Hierarchical Bayesian Analysis of the Spiny

Lobster Fishery in California

Brian Kinlan, Steve Gaines, Deborah McArdle, Katherine Emery

UCSB

The Original Data – An Exceptionally Long Catch-Effort Time Series

Goals

• Estimate dynamics of lobster population (including recruits and sublegals) over history of fishery

• Evaluate alternative models for population replenishment

• Examine interactive effects of variation in effort, climate, and population dynamics over long time series

• NEED: Model linking catch-effort time series to population dynamics

Methods Overview

• Size-Structured State-Space Model

• Length-Weight relationships used to link biomass catch data to abundance in underlying model

• Development of priors on growth, mortality, and size structure

• Implementation in WinBUGS, analysis in Matlab

Components of an Hierarchical Bayesian Model

• Data

• Likelihood model for data (Observation Error – assumed lognormal)

• Process Model

• Prior distributions for parameters

• Logical links specifying functional form of deterministic relationships among parameters

Ns,y=[Ns−1,y−1 * exp(−M) − Ca−1,y−1* exp(−0.5M)] * Gs,s-1,y

Process Equations

log(CPUEy) = log(q) + log(Ny)

Abundance

Catch

N1,y=Ry

Ry ~ LogNormal(μrecruits,σ2recruits)

Ly=Ly-1+B0 exp(B1 Ly-1)

Growth

Results

• Posterior distributions of parameters summarized by their mean

• Evaluation of Model Fit

• Patterns emerging from model – stock-recruitment relationships, climate correlations

1900 1920 1940 1960 1980 2000

0

0.5

1

1.5

2

2.5

3

x 106 legal stock

Year

No.

of

Lob

ste

rs

total including catch

escapement

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 107

Year

No

. o

f L

ob

ste

rstotal stock(1-8)sublegal stock(1-3)legal stock(4-8)reproductive stock(3-8)

1900 1920 1940 1960 1980 2000

0

0.2

0.4

0.6

0.8

1

exploitation rate

Year

Fra

ctio

n E

xplo

ited

exploitation rate

1900 1920 1940 1960 1980 2000

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

weight of avg legal lobster (lbs)

Year

Wei

gh

t (L

bs)

1900 1920 1940 1960 1980 2000

1

2

3

4

5

6

7

8

9

10

11

x 106 recruitment

Year

No.

of

Lob

ste

rs

1900 1920 1940 1960 1980 2000

27

28

29

30

31

32

33

34

35

growth parameter

Year

Asy

mp

totic

Gro

wth

Ra

te (

mm

ind

iv-

1 y

-1

)

Comparison with simpler standard fisheries models

• DeLury depletion model (abundance)

• Shaeffer surplus production model (biomass)

• Both assume constant r, K, q and fit unknown No; model estimated by least-squares or MLE

Comparison with Standard Fisheries ModelsBiomass Comparison: Schaeffer, De Lury and Bayesian (1888-2005)

YBayesian total biomassSchaeffer biomass (7 outliers)Schaefer biomass (12 outliers)DeLury biomass

0

1000000

2000000

3000000

4000000

5000000

6000000

7000000

8000000

9000000

Y

1880 1900 1920 1940 1960 1980 2000

Year

Model Fit and Residuals

• Model vs. Predicted Total Catch

• Model vs. Predicted CPUE

• Residuals – Effect of Constant Catchability Assumptions

0 2 4 6 8 10 12

x 105

0

2

4

6

8

10

12

x 105

Cat

ch(L

bs)

Obs

erve

d

Catch(Lbs) Model

Actual by Predicted Catch in Lbs

Data

Regression

1:1 Line

10 1 10 2 10 310 1

10 2

10 3

10 4

CP

UE

(Lbs

/Tra

p)

Obs

erv

ed

Expected CPUE = q*N[y]*P(g|s)*exp(-M/2)

Catch-Effort Model Fit (Test Constant q Assumption)

Data

Regression

1:1 Line

Da

ta P

oin

ts C

olo

r C

od

ed

by

Ye

ar

1895

1950

2005

‘Empirical’ Stock-Recruitment Relationships

• No assumptions or priors specifying a relationship between stock and recruitment were included in model

• Recruitment was fit based on Catch, Effort, and the dynamic state equations

• Does an ‘empirical’ relationship arise in the model fit?

500000 1000000 5000000

2000000

3000000

4000000

5000000

6000000

7000000

8000000

9000000

10000000

Stock-Recruitment Relationship

Reproductive Stock in Year Y-1 (No. of lobsters)

Re

cru

itm

en

t in

Ye

ar

Y (

No

. o

f lo

bs

ters

)

0 1 2 3 4 5 6

x 10 6

1.5

2

2.5

3

3.5

4

4.5

5

5.5Recruits per Adult vs. No. of Adults

No. of Adults in Year Y-1

Rec

ruit

s in

Yea

r Y

per

Ad

ult

in

Yea

r Y

-1

Future Model Directions

• allow time-dependency of catchability, time+size dependent mortality

• additional growth, mortality, size info via priors

• age-structured version with explicit modeling of cohort growth-in-length

• ocean climate covariates

• Spatial Model

Future Model Directions

• Spatial Model – use regional (port-based) catch-effort data– compare alternative models of connectivity via

larval movement and/or juvenile migration– will help clarify the population dynamic

mechanism underlying the compensatory recruits-per-spawner relationship (pre- or post-dispersal density dependence)

Applications in Context of the Sustainable Fisheries Group

• Evaluate forecast and hindcast scenarios of changing temporal (and spatial) patterns of effort

• Incorporate process and observation uncertainty explicitly using bayesian posteriors

• Assess value of information in this fishery