extending length-based models for data-limited fisheries into a state-space framework merrill b....

Extending length-based models for data-limited fisheries into a

state-space frameworkMerrill B. Rudd* and James T. Thorson

*PhD Student, School of Aquatic and Fishery Sciences, University of Washington

CAPAM Workshop on Data Weighting

October 22, 2015

Length-based methods for data-limited fisheries

• Easy and straightforward to take length measurements

• Length-based spawning potential ratio (Hordyk et al. 2015)

• Mean-length estimators of fishing mortality (Beverton and Holt 1957, Ault et al. 2005, Gedamke and Hoenig 2006, Nadon et al. 2015)

• Assume equilibrium conditions or set breakpoints to represent changes in fishing mortality over time

Nadon et al. 2015, PLoS ONE

Mean length reflects changes in fishing mortality

Data sources:

1) Life history information compiled from the literature

2) Diver surveys

3) Commercial fishery trip reports

Figure 3. Nadon et al. 2015Time series of average lengths in the exploited phase of the population.

Coral reef fishery example 1: Hawaii

Coral reef fishery example 1: Hawaii

Nadon et al. 2015, PLoS ONE

Mean length reflects changes in fishing mortality

Data sources:

1) Life history information compiled from the literature

2) Diver surveys

3) Commercial fishery trip reports

Figure 3. Nadon et al. 2015Time series of average lengths in the exploited phase of the population.

Coral reef fishery example 2: Kenya

Figure 1. From Hicks and McClanahan 2012 PLoS ONE

Hicks and McClanahan 2012, PLoS ONE

Catch curve and Beverton-Holt mean length estimator will be sensitive to changes in recruitment

- Short lived fisheries and heavily exploited

Data sources:

1) Life history information compiled from literature

2) Port surveys of length composition and effort

Potential need for direct consideration of recruitment variationChanges in fishing mortality and recruitment are confounded

Changes in fishing mortality and recruitment are confounded

Potential need for direct consideration of recruitment variation

Goal of this study

Alternative to equilibrium-based methods in data-poor situations mostly reliant on length composition data

Development

1) State-space model to account for recruitment variation

2) Tested under varying data availability scenarios

Operating modelAge-converted to length-structured population dynamics

1, 1,max1, 1

if a=0

if 0a t

t

a t Za t

RN

a aN e

Abundance

max

ln(0.01)a

M

,a t a tZ M S F Mortality

,

, ,,

(1 )a tZa ta t a t

a t

S FC N e

Z

(Hordyk et al. 2015)

1, 1,max1, 1

if a=0

if 0a t

t

a t Za t

RN

a aN e

Abundance

max

ln(0.01)a

M


,

, ,,

(1 )a tZa ta t a t

a t

S FC N e

Z

Slow-growing:k = 0.1L∞ = 60 cmM = 0.184Amax = 26

Fast-growing:k = 0.2L∞ = 30 cmM = 0.37Amax = 13



(Thorson and Cope 2015)

0

50

3( )log( )

L L

LA

k

50( )

1

1a a AMate

Maturity

(Williams and Shertzer 2003)

Growth

1, 1,max1, 1

if a=0

if 0a t

t

a t Za t

RN

a aN e

Abundance

max

ln(0.01)a

M


,

, ,,

(1 )a tZa ta t a t

a t

S FC N e

Z

0

50

3( )log( )

L L

LA

k

Growth

50( )

1

1a a AMate

Maturity

50( )

1

1 slopea S a ASe

Selectivity



(Williams and Shertzer 2003)

Slow-growing:k = 0.1L∞ = 60 cmM = 0.184Amax = 26

Fast-growing:k = 0.2L∞ = 30 cmM = 0.37Amax = 13

(Thorson and Cope 2015)

Operating model – fishing and recruitment dynamics

Operating model – generating length composition

Probability of being in a length bin given age

1

1

if i=1

( | ) if 1<i

if i=I1

highi a

L

high highi a i a

L L

highi a

L

i L

i L i LP i a I

i L



1

1

if i=1

( | ) if 1<i

if i=I1

highi a

L

high highi a i a

L L

highi a

L

i L

i L i LP i a I

i L

Probability of harvesting in a given length bin,( ) t a a

at

N SP C

N



1

1

if i=1

( | ) if 1<i

if i=I1

highi a

L

high highi a i a

L L

highi a

L

i L

i L i LP i a I

i L


at

N SP C

N

Probability of sampling a given length bin( ) ( | )* ( )i aP C P i a P C



1

1

if i=1

( | ) if 1<i

if i=I1

highi a

L

high highi a i a

L L

highi a

L

i L

i L i LP i a I

i L


at

N SP C

N

Probability of sampling a given length bin( ) ( | )* ( )i aP C P i a P C

, ~ ( , ( ))t i t iC Multinomial C P C

Data Scenario Catch Index Length Composition

Ultra-rich Full catch known Full effort known (CPUE index) 10,000 samples annually

Rich Full catch known Full effort known (CPUE index) 2,000 samples annually

Moderate 20% of catch accounted for 20% of effort accounted for (CPUE index) 500 samples annually

Poor A 20% of catch accounted for 20% of effort accounted for (CPUE index) 50 samples annually

Poor B Catch not accounted for Fishery-independent index 500 samples in final year

Poor C Catch not accounted for No index 2,000 samples in final year

Operating model – data generation

Operating model – data generation

Data Scenario Catch Index Length Composition

Ultra-rich Full catch known Full effort known (CPUE index) 10,000 samples annually

Rich Full catch known Full effort known (CPUE index) 2,000 samples annually

Moderate 20% of catch accounted for 20% of effort accounted for (CPUE index) 500 samples annually

Poor A 20% of catch accounted for 20% of effort accounted for (CPUE index) 50 samples annually

Poor B Catch not accounted for Fishery-independent index 500 samples in final year

Poor C Catch not accounted for No index 2,000 samples in final year

Reference point:Spawning potential ratio (SPR)

max

,1

a

t a t a aa

SB N W Mat

Annual Biomass

max

0 00

aaM

a aa

SB R e W Mat

Unfished biomass

0

currentSBSPR

SB

Spawning Potential Ratio (SPR)

(Nadon et al. 2015, Ault et al. 2008)

Inputs

Fixed parameters1) Von Bertalanffy Linf and k2) Maturity curve3) Natural mortality4) CV for length-at-age5) CV for observed catch and

index

Data inputs6) Length composition7) Catch time series8) Abundance index time series

Estimation model – implemented using Template Model Builder

Estimation model – implemented using Template Model Builder

Outputs

Estimated1) Annual fishing mortality (fixed effect)2) Global mean recruitment3) Random effects on annual

recruitment4) Recruitment variation (σR)5) Catchability coefficient6) Logistic selectivity parameters

Performance measure- SPR

Inputs

Fixed parameters1) Von Bertalanffy Linf and k2) Maturity curve3) Natural mortality4) CV for length-at-age5) CV for observed catch and

index

Data inputs6) Length composition7) Catch time series8) Abundance index time series

Estimation model – age-converted to length-structured

Recruitment

2ˆˆ2ˆ R

t tb

tR e

2

2

ˆ1

ˆt

tR

SEb

ˆ~ (0, )t RN

(Methot and Taylor 2011)

Estimation model – age- converted to length-structured

Recruitment

2ˆ

2ˆ Rt tb

tR e

2

2

ˆ1 t

tR

SEb

~ (0, )t RN

Abundance

,ˆˆ ˆ

a t a tZ M S F Mortality

,ˆ

, ,

,

ˆ ˆˆ ˆ (1 )

ˆa tZa t

a t a t

a t

S FC N e

Z

1, 1, ˆ

max1, 1

ˆ if a=0ˆˆ if 0a t

t

a t Za t

RN

a aN e


Estimation model – age- converted to length-structured

Recruitment

2ˆ

2ˆ Rt tb

tR e

2

2

ˆ1 t

tR

SEb

~ (0, )t RN

50ˆ ˆ( )

1ˆ1 slope

a S a AS

e

Selectivity

Abundance

,ˆˆ ˆ

a t a tZ M S F Mortality

,ˆ

, ,

,

ˆ ˆˆ ˆ (1 )

ˆa tZa t

a t a t

a t

S FC N e

Z

1, 1, ˆ

max1, 1

ˆ if a=0ˆˆ if 0a t

t

a t Za t

RN

a aN e


TMB

Estimation model – maximum penalized likelihood

, ,1

( , ( ))T

lengthcomp t i t it

L Multinomial C P C

TMB


, ,1

( , ( ))T

lengthcomp t i t it

L Multinomial C P C

1

ˆ( , , )

ˆ

T

index t t It

I t I

L Normal I I

I CV

ˆ ˆt tI qN

TMB


, ,1

( , ( ))T

lengthcomp t i t it

L Multinomial C P C

1

ˆ( , , )

ˆ

T

catch t t ct

c t I

L Normal C C

C CV

1

ˆ( , , )

ˆ

T

index t t It

I t I

L Normal I I

I CV

ˆ ˆt tI qN

TMB


1ˆ ˆlog( ) ~ ( , )t t FF Normal F Penalty on fishing

mortality

TMB


1ˆ ˆlog( ) ~ ( , )t t FF Normal F Penalty on fishing

mortality

Penalty on depletion in initial year

1log( ) ~ (1.0, )DD Normal

Model fits- Endogenous F, Constant R

Ultra-Rich Data Scenario

Biomass Recruitment

Fishing mortality

Depletion

Mean Length Catch

Index

Model fits- Endogenous F, Constant R

Moderate Data Scenario

Biomass Recruitment

Fishing mortality

Depletion

Mean Length Catch

Index

Model fits - Endogenous F, Constant RRecruitment

Ultra Rich

Moderate Poor A

Poor B Poor C

Model fits - Endogenous F, Constant RRecruitment Fishing Mortality

Ultra Rich

Moderate Poor A

Poor B Poor C

Ultra Rich

Moderate Poor A

Poor B Poor C

Relative error in SPR – fast-growing, 20 years data, σR=0.5

Alternative data scenario

• Snapshot of length composition

• Prior/penalty on catch time series and index based on local expert knowledge

(Variation on Poor C data scenario – which did not include any information on catch and effort index)

Sensitivity analyses

- Fixed parameters: growth curve, natural mortality, maturity

- Parameter starting values: sigmaR, sigmaF, selectivity

- Model structure

Set effective sample size appropriately

- Number of vessels

Move away from multinomial

Monthly time step for coral reef fish

Next steps

Concluding thoughts

• Sensitivity analysis required: ability to estimate terminal year depletion for data-poorest scenarios likely based on fixed parameter values

• Potential as another option for coral reef fisheries where equilibrium is unlikely

• Must be considered against equilibrium methods – is there a benefit to management from adding complexity?

Discussion topics

Where to consider data weighting and conflict?

1) Length composition data

2) Effective sample size

3) Exclusion of data due to representativeness

4) Weight of expert insight

Thank you

Wildlife Conservation Society

SNAP Data-Limited Fisheries working group

NSF IGERT Program on Ocean Change

School of Aquatic and Fishery Sciences

Trevor Branch

Hilborn & Branch labs

Kenyan coral reef fisheries

Lethrinus lentjan• 1 of 3 species that represent

60% of the total catch• Evidence of growth and

recruitment overfishing from equilibrium methods

extending length-based models for data-limited fisheries into a state-space framework merrill b....

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