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

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)

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

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”)

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)?

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”

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)

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

4 per recruit analysis.xlsx sheet YPR and SBPR

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

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

4 per recruit analysis.xlsx sheet YPR by u

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

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

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

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

MSY from age-structured models

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)

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

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

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

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

4 MSY Bmsy.xlsx sheet “MSY Bmsy”

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

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

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”