Biodiversity of Fishes
Stock-Recruitment Relationships
Rainer Froese, 15.01.2015
Typical S-R Data
Recruits
(N)
Spawning stock biomass
(tonnes)
Distribution of R (recruits)
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 (spawners)
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
2.178Rmax
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
2.178e 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
Use of Hockey-Stick in Management
Conceptual drawing of the hockey stick relationship between spawning stock size and recruitment.
SSBlim marks the border below which recruitment declines, SSBpa marks a precautionary distance to
SSBlim, and 2 * SSBpa can be used as a proxy for SSBmsy, the stock size that can produce the maximum
sustainable catch [ContHS.xlsx]. (Froese et al. 2014.)
How to Fit a Hockey-Stick
Fitting a rule-based hockey stick: (1) calculate geometric mean of recruits in upper half of biomass range:
gmean R = exp(average(log(R)) for R at SSB > 383 = 373
How to Fit a Hockey-Stick
Blim
Fitting a rule-based hockey stick: (2) Extend shaft to lowest biomass with same or higher recruitment.
This gives Blim.
How to Fit a Hockey-Stick
Blim
Bpa
Fitting a rule-based hockey stick: (3) Multiply Blim = 184 with 1.4 to get a precautionary Bpa = 258 .
How to Fit a Hockey-Stick
Blim
Bpa
Fitting a rule-based hockey stick: (4) Connect Blim to origin, check fit with low recruitment.
How to Fit a Hockey-Stick
Blim
Bpa
Fitting a rule-based hockey stick: (5) Use 2 * Bpa = 516 as proxy for Bmsy.
Bmsy
2014
Abstract
The appropriateness of three official fisheries management reference points used in
the north-east Atlantic was investigated: (i) the smallest stock size that is still
within safe biological limits (SSBpa), (ii) the maximum sustainable rate of exploitation
(Fmsy) and (iii) the age at first capture. As for (i), in 45% of the examined
stocks, the official value for SSBpa was below the consensus estimates determined
from three different methods. With respect to (ii), the official estimates of Fmsy
exceeded natural mortality M in 76% of the stocks, although M is widely regarded
as natural upper limit for Fmsy. And regarding (iii), the age at first capture was
below the age at maturity in 74% of the stocks. No official estimates of the stock
size (SSBmsy) that can produce the maximum sustainable yield (MSY) are available
for the north-east Atlantic. An analysis of stocks from other areas confirmed that
twice SSBpa provides a reasonable preliminary estimate. Comparing stock sizes in
2013 against this proxy showed that 88% were below the level that can produce
52%ofofthe
thestocks
stockswere
wereoutside
outsideofofsafe
safebiological
biologicallimits,
limits,and
and12%
12%were
were
MSY. Also, 52%
severely depleted.
depleted. Fishing mortality in 2013 exceeded natural mortality in 73% of
severely
the stocks, including those that were severely depleted. These results point to the
urgent need to re-assess fisheries reference points in the north-east Atlantic and to
implement the regulations of the new European Common Fisheries Policy regarding
sustainable fishing pressure, healthy stock sizes and adult age/size at first capture.
Exercises
Go to www.ices.dk, Community, Advisory process, Latest
advice
Select a stock with a stock-recruitment graph, e.g.:
Western Baltic cod (cod-2224)
Western Baltic herring (her-3a22)
North Sea cod (cod-347d)
• Fit hockey-stick to stock-recruitment plot
• Discuss Blim, Bpa and Bmsy compared to official estimates
• What is the current status of the stock?