assessment & management of fisheries based on mean-size...

Assessment & Management of Fisheries Based on Mean-Size

Statistics: an expansive methodology

John M. Hoenig, Amy Y.-H. Then, & Todd Gedamke, Meaghan Bryan, Jon Brodziak,

Quang C. Huynh, John F. Walter and Clay Porch

Age

Fre

quency

F = 0.2

F = 0.4

More Fishing Less Older Fish

Pauly (1984 p. 71)after Powell (1979)

LengthLc Linf

Nu

mb

er o

f fi

sh

Z/K ~ 1

Z/K ~ 2

Z/K > 2

Z/K < 1

More Fishing Less Large Fish

What to do with length composition data?

- Convert to ages, then analyze age data

Mixture analysis Cohort slicing(Ailloud et al. 2015)

length converted catch curve

0

1

2

3

4

5

6

0 0.5 1 1.5 2 2.5

relative age, t'

ln(n

um

ber)

ln(number)

regression

What to do with length composition data?

- Convert to ages, then analyze age data

)1log('infL

Lt Relative age

Pauly (1984)banded grouper (Epinephelus sexfasciatus)

Slope = 1 – Z/K

ln(n

um

ber

)

What to do with length composition data?

- Convert to ages, then analyze age data

- Analyze multiple years simultaneously in integrated stock assessment model

What to do with length composition data?

- Convert to ages, then analyze age data

- Analyze multiple years simultaneously in integrated stock assessment model

These approaches look at SHAPE of the length frequency distribution

Look at (just) a summary, the mean length

Alternative:

cLL

LLKZ

)(

Beverton-Holt mean length mortality estimator

growth rate maximum length

total mortality

mean lengthlength where all

animals fully

vulnerable

c

c

t tt

tt

N L dtL

N dt

LLL

Lc

KZ

ZL 1

Assumptions,Sophistication,Detail

Data & poorer quality,Complex,Sensitive to assumptions

Tradeoffs

More Less

• Relaxing assumptions

• Diagnostics

• Incorporating more types of data

• Bridges to (more) data-rich methods

Desiderata for mean-length-based methods

• Relaxing assumptions

• Diagnostics

• Incorporating more types of data

• Bridges to (more) data-rich methods

Desiderata for mean-length-based methods

Assumptions• Growth known• No variability

in size-at-age• Z constant

w/age• Z constant w/

time• constant

recruitment

• Relaxing assumptions

• Diagnostics

• Incorporating more types of data

• Bridges to (more) data-rich methods

Desiderata for mean-length-based methods

Data Types• mean length• catch rate

(cpue)• recruit index

(cpue by size)• effort• catch & cpue• other species

5 assumptions:

1. Asymptotic growth, K and L known &

constant over time

2. No individual variability in growth

3. Mortality constant with age (Selectivity, M)

4. Mortality constant over time Population in

equilibrium (mean length reflects mortality)

5. ‘Constant’ & continuous recruitment over time

Beverton-Holt mean length mortality estimator

cLL

LLKZ

)(

Queen Snapper

(hook and line in

Puerto Rico)

Parameter Lower bound Base Upper bound

Lc 335mm 365mm 465mm

VBK 0.25 0.45 0.65

L∞ 846mm 888mm 906mm

Growth known: What if life

history parameters

questionable??

Year

Mea

n le

ngt

h

Using Proportional Change in Mortality

11

1

( )

c

K L LZ

L L

22

2

( )

c

K L LZ

L L

2 1

1

Proportional Change in Z = Z Z

Z

2 1

2 1prop. change

1

1

( ) ( )

( )c c

c

K L L K L L

L L L LZ

K L L

L L

5 assumptions:

1. Asymptotic growth, K and L known &

constant over time

2. No individual variability in growth

3. Mortality constant with age (Selectivity, M)

4. Mortality constant over time Population in

equilibrium (mean length reflects mortality)

5. ‘Constant’ & continuous recruitment over time

Beverton-Holt mean length mortality estimator

cLL

LLKZ

)(

Ehrhardt-Ault (EA) mortality estimator

c

c

t tt

tt

N L dtL

N dt

tl

tl

Lλ?

Lc

Length

Age

tλSimulation approachIntroduction ● Method Results Conclusions

Chapter 2

Age

Length

Z, K, σ, tλ

EA: Varying imposed LλAge:

‘Actual’ tλ

Length

Age

tλSimulation approachIntroduction ● Method Results Conclusions

Chapter 2

Age

Length

Z, K, σ, tλ

EA: Varying imposed LλAge:

‘Actual’ tλ

‘Actual’ Lλ

Length

Age

tλSimulation approachIntroduction ● Method Results Conclusions

Chapter 2

Age

Length

Z, K, σ, tλ

EA: Varying imposed LλAge:

‘Actual’ tλ

‘Actual’ Lλ

Undertruncation

Overtruncation

Z = 0.1 Z = 0.25 Z = 0.5 Z = 1.0

K = 0.1, 0.4, 0.7, 1.0

varying imposed Lλ, Z & K (σ = 7)

A, B, C, D

Imposed tλ

BH EA

Data analyst specifies a tλ which corresponds to an Lλ

Take Home Message :

- Use not advised (Then et al. in revision)

growth variability not a problem for BH;

Truncation overestimate of Z

EA Model:- complex behavior, unpredictable bias

- RMSE much better or much worse

than BH, depending on Z, σ, & Lλ

Dome-shaped selectivity

- Live with positive bias- Evaluate sensitivity to suspected

selectivity pattern- Treat as Index of Z & Look at relative

change in Z- Incorporate selectivity into the model

no general solution

5 assumptions:

1. Asymptotic growth, K and L known &

constant over time

2. No individual variability in growth

3. Mortality constant with age (eg. Selectivity, M)

4. Mortality constant over time Population in

equilibrium (mean length reflects mortality)

5. ‘Constant’ & continuous recruitment over time

Beverton-Holt mean length mortality estimator

cLL

LLKZ

)(

First expansion: multiple years

Change in Z causes gradual (transitional) change in mean length to new equilibrium value

Year

The derivation

Younger animals Older animals

d

1 2 1 2 1 2

1 2 1 2 1 2

( ){ ( )exp( ( ) )}

( )( )( ( )exp( ))

cZ Z L L Z K Z Z Z K dL L

Z K Z K Z Z Z Z d

Simplified Equation (at least relatively)

d = number of years since change in mortality

d

Maximum likelihood estimation

Means distributed normally (Central Limit Theorem) “simplified

equation”

Year

1960 1970 1980 1990 2000

Me

an

Len

gth

(cm

)

40

50

60

70

80

Z1 = 0.15 Z2 = 0.48

Goosefish (Lophius americanus)Gedamke and Hoenig (2006)

Year

Mea

n le

ngt

h

Year

1960 1970 1980 1990 2000

Me

an

Len

gth

(cm

)

40

50

60

70

80

Z1 = 0.15 Z2 = 0.48

Diagnostic – pattern to residuals

Goosefish (Lophius americanus)Gedamke and Hoenig (2006)

Year

Mea

n le

ngt

h

Year

1960 1970 1980 1990 2000

Me

an

Len

gth

(cm

)

40

50

60

70

80

Z1 = 0.15 Z2 = 0.48

Year

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

Mean L

ength

(cm

)

40

50

60

70

80

Z1 = 0.14 Z2 = 0.29 Z3 =.55

Goosefish (Lophius americanus)Gedamke and Hoenig (2006)

Commercial Landings Northern Management Region

Barndoor Skate(Gulf of Maine, Southern New England. NMFS Survey)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

Year

Nu

mb

er

Cau

gh

t p

er

To

w

Barndoor skate NMFS survey

Barndoor skate (Dipturus laevis) Gedamke et al. (2008)

2nd expansion: Using an index of recruitment

Equate observed mean length with predicted; predicted computed as sum (not integral) and constant recruitment replaced by year-specific index

0

0.05

0.1

0.15

0.2

0.25

0.3

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

Year

Recru

its C

au

gh

t p

er

To

w

What if you don’t have an index of recruitment?

Overestimate Z when recruitment is increasing & underestimate when declining

3rd expansion: Incorporating effort data THoG model (Then, Hoenig, Gedamke)

Simply replace Z by q ft + M in “simplified equation”

same likelihood function 𝐿𝑝𝑟𝑒𝑑 − 𝐿𝑜𝑏𝑠2

Estimate catchability, q, & natural mortality, M

year-specific fishing & total mortality rates, F & Z

r2=-0.892 r2=-0.887 r2=-0.892

q

M

0.025, 0.975 Quantiles

True Values

σ’ = 1 σ’ = 3 σ’ = 5

Base Case: q & M estimates

Example: Z1 = 0.6, Z2 = 1.0, K = 0.4, Linf = 100

s’ depends on sample size & recruitment variability

Chapter 3

Z1

^Z2

σ’ = 1 σ’ = 3 σ’ = 5

true Z2/Z1 = 1.67

Z2/Z1

Z2/Z1

Base Case: Z1 & Z2 estimates & ratios

4th expansion – use total catch & cpue data

fishing effort

Then-Hoenig-Gedamke (THoG)

q, M, Ft, Zt

THoG

total catch, cpue

more years surplus production model

q, M, Ft, Zt MSY, BMSY, FMSY, Ft, Bt

abundance = cpue/q

effort = f = catch/cpue

Z = q catch/cpue + M

mortality increases fewer fish

5th expansion – use of cpue data

Ratio of observed CPUE inverse to relative change in Z

𝐶𝑃𝑈𝐸2

𝐶𝑃𝑈𝐸1=

𝑍1

𝑍2

cpue same transitional behavior as mean length

Year

Mo

rtal

ity

rate

, Zcp

ue

(denominator in derivation of “simplified equation”)

Fit model to data

• Estimate total mortality in stanzas

• Specify number of changes in mortality

• Obtain maximum likelihood estimates of Z, year(s) of change:

log Λ = log Λ𝐿 + logΛ𝐶𝑃𝑈𝐸

both parts have S(observed – predicted)2

• use AIC to find the best fitting model.

cpue Constrained Gedamke-Hoenig

period Z

cpue2 / cpue1 = Z1 / Z2 in equilibrium

cpue constrains magnitude of change in Z, not absolute levels

38

42

46

50

Me

an

Le

ng

ths

1

2

0 4 8 12 16 20 24

CP

UE

Year

0

1

2

Recru

itm

ent• Change in

recruitment changes mean length & cpue

Effects of recruitment

Expect:

Negative correlation of residuals & long runs

• Multiple species information on shared parameters

• Species ‘complexes’ subject to similar effort patterns

Example: 4 species w/ 3 changes in mortality

• Each species individually 32 parameters• Common change years 23 parameters• Common change years & proportional changes in

F 14 parameters

6th expansion – multi-species approach

Multispecies w/ Common Proportional Change in FCommon Year of Change = 1998.03

Common Proportional change in F = 0.57

200

220

240

260

280

300

320

340

360

380

1980 1983 1986 1989 1992 1995 1998 2001 2004 2007 2010

Mean

Len

gth

(m

m)

Blackfin

Silk

Vermillion

Derivation of overfishing limit (OFL)

(1) 𝑂𝐹𝐿 = 𝐹𝑀𝑆𝑌𝑁𝑐𝑢𝑟𝑟𝑒𝑛𝑡

• 𝐹𝑀𝑆𝑌 & 𝑁𝑐𝑢𝑟𝑟𝑒𝑛𝑡 difficult to obtain

• If recent (reference) average catch & F known:

(2) 𝐶𝑟𝑒𝑓 = 𝐹𝑟𝑒𝑓𝑁𝑟𝑒𝑓 𝑁𝑟𝑒𝑓 = 𝐶𝑟𝑒𝑓 / 𝐹𝑟𝑒𝑓

(3) 𝑂𝐹𝐿 = 𝐹𝑀𝑆𝑌𝐶𝑟𝑒𝑓

𝐹𝑟𝑒𝑓

• Use a per-recruit statistic as proxy for 𝐹𝑀𝑆𝑌:

(4) 𝑂𝐹𝐿 =𝐹𝑏𝑒𝑛𝑐ℎ𝑚𝑎𝑟𝑘

𝐹𝑟𝑒𝑓𝐶𝑟𝑒𝑓

Flow diagram of approach

Mean length Z

(Gedamke-

Hoenig)

Per recruit

analysis

𝐿, 𝐿𝑐 ,k, 𝐿∞

Z by period, Fbenchmark (e.g., F0.1, Fmax, F30%)

External values:M, wa, age-length, fecundity

Fref & Cref

OFL = Fbenchmark

FrefCref

(Most recent year of change to present)

External M

Example: Queen snapper in Puerto RicoM

ean

len

gth

Year

Data:

lengths

catches

F30% Fcur

Fre

qu

en

cy

0 5 10 15 20

0

5

10

15

20

25

20%

0

20

40

60

80

100

Cu

mu

lativ

e P

erc

en

t

Describe uncertainty in growth by simulation

F30% / Fref

Freq

uen

cy

Cu

mu

lati

ve %

Overfishing not occurring

• Mean length BH-Z multiple years GH (changing Z)

• Add additional data:• recruit index• effort• catch rate (cpue)• catch & cpue• other species

• Estimate q, M, Ft, Zt; relax assumptions, obtain diagnostics• Combine results with other analyses: yield per recruit,

abundance estimation OFL• Bridge to production models & other models

Bottom line:

Acknowledgements

NMFS Southeast Fisheries Science Centerthrough University of Miami

NMFS Stock Assessment Improvement

Alaska Sea Grant

U of Alaska Fairbanks at Juneau & Terry

Thank you!