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1Monday, February 17, 14
A Perspective On Steepness and Its Implications for Fishery Management
Marc MangelCenter for Stock Assessment Research and Department
of Applied Mathematics & Statistics
UCSC
PERSPECTIVE
A perspective on steepness, reference points, and stock assessmentMarc Mangel, Alec D. MacCall, Jon Brodziak, E.J. Dick, Robyn E. Forrest, Roxanna Pourzand, and Stephen Ralston
Abstract: We provide a perspective on steepness, reference points for fishery management, and stock assessment. We firstreview published data and give new results showing that key reference points are fixed when steepness and other life historyparameters are fixed in stock assessments using a Beverton–Holt stock–recruitment relationship. We use both production andage-structuredmodels to explore these patterns. For the productionmodel, we derive explicit relationships for steepness and lifehistory parameters and then for steepness andmajor reference points. For the age-structuredmodel, we are required to generallyuse numerical computation, and sowe provide an example that complements the analytical results of the productionmodel.Wediscuss what it means to set steepness equal to 1 and how to construct a prior for steepness. Ways out of the difficult situationraised by fixing steepness and life history parameters include not fixing them, using a more complicated stock–recruitmentrelationship, and being more explicit about the information content of the data and what that means for policy makers. Wediscuss the strengths and limitations of each approach.
Résumé : Nous offrons une perspective sur l’inclinaison, les points de référence pour la gestion des pêches et l’évaluation desstocks. Nous passons d’abord en revue les résultats publiés et présentons de nouveaux résultats qui démontrent que des pointsde référence clés sont fixés quand l’inclinaison et d’autres paramètres du cycle biologique sont fixés dans les évaluations desstocks reposant sur une relation stock–recrutement de type Beverton–Holt. Nous utilisons des modèles de production etstructurés par âge pour explorer ces situations. En ce qui concerne le modèle de production, nous obtenons des relationsexplicites pour l’inclinaison et les paramètres du cycle biologique, puis pour l’inclinaison et les principaux points de référence.Pour le modèle structuré par âge, nous devons généralement utiliser une approche numérique et présentons un exemple quicomplémente les résultats analytiques du modèle de production. Nous discutons de ce que signifie le fait de fixer la valeur del’inclinaison a 1 et de la manière d’établir un a priori pour l’inclinaison. Parmi les moyens pour contourner la difficulté soulevéepar la fixation de l’inclinaison et des paramètres du cycle biologique figurent le fait de ne pas fixer ces valeurs, l’utilisation d’unerelation stock-recrutement plus complexe et une approche plus explicite en ce qui concerne le contenu en information desdonnées et ce que cela signifie pour les responsables de l’élaboration de politiques. Nous abordons les forces et les limites dechacune de ces approches. [Traduit par la Rédaction]
IntroductionModels for the stock–recruitment relationship (SRR) involving
two parameters almost entirely dominate in age-structured stockassessments. These SRRs can bewritten in the form R(B) = !Bf(B, "),where R(B) is a measure of the recruitment produced by spawningbiomass B, ! is maximum productivity per unit spawning bio-mass, and f(B, ") characterizes the form of density dependence,with the parameter " measuring the intensity of the density de-pendence in the prerecruit phase (Walters and Martell 2004;Mangel 2006) and f(B, ") ¡ 1 as B declines. Two of the mostcommonly used SRRs are the Beverton–Holt SRR (BH-SRR;Beverton and Holt 1957)
(1) R(B) #!B
1 $ "B
for which recruitment increases asymptotically to its maximumvalue !/", and the Ricker SRR (R-SRR; Ricker 1954)
(2) R(B) # !B exp(–"B)
for which there is a possibility of a decline in recruitment at highspawning stock abundance.WhenB is very small, so that the denom-inator in eq. 1 or the exp in eq. 2 are well approximated by 1, both ofthese SRRs are of the form R(B) ! !B (see Sissenwine and Shepherd1987); when "B !! 1 so that the denominator or exponential can beTaylor–expanded, they are of the form R(B) ! !B(1 % "B), giving thefamiliar logistic equation.
Although the shape of the SRR is important, it is often ambig-uous what the appropriate SSR is given the available stock assess-ment information. For example, Dorn (2002) concluded in ameta-analysis of rockfish (Sebastes spp.) stock and recruitmentdata that neither the R-SRR nor the BH-SRR can be distinguishedas a preferred model on the basis of statistical goodness of fit.Brodziak (2002) reached a similar conclusion in an analysis of USAwest coast groundfish stock–recruitment data. Alternatively, in ameta-analysis of 128 diverse fish stocks, Punt et al. (2005) con-cluded that as a general model, the BH-SRR is strongly preferredover the R-SRR, but that “there are also indications that other(more complicated) forms may provide better representations ofthe existing data” (p. 76). This conclusion is not unexpected be-cause models with more parameters can approximate variabilityin the observations used to estimate the SRRmore accurately than
Received 24 August 2012. Accepted 5 April 2013.
Paper handled by Associate Editor Kenneth Rose.
M. Mangel. Center for Stock Assessment Research MS E-2, University of California, Santa Cruz, CA 95064, USA. Department of Biology, University of Bergen, Bergen 5020, Norway.A.D. MacCall, E.J. Dick, and S. Ralston. National Marine Fisheries Service, Southwest Fisheries Science Center, 110 Shaffer Road, Santa Cruz, CA 95060, USA.J. Brodziak. National Marine Fisheries Service, Pacific Islands Fisheries Science Center, 2570 Dole Street, Honolulu, HI, 96822.R.E. Forrest. Fisheries and Oceans Canada Pacific Biological Station, 3190 Hammond Bay Road, Nanaimo, BC V9T 6N7, Canada.R. Pourzand.* Center for Stock Assessment Research, MS E-2, University of California, Santa Cruz, CA 95064, USA.Corresponding author: Marc Mangel (e-mail: [email protected]).*Present address: Oracle Corporation, 475 Sansome St., San Francisco, CA 94101, USA.
930
Can. J. Fish. Aquat. Sci. 70: 930–940 (2013) dx.doi.org/10.1139/cjfas-2012-0372 Published at www.nrcresearchpress.com/cjfas on 9 April 2013.
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2Monday, February 17, 14
Outline• Density Dependence and Population Biology• What is Steepness?• Strategic Fishery Management: The Stock Assessment Process• Connections Between Steepness and Reference Points: Previous Observations• The Reproductive Biology of Steepness• Implications for Reference Points• Fixing Steepness at 1• Moving Forward• Conclusions
3Monday, February 17, 14
Outline• Density Dependence and Population Biology• What is Steepness?• Strategic Fishery Management: The Stock Assessment Process• Connections Between Steepness and Reference Points: Previous Observations• The Reproductive Biology of Steepness• Implications for Reference Points• Fixing Steepness at 1• Moving Forward• Conclusions
4Monday, February 17, 14
Density Dependence and Population Biology
Fundamental Law of Population Biology
Next year’s population = This year’s population + Reproduction + Immigration - Death - Emmigration
5Monday, February 17, 14
Density Dependence and Population Biology
Fundamental Law of Population Biology
Next year’s population = This year’s population + Reproduction + Immigration - Death - Emmigration
Reproduction (“Recruitment”)
= Function of this year’s population size and reproductively active individuals (“Spawners”)
6Monday, February 17, 14
Because of Density Dependence There Are Points in Biomass- Recruitment Space That Lie Above the Replacement Line
7Monday, February 17, 14
0
750
1500
2250
3000
0 375 750 1125 15000
17.5
35.0
52.5
70.0
0 10.0 20.0 30.0 40.0
Alaskan Pink SalmonWalleye Pollock (Kamchatka)
Females
Fry
106 Fish Age 7+
Rec
ruits
(106
Age
6)
Data from R.A.Myers, J.Bridson, and N.J.Barrowman (1995) Summary of Worldwide Spawner and Recruitment Data. Can. Tech. Rep. Fish. Aquat. Sci. 2020
But The Data Are Noisy...
8Monday, February 17, 14
Classical Solution: Parametric Stock-Recrument Relationships -- The Beverton Holt SRR
Spawning stock biomass, B
Recruits, R
Biomass
Recruits
9Monday, February 17, 14
Classical Solution: Parametric Stock-Recrument Relationships
Spawning stock biomass, B
Recruits, R
Beverton
Biomass
Recruits
10Monday, February 17, 14
Classical Solution: Parametric Stock-Recrument Relationships
Spawning stock biomass, B
Recruits, R
Biomass
Recruits
11Monday, February 17, 14
The Ricker SRR
Spawning stock biomass
Recruits
S
R
12Monday, February 17, 14
Outline• Density Dependence and Population Biology• What is Steepness?• Strategic Fishery Management: The Stock Assessment Process• Connections Between Steepness and Reference Points: Previous Observations• The Reproductive Biology of Steepness• Implications for Reference Points• Fixing Steepness at 1• Moving Forward• Conclusions
13Monday, February 17, 14
What is Steepness: Mace and Doonan (1988)
Steepness: Fraction of the recruitment at the unfished the spawning biomass when the spawning biomass is 20% of the unfished size
Mace, P. and I.J. Doonan. 1988. A Generalised Bioeconomic Simulation Model for Fish Population Dynamics. New Zealand Fishery Assessment Research Document 88/4, Fisheries Research Centre, MAFFish, POB 297, Wellington, NZ
h = R(0.2B0 )R(B0 )
Biomass
Recruits
Spawning Biomass, B
Recruits,R
R(B0 )
R(0.2B0 )
0.2B0 B0
14Monday, February 17, 14
What is Steepness: Mace and Doonan (1988)
Steepness: Fraction of the recruitment at the unfished the spawning biomass when the spawning biomass is 20% of the unfished size
Mace, P. and I.J. Doonan. 1988. A Generalised Bioeconomic Simulation Model for Fish Population Dynamics. New Zealand Fishery Assessment Research Document 88/4, Fisheries Research Centre, MAFFish, POB 297, Wellington, NZ
h = R(0.2B0 )R(B0 )
Biomass
Recruits
Spawning Biomass, B
Recruits,R
R(B0 )
R(0.2B0 )
0.2B0 B0
Parameters of the SRR can be related to steepness and unfished biomass and recruitment
15Monday, February 17, 14
Outline• Density Dependence and Population Biology• What is Steepness?• Strategic Fishery Management: The Stock Assessment Process• Connections Between Steepness and Reference Points: Previous Observations• The Reproductive Biology of Steepness• Implications for Reference Points• Fixing Steepness at 1• Moving Forward• Conclusions
16Monday, February 17, 14
Strategic Fishery Management: The Stock Assessment Process
•Organize all of the data (catch, survey, fishery dependent, fishery independent)
•Fit the data to a model of population dynamics
•Use the fitted model to investigate state of the stock and management options
17Monday, February 17, 14
18
Reference Points of Strategic Fishery Management
Unfished Biomass
Biomass giving Maximum Sustainable Yield (MSY)
Biomass giving Maximum Net Productivity (MNP)
Rate of Fishing Mortality giving MSY
Spawning biomass per recruit when fished at rate giving MSY divided by that when unfished
B0BMSY
BMNP
FMSYSPRMSY
18Monday, February 17, 14
19
Reference Points of Strategic Fishery Management
Maximum Excess Recruitment
Ratio of MER to that in an unfished population
MERSPRMER
19Monday, February 17, 14
0
500
1000
1500
2000
1947
1952
1957
1962
1967
1972
1977
1982
1987
1992
1997
2002
Beginning biomass (m tons)
0
10000
20000
30000
40000
50000
Total biomass
Eggs
Eggs
YEAR
The Stock Assessment Process: Reconstruct the Trajectories of Biomass and Reproduction
20Monday, February 17, 14
Year
Spaw
ning
bio
mas
s (eg
gs x
105 )
0
10,000
20,000
30,000
40,000
50,000
60,000
70,000
80,000
90,000
2000
2004
2008
2012
2016
2020
2024
2028
2032
2036
2040
2044
2048
2052
2056
2060
2064
2068
2072
2076
2080
2084
2088
2092
2096
10th
25th
50th
75th
90th
The Stock Assessment Process: Forward Project the Fate of the Stock
No fishing
21Monday, February 17, 14
Total biomass (metric tons)
Freq
uenc
y
0
50
100
150
200
250
0 1000 2000 3000 4000 5000 6000
F=0
F=0.2
F=0.5
The Stock Assessment Process: Frequency Distribution of Biomass Projected Under Different Scenarios About Fishing
22Monday, February 17, 14
Total biomass (metric tons)
Freq
uenc
y
0
50
100
150
200
250
0 1000 2000 3000 4000 5000 6000
F=0
F=0.2
F=0.5
The Stock Assessment Process: Frequency Distribution of Biomass Projected Under Different Scenarios About Fishing
Steepness of the SRRis clearly important in these predictions.
23Monday, February 17, 14
KA Rose and JH Cowan, Jr. 2003. Annual Review of Ecology andSystematics 34:127
Steepness is Often Fixed In Stock Assessments
24Monday, February 17, 14
KA Rose and JH Cowan, Jr. 2003. Annual Review of Ecology andSystematics 34:127
Steepness is Often Fixed In Stock Assessments
25Monday, February 17, 14
Outline• Density Dependence and Population Biology• What is Steepness?• Strategic Fishery Management: The Stock Assessment Process• Connections Between Steepness and Reference Points: Previous Observations• The Reproductive Biology of Steepness• Implications for Reference Points• Fixing Steepness at 1• Moving Forward• Conclusions
26Monday, February 17, 14
27
Connections Between RPs and Steepness Have Been Noted
Williams (NA J Fish Mang. 2002)
MSY increases with steepness but depends on age at maturity and selectivity.SPRMSY determined by steepness but independent of
age at maturity and selectivity
27Monday, February 17, 14
28
Connections Between RPs and Steepness Have Been Noted
Williams (NA J Fish Mang. 2002)
MSY increases with steepness but depends on age at maturity and selectivity.SPRMSY
Punt et al (Fish. Res. 2008)
determined by steepness but independent of
age at maturity and selectivity
FMSY is an increasing function of steepness but
depends on life history parameters and selectivity
BMSY / B0 and SPRMSY nearly perfectly predicted by
steepness
28Monday, February 17, 14
29
Connections Between RPs and Steepness Have Been Noted
Brooks et al (ICES J. Mar. Sci 2010)
SPRMER =121− hh
29Monday, February 17, 14
30
Connections Between RPs and Steepness Have Been Noted
Brooks et al (ICES J. Mar. Sci 2010)
SPRMER =121− hh
Begs us to answer three questions
30Monday, February 17, 14
31
Connections Between RPs and Steepness Has Been Noted
Brooks et al (ICES J. Mar. Sci 2010)
SPRMER =121− hh
Begs us to answer three questions
1) Why is this so?
31Monday, February 17, 14
32
Connections Between RPs and Steepness Has Been Noted
Brooks et al (ICES J. Mar. Sci 2010)
SPRMER =121− hh
Begs us to answer three questions
1) Why is this so?
2) What does it mean for the process of stock assessment?
32Monday, February 17, 14
33
Connections Between RPs and Steepness Has Been Noted
Brooks et al (ICES J. Mar. Sci 2010)
SPRMER =121− hh
Begs us to answer three questions
1) Why is this so?
2) What does it mean for the process of stock assessment?
3) What should we do given the answer to 2)?
33Monday, February 17, 14
34
First: Some Other Examples
"Some other observations are worth noting .... For the 14 populations, there was strong negative relationship between h and at [the MSY harvest fraction] under both Beverton-Holt (r=-0.96) and Ricker (r=-0.94) recruitment (Figs 8a, 8b). This might be expected, as populations with stronger recruitment compensation are expected to be able to be sustained at lower levels of spawning biomass."
CJFAS 65:286 (2008)
34Monday, February 17, 14
The Results of Forrest et al Are Nearly Perfectly Predicted by Steepness
a)##
###b)#
##c)##
#
0.15
0.2
0.25
0.3
0.35
0.4
0.15 0.2 0.25 0.3 0.35 0.4
Rep
orte
d B
MS
Y/B
0 by
Forr
est e
t al (
2010
)
Predicted BMSY
/B0 from Steepness
2
Mangel'et'al'Figure'1b'7'
'8'
'9'
'10'
'11'
'12'
'13'
'14'
3
Mangel'et'al'Figure'1c'15'
'16'
17'
0
0.2
0.4
0.6
0.8
1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
SP
RM
SY
Steepness, h
Points: Forrest et al
Line: Theory (to be explained)
35Monday, February 17, 14
And Other Results of Forrest et al Are Nearly Perfectly Predicted by Steepness Alone
Points: Forrest et al Line: Theory (to be explained)
36Monday, February 17, 14
Some Stock Assessments with BH-SRR and Fixed Steepness
37Monday, February 17, 14
Some Stock Assessments with BH-SRR and Fixed Steepness
Points: SPR estimated from complicated stock assessments.
Line: The theory (to be explained)
38Monday, February 17, 14
Outline• Density Dependence and Population Biology• What is Steepness?• Strategic Fishery Management: The Stock Assessment Process• Connections Between Steepness and Reference Points: Previous Observations• The Reproductive Biology of Steepness• Implications for Reference Points• Fixing Steepness at 1• Moving Forward• Conclusions
39Monday, February 17, 14
A Theory for The Reproductive Ecology of Steepness: The Production Model
Dynamics of Biomass
Steady state in the absence of fishing
dBdt
=α pB1+ βB
− (M + F)B
B0 =1β
α p
M−1⎛
⎝⎜⎞⎠⎟
40Monday, February 17, 14
So that
is a dimensionless (Beverton) number of the life history
Steepness is
α p
M
h = 0.2 ⋅ 1+ βB01+ 0.2βB0
41Monday, February 17, 14
and h = 0.2 ⋅ 1+ βB01+ 0.2βB0
B0 =1β
α p
M−1⎛
⎝⎜⎞⎠⎟
42Monday, February 17, 14
and
So that
h = 0.2 ⋅ 1+ βB01+ 0.2βB0
B0 =1β
α p
M−1⎛
⎝⎜⎞⎠⎟
βB0 =α p
M−1⎡
⎣⎢⎤⎦⎥
43Monday, February 17, 14
and
So that
And steepness is
Constant for constant Beverton number
h = 0.2 ⋅ 1+ βB01+ 0.2βB0
B0 =1β
α p
M−1⎛
⎝⎜⎞⎠⎟
βB0 =α p
M−1⎡
⎣⎢⎤⎦⎥
h =
α p
M
4 +α p
M
44Monday, February 17, 14
N(1,t)
N(2,t)
N(3,t)
N(amax,t)
N(1,t-1)
N(2,t-1)N(1,t-2) }. . .
. . .
N(0,t-1)
N(0,t-2)
N(0,t-3)
N(0, t-amax) N(0,t)
e-M(0)
e-M(0)
e-M(0)
e-M(0)
e-M(1)
e-M(1) e-M(2)
e-M(amax-1)
The Age Structured Model
B = WaPaN(a,t)a∑
f (B)
45Monday, February 17, 14
The Age Structured Model
N(a,t) = N(a −1,t −1)e−Z (a−1)Population dynamics, beyond eggs/larvae
46Monday, February 17, 14
The Age Structured Model
N(a,t) = N(a −1,t −1)e−Z (a−1)Population dynamics, beyond eggs/larvae
Spawning biomass
Bs (t) = N(a,t)Wf (a)pf ,m (a)a=1
amax
∑
47Monday, February 17, 14
The Age Structured Model
N(a,t) = N(a −1,t −1)e−Z (a−1)
N(0,t) = α sBs (t)1+ βBs (t)
Population dynamics, beyond eggs/larvae
Spawning biomass
Recruits
Bs (t) = N(a,t)Wf (a)pf ,m (a)a=1
amax
∑
48Monday, February 17, 14
In the steady state
N(a) = S(a) ⋅ R0
R0 =α sB01+ βB0
B0 = N(a)Wf (a)pf ,m (a)a=1
amax
∑
49Monday, February 17, 14
In the steady state
N(a) = S(a) ⋅ R0
R0 =α sB01+ βB0
Define
Wf = S(a)Wf (a)pf ,m (a)a=1
amax
∑
B0 = N(a)Wf (a)pf ,m (a)a=1
amax
∑
Average mass of a spawning fish50Monday, February 17, 14
R0 =α sR0Wf
1+ β ⋅R0Wf
51Monday, February 17, 14
Steepness is
R0 =α sR0Wf
1+ β ⋅R0Wf
h =
α s ⋅0.2R0Wf
1+ β ⋅0.2R0Wf
R0
52Monday, February 17, 14
Steepness is
But
R0 =α sR0Wf
1+ β ⋅R0Wf
h =
α s ⋅0.2R0Wf
1+ β ⋅0.2R0Wf
R0
βR0Wf =α sWf −1
53Monday, February 17, 14
Steepness is
But
So that
R0 =α sR0Wf
1+ β ⋅R0Wf
h =
α s ⋅0.2R0Wf
1+ β ⋅0.2R0Wf
R0
βR0Wf =α sWf −1
h =0.2α sWf
1+ 0.2 α sWf −1⎡⎣ ⎤⎦=
α sWf
4 +α sWf
54Monday, February 17, 14
ConnectingAge-Structuredand ProductionModels
These are the same if
Age-structured Production
α sWf
4 +α sWf
≈
α p
M
4 +α p
M
55Monday, February 17, 14
ConnectingAge-Structuredand ProductionModels
These are the same if
S(a) = e−Ma
Age-structured Production
Constant mortality
α sWf
4 +α sWf
≈
α p
M
4 +α p
M
56Monday, February 17, 14
ConnectingAge-Structuredand ProductionModels
These are the same if
S(a) = e−Ma
1− e−Mamax ≈ 1
Age-structured Production
Constant mortality
Long-lived
α sWf
4 +α sWf
≈
α p
M
4 +α p
M
57Monday, February 17, 14
ConnectingAge-Structuredand ProductionModels
These are the same if
S(a) = e−Ma
1− e−Mamax ≈ 11
1− e−M≈1M
Age-structured Production
Constant mortality
Long-lived
M is small
α sWf
4 +α sWf
≈
α p
M
4 +α p
M
58Monday, February 17, 14
ConnectingAge-Structuredand ProductionModels
These are the same if
S(a) = e−Ma
1− e−Mamax ≈ 11
1− e−M≈1M
Age-structured Production
Wf (a) ~ constant
Constant mortality
Long-lived
M is small
Mass at age constant
α sWf
4 +α sWf
≈
α p
M
4 +α p
M
59Monday, February 17, 14
Outline• Density Dependence and Population Biology• What is Steepness?• Strategic Fishery Management: The Stock Assessment Process• Connections Between Steepness and Reference Points: Previous Observations• The Reproductive Biology of Steepness• Implications for Reference Points• Fixing Steepness at 1• Moving Forward• Conclusions
60Monday, February 17, 14
Looking at Some Reference Points for the Production Model
dBdt
=α pB1+ βB
− (M + F)B
61Monday, February 17, 14
Looking at Some Reference Points for the Production Model
dBdt
=α pB1+ βB
− (M + F)B B(F) = 1β
α p
M + F−1⎛
⎝⎜⎞⎠⎟
62Monday, February 17, 14
Y (F) = FB(F)
Looking at Some Reference Points for the Production Model
dBdt
=α pB1+ βB
− (M + F)B B(F) = 1β
α p
M + F−1⎛
⎝⎜⎞⎠⎟
63Monday, February 17, 14
Y (F) = FB(F)
What is FMSY ?
Looking at Some Reference Points for the Production Model
dBdt
=α pB1+ βB
− (M + F)B B(F) = 1β
α p
M + F−1⎛
⎝⎜⎞⎠⎟
64Monday, February 17, 14
Y (F) = FB(F)
What is FMSY ?
Looking at Some Reference Points for the Production Model
dBdt
=α pB1+ βB
− (M + F)B B(F) = 1β
α p
M + F−1⎛
⎝⎜⎞⎠⎟
FMSYM
=α p
M−1
65Monday, February 17, 14
Y (F) = FB(F)
What is FMSY ?
But
Looking at Some Reference Points for the Production Model
dBdt
=α pB1+ βB
− (M + F)B B(F) = 1β
α p
M + F−1⎛
⎝⎜⎞⎠⎟
FMSYM
=α p
M−1
h =
α p
M
4 +α p
M66Monday, February 17, 14
Y (F) = FB(F)
What is FMSY ?
But
Looking at Some Reference Points for the Production Model
so that
dBdt
=α pB1+ βB
− (M + F)B B(F) = 1β
α p
M + F−1⎛
⎝⎜⎞⎠⎟
FMSYM
=α p
M−1
h =
α p
M
4 +α p
M
α p
M= 4h1− h
67Monday, February 17, 14
FMSYM
=4h1− h
−1
• Only two of the three parameters natural mortality, steepness, and fishing mortality giving MSY can be treated as freely estimated parameters in an estimation procedure.
Thus, elementary calculus, used wisely, shows that:
68Monday, February 17, 14
FMSYM
=4h1− h
−1
• Only two of the three parameters natural mortality, steepness, and fishing mortality giving MSY can be treated as freely estimated parameters in an estimation procedure.
• It is logically inconsistent to try to estimate all three of them in a stock assessment based on the BH-SRR. This inconsistency may show up as poor model fit and be modeled as ‘noise’.
Thus, elementary calculus, used wisely, shows that:
69Monday, February 17, 14
FMSYM
=4h1− h
−1
• Only two of the three parameters natural mortality, steepness, and fishing mortality giving MSY can be treated as freely estimated parameters in an estimation procedure.
• It is logically inconsistent to try to estimate all three of them in a stock assessment based on the BH-SRR. This inconsistency may show up as poor model fit and be modeled as ‘noise’.
• Furthermore, note that setting steepness equal to 1, makesFMSY infinite.
Thus, elementary calculus, used wisely, shows that:
70Monday, February 17, 14
More On Reference Points: Another common management strategy is to choose the fishing mortality rate that produces a steady state biomass that is x per-cent of the unfished biomass (Fx).
1β
α p
M + Fx−1
⎛⎝⎜
⎞⎠⎟= xβ
α p
M−1
⎛⎝⎜
⎞⎠⎟
71Monday, February 17, 14
More On Reference Points: Another common management strategy is to choose the fishing mortality rate that produces a steady state biomass that is x per-cent of the unfished biomass (Fx).
Rearranging gives
1β
α p
M + Fx−1
⎛⎝⎜
⎞⎠⎟= xβ
α p
M−1
⎛⎝⎜
⎞⎠⎟
α p
M1+ Fx
M
−1= xα p
M−1
⎛⎝⎜
⎞⎠⎟
72Monday, February 17, 14
More On Reference Points: Another common management strategy is to choose the fishing mortality rate that produces a steady state biomass that is x per-cent of the unfished biomass (Fx).
Rearranging gives
1β
α p
M + Fx−1
⎛⎝⎜
⎞⎠⎟= xβ
α p
M−1
⎛⎝⎜
⎞⎠⎟
α p
M1+ Fx
M
−1= xα p
M−1
⎛⎝⎜
⎞⎠⎟
α p
M= 4h1− h
73Monday, February 17, 14
More On Reference Points: Another common management strategy is to choose the fishing mortality rate that produces a steady state biomass that is x per-cent of the unfished biomass (Fx).
Rearranging gives
FxM
is completely determined by x andsteepness
1β
α p
M + Fx−1
⎛⎝⎜
⎞⎠⎟= xβ
α p
M−1
⎛⎝⎜
⎞⎠⎟
α p
M1+ Fx
M
−1= xα p
M−1
⎛⎝⎜
⎞⎠⎟
α p
M= 4h1− h
74Monday, February 17, 14
More Results Another primary management reference point is the biomass that gives maximum net recruitment, BMNP.
Thus, the single parameter steepness determines both of the major reference management points
BMNP
B0=
α p
M−1
α p
M−1
=
4h1− h
−1
4h1− h
−1
75Monday, February 17, 14
An Example
Suppose in a data-poor stock assessment we assert that h=0.8, M=0.15.
76Monday, February 17, 14
An Example
Suppose in a data-poor stock assessment we assert that h=0.8, M=0.15. Then we have no choice but to conclude that
B0 = 15 / β;FMSY = 3M ;BMNP = 0.2B0MSY = 0.09B0
77Monday, February 17, 14
An Example
Suppose in a data-poor stock assessment we assert that h=0.8, M=0.15. Then we have no choice but to conclude that
B0 = 15 / β;FMSY = 3M ;BMNP = 0.2B0MSY = 0.09B0
The only free parameter that can be estimated in this stock assessment is unfished biomass (or alternatively the strength of density dependence of recruitment), and it is clear that its estimated value is strongly conditional on assumed values of M and/or h.
78Monday, February 17, 14
.
“Okay, that is true for the production model, but not for the age-structured model because of selectivity curves”
79Monday, February 17, 14
Production Model
.
“Okay, that is true for the production model, but not for the age-structured model because of selectivity curves”
h =
α p
M
4 +α p
M
80Monday, February 17, 14
Production Model
FMSYM
=4h1− h
−1 .
“Okay, that is true for the production model, but not for the age-structured model because of selectivity curves”
h =
α p
M
4 +α p
M
81Monday, February 17, 14
h =α sWf
4 +α sWf
Production Model Age Structured Model
FMSYM
=4h1− h
−1 .
“Okay, that is true for the production model, but not for the age-structured model because of selectivity curves”
h =
α p
M
4 +α p
M
82Monday, February 17, 14
h =α sWf
4 +α sWf
Production Model Age Structured Model
FMSYM
=4h1− h
−1 .
Hypothesis:
FMSYWf = g(h)
“Okay, that is true for the production model, but not for the age-structured model because of selectivity curves”
h =
α p
M
4 +α p
M
83Monday, February 17, 14
Faster Growth Makes the Age Structured Model More and More Like a Production Model
84Monday, February 17, 14
The Test of the Hypothesis: Scaled Results
Constant mortality
Long-lived
M is small
Mass at age constant
!
4
'Mangel'et'al,'Figure'2'18'
'19'
' '20'
Right&hand&side&of&eq.&15&(dots)&
FMSYWf (lines)&
Points: Production ModelLines: Age Structured Model
FMSYWf
(lines)
4h1− h
−1
(points)
85Monday, February 17, 14
“Your Results Depend on the Fishery Selectivity Curve and Fishing Mortality Giving MSY So Do Not Generalize”
Spawning Biomass Per Recruit (SBR) at Fishing Mortality F
SBR(F) = Na=1
20
∑ (a)W (a)pm (a) e−M (a)− s(a)Fa '=0
a−1
∏
86Monday, February 17, 14
“Your Results Depend on the Fishery Selectivity Curve and Fishing Mortality Giving MSY So Do Not Generalize”
Spawning Biomass Per Recruit (SBR) at Fishing Mortality F
SBR(F) = Na=1
20
∑ (a)W (a)pm (a) e−M (a)− s(a)Fa '=0
a−1
∏
Spawning Potential Ratio (also %MSP)
SPRMSY =SBR(FMSY )SBR(0)
87Monday, February 17, 14
“Your Results Depend on the Fishery Selectivity Curve and Fishing Mortality Giving MSY So Do Not Generalize”
Spawning Biomass Per Recruit (SBR) at Fishing Mortality F
SBR(F) = Na=1
20
∑ (a)W (a)pm (a) e−M (a)− s(a)Fa '=0
a−1
∏
Spawning Potential Ratio (also %MSP)
SPRMSY =SBR(FMSY )SBR(0)
88Monday, February 17, 14
Spawning Potential Ratio for the Production Model
SPRMSY =SBR(FMSY )SBR(0)
=Mα
=1− h4h
Note
SPRMSY → 0
as
h→1
89Monday, February 17, 14
90
5
Mangel'et'al,'Figure'3a 21''22'
23'
Another Example: Three Selectivities, Six Ages at Maturity
90Monday, February 17, 14
91
Another Example: Three Selectivities, Six Ages at Maturity
9
''33'
Mangel'et'al,'Figure'3e'34''35'
'36''37''38'
39'
91Monday, February 17, 14
Outline• Density Dependence and Population Biology• What is Steepness?• Strategic Fishery Management: The Stock Assessment Process• Connections Between Steepness and Reference Points: Previous Observations• The Reproductive Biology of Steepness• Implications for Reference Points• Fixing Steepness at 1• Moving Forward• Conclusions
92Monday, February 17, 14
93
Fixing Steepness at 1: Biological Interpretation
h =
α p
M
4 +α p
M
h→1 as α p
M→∞
The stock is either infinitely productiveor infinitely long-lived (or both)
93Monday, February 17, 14
94
Fixing Steepness at 1: Biological Interpretation
There is no evidence yet of Darwinian demons in fish
h =
α p
M
4 +α p
M
h→1 as α p
M→∞
The stock is either infinitely productiveor infinitely long-lived (or both)
94Monday, February 17, 14
95
Fixing Steepness at 1: Biological Interpretation (continued)
h→1As
FMSYM
=4h1− h
−1
FMSYM
→∞
Since the stock is infinitely productive, we can fish it infinitely hard!!!
95Monday, February 17, 14
96
Fixing Steepness at 1: Probabilistic Interpretation
h = 1 means
Pr{R(0.2B0 ) = R0} = 1
With certainty, recruitment at 20% of unfished biomass is unfished recruitment. Is this what we mean?
96Monday, February 17, 14
97
Fixing Steepness at 1: Probabilistic Interpretation
h = 1 means
Pr{R(0.2B0 ) = R0} = 1
With certainty, recruitment at 20% of unfished biomass is unfished recruitment. Is this what we mean?
What if recruitment at 20% of unfished biomass can take any value between 20% of unfished recruitment and unfished recruitment?
97Monday, February 17, 14
98
Fixing Steepness at 1: Probabilistic Interpretation
h = 1 means
Pr{R(0.2B0 ) = R0} = 1
With certainty, recruitment at 20% of unfished biomass is unfished recruitment. Is this what we mean?
What if recruitment at 20% of unfished biomass can take any value between 20% of unfished recruitment and unfished recruitment?
Steepness ranges between 0.2 and 1.0 (tied down at both ends by biology)
98Monday, February 17, 14
Michielsens and McAllister.2004. CJFAS 61:1032-1047
Thus, steepness should have a probability distribution
Harley et al. 2009.Bigeye Tuna Assessment
99Monday, February 17, 14
Outline• Density Dependence and Population Biology• What is Steepness?• Strategic Fishery Management: The Stock Assessment Process• Connections Between Steepness and Reference Points: Previous Observations• The Reproductive Biology of Steepness• Implications for Reference Points• Fixing Steepness at 1• Moving Forward• Conclusions
100Monday, February 17, 14
Three Options for Moving Forward
101
Do Not Fix Steepness and Mortality Rate
101Monday, February 17, 14
Three Options for Moving Forward
102
Do Not Fix Steepness and Mortality Rate
Replace the BH-SRR by a SRR that Avoids the Problem
102Monday, February 17, 14
Three Options for Moving Forward
103
Do Not Fix Steepness and Mortality Rate
Replace the BH-SRR by a SRR that Avoids the Problem
Be fully honest about the limitations of the data and the stock assessment
103Monday, February 17, 14
104
Do Not Fix Steepness and Mortality Rate
When will data be informative?
Use Simulation Methods to determine what kinds of data are necessary so that steepness and natural mortality can be estimated in the stock assessment
104Monday, February 17, 14
105
Do Not Fix Steepness and Mortality Rate
When will data be informative?
Use Simulation Methods to determine what kinds of data are necessary so that steepness and natural mortality can be estimated in the stock assessment
Already started:
Lee, H.-H. et al. 2011. Estimating natural mortality within a fisheries stock assessment model: An evaluation using simulation analysis based on twelve stock assessments. Fisheries Research 109: 89-94.
Lee, H.-H. et al. 2012. Can steepness of the stock–recruitment relationship be estimated in fishery stock assessment models? Fisheries Research 125-126: 254-261.
105Monday, February 17, 14
Replace the BH-SRR by a SRR that Avoids the Problem
An example: Maynard Smith/Shepherd model
dBdt
=α pB
1+ βB1n
− (M + F)B
106Monday, February 17, 14
Replace the BH-SRR by a SRR that Avoids the Problem
An example: Maynard Smith/Shepherd model
dBdt
=α pB
1+ βB1n
− (M + F)B
107Monday, February 17, 14
n>1, Cushing-like
n=1, Beverton-Holt
n<1, Ricker-like
dBdt
=α pB
1+ βB1n
− (M + F)B
108Monday, February 17, 14
109
For Maynard Smith Shepherd SRR, steepness is
109Monday, February 17, 14
110
For Maynard Smith Shepherd SRR, steepness is
so that
110Monday, February 17, 14
111
and most importantly
For Maynard Smith Shepherd SRR, steepness is
so that
111Monday, February 17, 14
112
and most importantly
There is still an undetermined parameter for the analysis -- the data can tell us something! - orwe can integrate over the potential range of n
For Maynard Smith Shepherd SRR, steepness is
so that
112Monday, February 17, 14
113
Application to Cowcod (Dick, WGC 2012)
BH SRR�
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0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.2
0.4
0.6
0.8
1.0
Fmsy / M
Bmsy
/ B0
113Monday, February 17, 14
114
Application to Cowcod (Dick, WGC 2012)
BH SRR (h fixed at 0.6)
3 parameter SRR
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X
114Monday, February 17, 14
We Can Do a Meta-analysis on the Parameter, And ItMight Look Like This
Distribution ofn
115Monday, February 17, 14
Or The Meta-analysis Could Yield This
116Monday, February 17, 14
Or This
117Monday, February 17, 14
But Not Likely This
118Monday, February 17, 14
119
Be fully honest about the limitations of the data and the stock assessment
• Management policy with biomass targets or rebuilding plans on a fixed time table with specified probability may often overstep what can realistically be expected from a defensible assessment of an individual stock.
119Monday, February 17, 14
120
Be fully honest about the limitations of the data and the stock assessment
• Management policy with biomass targets or rebuilding plans on a fixed time table with specified probability may often overstep what can realistically be expected from a defensible assessment of an individual stock.
• The community of stock assessment scientists needs to agree on workable protocols for several classes of life history parameters, ecosystem types, fishery histories that are reasonably robust in achieving management objectives in the face of scientific uncertainty. Developing good proxies and protocols is a scientific matter. It should be based on meta analyses and management strategy evaluation.
120Monday, February 17, 14
Outline• Density Dependence and Population Biology• What is Steepness?• Strategic Fishery Management: The Stock Assessment Process• Connections Between Steepness and Reference Points: Previous Observations• The Reproductive Biology of Steepness• Implications for Reference Points• Fixing Steepness at 1• Moving Forward• Conclusions
121Monday, February 17, 14
122
Conclusions, Part 1: A scien3fic theory should be as simple as possible, but no simpler.
122Monday, February 17, 14
123
Conclusions, Part 1: A scien3fic theory should be as simple as possible, but no simpler.
This is a case of a too simple theory
123Monday, February 17, 14
Conclusions, Part 2
• As soon as we are able to develop a demographic model for the survival and reproduction of a cohort, we are able to obtain a point estimate for steepness (steepness cannot simply be fixed)
124Monday, February 17, 14
Conclusions, Part 2
• As soon as we are able to develop a demographic model for the survival and reproduction of a cohort, we are able to obtain a point estimate for steepness (steepness cannot simply be fixed)
•Allowing stochasticity in survival allows us to derive probability distributions, not just point estimates (see Mangel et al. 2010. Fish and Fisheries 11:89-104; NMFS Toolbox Code in Development)
125Monday, February 17, 14
Conclusions, Part 2
• As soon as we are able to develop a demographic model for the survival and reproduction of a cohort, we are able to obtain a point estimate for steepness (steepness cannot simply be fixed)
•Allowing stochasticity in survival allows us to derive probability distributions, not just point estimates (see Mangel et al. 2010. Fish and Fisheries 11:89-104; NMFS Toolbox Code in Development)
•In some cases, steepness is relatively high, but not 1.
126Monday, February 17, 14
Conclusions, Part 2
• As soon as we are able to develop a demographic model for the survival and reproduction of a cohort, we are able to obtain a point estimate for steepness (steepness cannot simply be fixed)
•Allowing stochasticity in survival allows us to derive probability distributions, not just point estimates (see Mangel et al. 2010. Fish and Fisheries 11:89-104; NMFS Toolbox Code in Development)
•In some cases, steepness is relatively high, but not 1.
•Age structure broadens the prior for steepness.
127Monday, February 17, 14
Conclusions, Part 2
• As soon as we are able to develop a demographic model for the survival and reproduction of a cohort, we are able to obtain a point estimate for steepness (steepness cannot simply be fixed)
•Allowing stochasticity in survival allows us to derive probability distributions, not just point estimates (see Mangel et al. 2010. Fish and Fisheries 11:89-104; NMFS Toolbox Code in Development)
•In some cases, steepness is relatively high, but not 1.
•Age structure broadens the prior for steepness.
•The early life history is key to understanding steepness
128Monday, February 17, 14
Conclusions, Part 2
• As soon as we are able to develop a demographic model for the survival and reproduction of a cohort, we are able to obtain a point estimate for steepness (steepness cannot simply be fixed)
•Allowing stochasticity in survival allows us to derive probability distributions, not just point estimates (see Mangel et al. 2010. Fish and Fisheries 11:89-104; NMFS Toolbox Code in Development)
•In some cases, steepness is relatively high, but not 1.
•Age structure broadens the prior for steepness.
•The early life history is key to understanding steepness
•Environmental forcing can be built into the early life history through fluctuations in mortality rate and into productivity through fluctuations in egg production.
129Monday, February 17, 14
Conclusions, Part 3: Se>ng Steepness Equal to 1 is
• Biologically naive
130Monday, February 17, 14
Conclusions, Part 3: Se>ng Steepness Equal to 1 is
• Biologically naive
• Very non-precautionary
131Monday, February 17, 14
Conclusions, Part 3: Se>ng Steepness Equal to 1 is
• Biologically naive
• Very non-precautionary
• Wrong scientific inference: Recruitment independent of stock size does not mean h=1 (with probability 1 the recruitment at 20% of spawning stock size is the same as R0). Rather it means any percentage of recruitment is possible:h has a uniform distribution (tied down at small values and near 1).
132Monday, February 17, 14
Conclusions, Part 3: Se>ng Steepness Equal to 1 is
• Biologically naive
• Very non-precautionary
• Wrong scientific inference: Recruitment independent of stock size does not mean h=1 (with probability 1 the recruitment at 20% of spawning stock size is the same as R0). Rather it means any percentage of recruitment is possible:h has a uniform distribution (tied down at small values and near 1).
133Monday, February 17, 14
The first responsibility of scientists is to the integrity of science and it is critical to be explicit about what is known and not known. And there is not a moment to be lost.
134Monday, February 17, 14
A Few Citations
He,X., Mangel, M. and A. MacCall. 2006. A prior for steepness in stock-recruitment relationships, based on an evolutionary persistence principle. Fishery Bulletin 104: 428-433
Mangel, M., Brodziak, J.K.T., and G. DiNardo. 2010. Reproductive ecology and scientific inference of steepness: a fundamental metric of population dynamics and strategic fisheries management. Fish and Fisheries 11:89-104.
Enberg, K., Jorgensen, C. , and M. Mangel. 2010. Fishing-induced evolution and changing reproductive ecology of fish: the evolution of steepness. Canadian Journal of Fisheries and Aquatic Sciences 67:1708-1719.
Mangel, M., MacCall, A.D., Brodziak, J., Dick, E.J., Forrest, R.E., Pourzand, R., and S. Ralston. 2013. A perspective on steepness, reference points, and stock assessment. Canadian Journal of Fisheries and Aquatic Sciences 70:930-940
135Monday, February 17, 14