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Hierarchical Bayesian Analysis of the Spiny
Lobster Fishery in California
Brian Kinlan, Steve Gaines, Deborah McArdle, Katherine Emery
UCSB
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
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)