Transcript Revisiting Stock-Recruitment Relationships

Revisiting
Stock-Recruitment Relationships
Rainer Froese
24.02.09
Mini-workshop on
Fisheries: Ecology, Economics and Policy
CAU, Kiel, Germany
Overview
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A review of S-R models and their properties
Estimating Sdecline
Estimating unfished biomass S0 and Smsy
Estimating annual reproductive rate αr
Estimating rmax
Estimating MSY and Fmsy
Estimating time to reach Smsy
MSY and Fmsy from ICES data
Some results
Typical S-R Data
(N)
(tonnes)
Distribution of R
roughly log-normal
Skewed
2500
2500
1500
Frequency (n)
Frequency (n)
Frequency (n)
2000
2000
1000
500
1500
1500
1000
1000
500
500
0
0
1
2
3
Rnorm
4
5
00
-5.00
-5
-2.50
-3
0.00
0
LNSnorm
Rnorm
2.50
3
5.00
5
Distribution of S
roughly log-normal
2500
2500
2000
2000
Frequency (n)
Frequency (n)
skewed
1500
1000
500
1500
1000
500
0
0
1
3
Snorm
4
5
0
-5.00
-2.50
0.00
LNSnorm
2.50
5.00
The Hump (Ricker, 1954)
R   S e

S



R  Se
Rmax  2.178

S
2.178 Rmax
ln R  ln   ln S 
where A = ln Rmax
Assumptions:
a) negative S-R relationship at high S
b) highest recruitment at intermediate S
S
2.178 e A
The Asymptote (Beverton & Holt 1957)
R
 S
1

S

R

S
S
1
Rmax
ln R  ln   ln S  ln(1 
where A = ln Rmax
Assumption:
Positive S-R relationship at high S
Rmax
S
eA
)
The Hockey-Stick
(Barrowman & Myers 2000)
Recruits (N)
R2  Rmax
R1    S
Spawners (N)
Assumptions:
a) Constant R/S at low S
b) Constant R at high S
The Smooth Hockey-Stick
(Froese 2008)
R  Rmax (1  e


Rmax
ln R  A  ln(1  e
where A = ln Rmax
Assumptions:
a) Practically constant R at high S
b) Gradually increasing R/S at lower S

S
)

e
A
S
)
Example Striped bass Morone saxatilis
S-R Model comparison for Morone saxatilis (striped bass) n=17 1982 --> 1998
[Stock: STRIPEDBASSUSA2]
25
20
15
Froese
Ricker
R
B&H
10
observed
5
0
0
10
20
30
40
50
60
S
Model
α
low
up
Rmax
low
up
r2
B&H
3.67
2.60
4.73
24.9
17.3
36.0
0.834
Froese
3.40
2.64
4.15
17.4
13.5
22.6
0.843
Ricker
3.22
2.64
3.81
19.8
16.5
23.9
0.846
Parameters and accounted variance not significantly different
Extrapolation VERY different
Example: 12 stocks of Atlantic cod Gadus morhua
Recruit abundance
10
1
0.1
0.01
0.01
0.1
1
10
Spawner abundance
Bold line is Smooth Hockey-Stick with n = 414, α = 4.5, Rmax = 0.85 Dotted
line the Hump with n = 414, α = 3.1, Rmax = 1.4. Data were normalized by
dividing both R and S by Rmax for the respective stock.
Conclusion of detailed comparison
(Froese et al. in prep.)
With regard to resilience of stocks to overfishing (α) and the
carrying capacity of the environment for recruits (Rmax)
• The Asymptote tends to overestimate both α and Rmax
• The Hump gives conservative estimates of α but tends to
overestimate Rmax
• The Piece-wise Hockey-Stick gives the most conservative
estimates of α and Rmax
• The Smooth Hockey-Stick tends towards intermediate
estimates of α and conservative estimates of Rmax.
When does R decline?
For the hockey-sticks:
S decline 
Rmax

Example: North-east Arctic Cod
10,000,000
SSlimdeclineSpa
S msy
Smax
R (thousands)
1,000,000
100,000
10,000
1,000
10,000
100,000
1,000,000
S (tons)
10,000,000
100,000,000
What is the number of recruits
surviving to maturity?
The mean maximum number of recruits surviving to
maturity (Rm) can be obtained from Rmax and the agespecific mortality rates of juveniles (Mt)
Rm  Rmaxe

tr
tm 1
Mt
where tr is the mean age at recruitment and tm is the mean age at first maturity
What is the unexploited
spawner biomass S0?
At S0, recruitment replaces deaths. If the mean mortality
rate (M) after mean age at maturity (tm) is known, then
the total number of individuals (SN0) can be obtained by
summing up annual survival
SN0  t Rm e  M (t tm )

m
e M  Mt m (e  M )tm
eM
 Rm
 Rm M
M
e 1
e 1
Multiplying SN0 with mean body weight Wmean gives S0
t max
Wmean 
M

c1 c
Wt Pt e

t 1
tr

t max
tr
M

c1 c
Pt e

t 1
Where Pt is the proportion of mature
individuals at age t and Mc is the
age-specific mortality rate
Example: North-east Arctic Cod
10,000,000
S decline
Smsy
S0
R (thousands)
1,000,000
100,000
10,000
1,000
10,000
100,000
1,000,000
S (tons)
10,000,000
100,000,000
What is the maximum number of
replacement spawners per spawner?
S decline 
1. For the hockey-sticks, a simple relationship between
maximum recruitment and spawner biomass is given by
t max
2. Dividing Sdecline by mean body weight gives the
number of respective (fished) spawners SNdecline
Wcur 

c1 ( M c  Fc )
W
P
e
 tt

t 1
tr

t max
tr
3. The maximum number of replacement spawners
at low spawner densities (αr) is then obtained as
Rmax
( M c  Fc )

c 1
Pt e

t 1
Rm
r 
SNdecline
Multiple spawners
Replacement spawner abundance
100
10
1
0.1
0.01
0.001
α=7
α=3
α=1
0.0001
0.0001
0.001
0.01
0.1
1
10
100
Spawner abundance (iteroparous)
Standardized replacement spawner abundance over spawner abundance for 56 stocks of 25 iteroparous
species. The curves are smoothed hockey sticks with Rmax = 1 and α as indicated. Median α = 2.1 (1.7 – 2.8).
One-time spawners
Replacement spawner abundance
100
10
1
0.1
0.01
α=4.2
0.001
0.0001
0.0001
0.001
0.01
0.1
1
Spawner abundance (semelparous)
Median α = 4.2 (3.6 – 5.2)
10
100
What is the intrinsic rate of
population increase rmax?
In semelparous species
(one-time spawners )
In iteroparous species
(multiple spawners)
(Myers & Mertz 1998)
ln( r )
rmax 
tm
ermax tm  ermax (tm 1)M   r  0
Estimating MSY and Fmsy
Fmsy  0.5rmax
MSY  Fmsy 0.5S0
Time to reach Smsy
S0
ln(
 1)
Scur
t 
2( Fmsy  Fcur )
where Scur is the current spawner biomass and Fcur is the current fishing mortality
MSY from ICES data
ICES gives the maximum yield per recruit
(Y/R)max and maximum recruitment Rmax can
be obtained as geometric mean of
recruitment at stock sizes beyond Spa.
Then
MSY = Rmax (Y/R)max
MSY rmax vs MSY (Y/R)max
1:1
1,000,000
MSY r max (tonnes)
100,000
10,000
1,000
100
100
1,000
10,000
100,000
M SY (Y/R) (tonnes)
1,000,000
10,000,000
rmax and Fmsy from ICES data
ICES provides a fishing mortality Fpa that stabilizes
the stock at a low size Spa
Fpa must then be smaller than but close to rmax
Fpa thus is a conservative estimate of rmax
rmax vs Fpa
10
1:1
r max
1
0.1
0.01
0.01
0.1
1
F pa ICES
10
Some Results
For 53 ICES stocks with available data
• 47 stock sizes are below Smsy
• These will need 0.7 – 22 years to reach Smsy if
fishing is halted (median = 6.0 years)
• Current fishing mortality in these stocks is much
larger than Fmsy
• Landings from these stocks could be 1 million
tonnes higher (+16%) at MSY
Relevance of MSY
• MSY and the related Biomass are goals
for overfished stocks and lower limits for
healthy stocks
• This is prescribed in the Law of the Sea.
• The Johannesburg Declaration of 2002 set
the deadline of 2015 to reach this
objective.
• The EC instead aims to reach Fmsy
Thank You