1. Age-structured models Yield-per-recruit, reference points, MSY

2. Today Review age-structured models
Yield-per-recruit analysis: SBPR, YPR
Reference points: F40%, Fmax, R0, SSB0, MSY, BMSY, uMSY
How to use the Table Function in Excel (Data->What If Analysis->Data Table)

3. Review: age structure year 1
Starting recruitment
Vulnerability depends on age
Exploitation rate depends on year
Natural survival rate
Numbers in plus group age n

4. Review: future years Fecundity (egg production) Egg production
Recruitment is a function of egg production e.g. Beverton-Holt
Recruitment
Exploitation rate
Vulnerability
Natural survival rate
Ages between 1 and n
The plus group
All individuals identical above the plus group age
Catch weight
Weight at age (I know, “mass”)

5. Yield-per-recruit analysis
Tracking one recruit (or one cohort)
At different exploitation rates (ut), what is the lifetime expected spawning biomass (egg production) of one recruit?
What is the lifetime expected yield (catch) from one recruit?
What exploitation rate would maximize yield?
What is MSY (maximum sustainable yield)?

6. SBPR and YPR One recruit (one individual) Vulnerability
Exploitation rate
Natural survival rate
Plus group age
Fecundity
Weight-at-age
Spreadsheet: “4 per recruit analysis.xlsx”

7. YPR is different from full age-structured model
One recruit (not R0)
Only analyzing one cohort, not impacts on multiple generations
No recruitment function (in effect, assumes recruitment is constant regardless of spawning biomass)

8. Numbers Cohort weight Age (years) Age (years) Weight at age
Spawning biomass
Age (years)
Age (years)
4 per recruit analysis.xlsx, sheet YPR and SBPR

9. 4 per recruit analysis.xlsx
sheet YPR and SBPR

10. Typical calculations YPR and SBPR as function of exploitation rate u
What is the impact of changing vulnerability through regulations?
Many reference points used in fisheries

11. YPR and SBPR as a function of exploitation rate u
Yield per recruit
SB per recruit
Exploitation rate
Exploitation rate
4 per recruit analysis.xlsx, sheet YPR by u

12. 4 per recruit analysis.xlsx
sheet YPR by u

13. Key issues in yield-per-recruit
The two most common YPR shapes are (1) asymptotic and (2) a curve that peaks and then gradual declines
Vulnerability to fishing determines which pattern occurs: when vulnerability occurs before growth has slowed, then YPR may rise and then decline, otherwise it is asymptotic

14. Reference points based on YPR and SBPR
Egg production: F40% is the fishing mortality rate at which SBPR is 40% of maximum (also F35%, F50%, etc.)
Fmax is the fishing mortality rate that maximizes YPR, if this exists
For many species where there is little concern about recruitment overfishing1, yield-per-recruit dominates (YPR assumes there is no change in recruitment as biomass declines)
1Recruitment overfishing is when fishing mortality is so high that spawning biomass cannot replenish itself

15. From yield-per-recruit, no stock-recruit relationship
Fmax and F40%
From yield-per-recruit, no stock-recruit relationship
SBPR0
Yield per recruit
SB per recruit
0.4×SBPR0
Exploitation rate
Exploitation rate
Fmax
F40%
4 per recruit analysis.xlsx, sheet YPR by u

16. Fmax can be undefined = Fmax 5 yr 4 yr 3 yr 2 yr Yield per recruit
Each curve has a different age at vulnerability to fishing, at 5 yr there is no defined Fmax
Exploitation rate
4 per recruit analysis.xlsx, sheet Fmax

17. MSY from age-structured models

18. Unfished spawning biomass
In the absence of harvest, spawning biomass per recruit SBPR0 is the same as the total egg production in the yield-per-recruit calculations
Therefore unfished spawning biomass (common symbols are SSB0, B0, SB0 or E0) is SBPR0 multiplied by recruitment in the unfished population (R0)

19. Spawning biomass under fishing
When exploitation rate is u, the spawning biomass produced per recruit is SBPR(u)
In equilibrium, by definition
Req recruits will produce spawning biomass Seq = Req×SBPR(u)
and spawning biomass Seq will produce exactly Req new recruits (from the stock-recruit function)
Can use these two facts to solve for Req

20. Equilibrium recruitment for exploitation rate u
Beverton-Holt equation, recruits depend on spawners
Spawners depend on recruits
At equilibrium (1) and (2) hold

21. Calculating MSY and BMSY
Unlike YPR calculations of Fmax, this needs the stock-recruit relation
At a given harvest rate, total yield = yield-per- recruit × recruitment, or C = YPR × R
Given this model we can calculate MSY and BMSY by using analytic formulae for the yield as a function of exploitation rate.
MSY is the highest yield, BMSY is the stock size that produces the highest yield

22. loop over different values of u calculate SBPR(u), YPR(u)
Equilibrium recruits
Equilibrium catch
Equilibrium spawning biomass
loop over different values of u
calculate SBPR(u), YPR(u)
calculate R(u), C(u), SSB(u)
end loop over values of u
MSY is maximum C(u)
SSBMSY is the spawning stock biomass at the u that produces MSY
uMSY is the exploitation rate u producing MSY

23. 4 MSY Bmsy.xlsx sheet “MSY Bmsy”

24. Equilibrium exploitation vs. catch
MSY
Sustainable yield
Unsustainable
Exploitation rate (u)
uMSY
4 MSY Bmsy.xlsx sheet “MSY Bmsy”

25. Spawning output vs. catch
MSY
SSBMSY at 26% of SSB0
Sustainable yield
SSB0
Spawning output (eggs)
SSBMSY
4 MSY Bmsy.xlsx sheet “MSY Bmsy”

26. Total biomass (weight)
Total biomass vs. catch
MSY
TBMSY at 32% of TB0
Sustainable yield B0
Total biomass (weight)
BMSY(or TBMSY)
4 MSY Bmsy.xlsx sheet “MSY Bmsy”