estimate of N(1)– Distribution of bootstrap replicates of N(1)

• Nonparametric• Parametric

– Samples from posterior distribution of N(1)

Bootstrap Algorithm

George Bank Haddock 2001 Spawning Biomass Distribution

Precision of 2001 SSB Estimate

SSB (thousand, mt)

50 55 60 65 70 75 80 85 90 95 100 105 110

Freq

uenc

y

0

20

40

60

80

100

120

140

160

Cum

ulat

ive

Prob

abilit

y (%

)

0

10

20

30

40

50

60

70

80

90

100

Input/Output Schematic Diagram

INPUT FILENAME

OUTPUT FILENAME

SYSTEM DATA

AGEPRO

SIMULATION DATA

BIOLOGICAL DATA

FISHERY DATA

INPUT FILE

PROJECTIONDESCRIPTION

HARVEST STRATEGY

SSB TRAJECTORY

SSB THRESHOLD*

RECRUITMENTTRAJECTORY

CATCH-AT-AGEINDEX*

LANDINGSTRAJECTORY*

MARKET CATEGORYTRAJECTORIES*

DISCARDTRAJECTORY*

REALIZED FTRAJECTORY*

OUTPUT FILE

MARKET CATEGORY FILE*

REALIZED LANDINGSBY CATEGORY*If applicable

NO

YES

NO

YES

NO

NO

YES

BEGIN

END

*If Applicable

Flowchart for AGEPRO software

DONEWITH ICLOOP?

DONEWITH SIM

LOOP?

SET INITIALPOPULATION VECTOR

DONEWITH TIME

LOOP?

SET VARIABLES THATVARY BY TIME PERIOD*

COMPUTECATCH AT AGE

ISQUOTA

FEASIBLE?

RECORD THATSIMULATION IS

INFEASIBLE

COMPUTE LANDINGS

COMPUTE SSB

GENERATE RANDOMRECRUITMENT

COMPUTE FUTUREPOPULATION VECTOR

FOR NEXT TIME PERIOD

COPY CURRENT SSB TOPREVIOUS SSB

COPY FUTUREPOPULATION VECTOR

TO CURRENT

George Bank Haddock FREBUILDfor 1999-2009 Time Horizon

Year

2002 2006 2010 2014 2018 2022 2026

Fish

ing

Mor

talit

y

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Fishing MortalityFMSY

George Bank Haddock FREBUILDSpawning Biomass Distribution

Year

2002 2006 2010 2014 2018 2022 2026

Spaw

ning

Bio

mas

s (0

00s

mt)

0

50

100

150

200

250

300

350

75th percentile Median Spawning Biomass25th percentileBMSY

George Bank Haddock FREBUILDLandings Distribution

Year

2002 2006 2010 2014 2018 2022 2026

Land

ings

(000

s m

t)

0

5

10

15

20

25

30

35

40

45

50

55

60

65

75th percentileMedian Landings25th percentile

“““Prediction is very difficult Prediction is very difficult Prediction is very difficult …especially about the future”…especially about the future”…especially about the future”

NielsNielsNiels BohrBohrBohr