1. Biodiversity of Fishes Stock-Recruitment Relationships
Rainer Froese,

2. Typical S-R Data
Recruits
(N)
Spawning stock biomass
(tonnes)

3. Distribution of R (recruits)
roughly log-normal
Skewed

4. Distribution of S (spawners)
skewed
roughly log-normal

5. The Hump (Ricker, 1954) Assumptions:
where A = ln Rmax
Assumptions:
a) negative S-R relationship at high S
b) highest recruitment at intermediate S

6. The Asymptote (Beverton & Holt 1957)
where A = ln Rmax
Assumption:
Positive S-R relationship at high S

7. The Hockey-Stick (Barrowman & Myers 2000)
Assumptions:
Constant R/S at low S
Constant R at high S

8. The Smooth Hockey-Stick (Froese 2008)
where A = ln Rmax
Assumptions:
Practically constant R at high S
Gradually increasing R/S at lower S

9. Example Striped bass Morone saxatilis
Model α
low up
Rmax 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
3.81
19.8
16.5
23.9
0.846
Parameters and accounted variance not significantly different
Extrapolation VERY different

10. Example: 12 stocks of Atlantic cod Gadus morhua
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.

11. Conclusion of detailed comparison (Froese 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.

12. When does R decline?
For the hockey-sticks:

13. Example: North-east Arctic Cod
Slim Spa
Smax
Froese & Proelss 2010)

14. 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 age- specific mortality rates of juveniles (Mt)
where tr is the mean age at recruitment and tm is the mean age at first maturity
Froese & Proelss 2010)

15. 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
Multiplying SN0 with mean body weight Wmean gives S0
Where Pt is the proportion of mature
individuals at age t and Mc is the
age-specific mortality rate
Froese & Proelss 2010)

16. Example: North-east Arctic Cod
Smsy

17. What is the maximum number of replacement spawners per spawner?
1. For the hockey-sticks, a simple relationship between
maximum recruitment and spawner biomass is given by
2. Dividing Sdecline by mean body weight gives the
number of respective (fished) spawners SNdecline
3. The maximum number of replacement spawners
at low spawner densities (αr) is then obtained as

18. Multiple spawners α > 7 leads to dangerous fluctuations
α ~ 1 would be expected for sharks
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).
Froese & Proelss 2010)

19. One-time spawners
Median α = 4.2 (3.6 – 5.2)

20. What is the intrinsic rate of population increase rmax?
In semelparous species
(one-time spawners )
In iteroparous species
(multiple spawners)
(Myers & Mertz 1998)

21. Estimating MSY and Fmsy

22. Time to reach Smsy
where Scur is the current spawner biomass and Fcur is the current fishing mortality
Froese & Proelss 2010)

23. 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

24. MSY rmax vs MSY (Y/R)max
Data from Froese & Proelss 2010)

25. Exercises
Go to and select one of these stocks: Western Baltic cod (cod-2224)
Baltic sprat (spr-2232)
Western Baltic herring (her-3a22)
Find stock-recruitment plot
Discuss which S-R curve would fit the data
Where woud S-decline fall?
If S-decline is 0.5 S-msy, what is S-msy?