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