For 2014 Show the line of R producing SSB, and SSB producing R, and how they would spiderweb to get to equilibrium R. Took a long time, did not get to the end stuff after Sustainable Fisheries Act.

Age-structured models part 2

Today Yield-per-recruit analysis: SBPR, YPR Reference points: B 40%, F max, R 0, SSB 0, MSY, B MSY, u MSY How to use the Table Function in Excel (Data->What If Analysis->Data Table) Discrete vs. continuous fishing

Review: Basic age-structured model sexes grouped The “plus” group Egg production Recruitment Catch weight a function of egg production e.g. Beverton-Holt Ages between 1 and n Mass at age Natural survival rate (0-1) Exploitation rate (0-1) Vulnerability (0-1) All individuals identical above the plus group age Fecundity

Review: starting conditions (t = 1) Starting recruitment Natural survival rate Exploitation rate Vulnerability Numbers in plus group age n

Yield-per-recruit analysis Tracking one recruit (or one cohort) At different exploitation rates (u t ), 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) Natural survival rate Exploitation rate Vulnerability Plus group age Fecundity Weight-at-age Spreadsheet: “4 per recruit analysis.xlsx”

YPR is different from full age- structured model One recruit (not R 0 ) Only analyzing one cohort, not impacts on multiple generations No recruitment function

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 Spreadsheet: “4 per recruit analysis.xlsx”

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

Reference points based on YPR and SBPR Egg production: F 40% is the fishing mortality rate at which SBPR is 40% of maximum (also F 35%, F 50%, etc.) F max is the fishing mortality rate that maximizes YPR, where this exists For many species where there is little concern about recruitment overfishing, yield-per-recruit dominates (since it is assumes there is no change in recruitment)

F max and F 40% From yield-per-recruit, no stock-recruit relationship 4 per recruit analysis.xlsx, sheet YPR by u F 40% SBPR 0 0.4×SBPR 0 F max

F max can be undefined Each curve has a different age for when fish are vulnerable to fishing, at 5 yr there is no defined F max 1 yr 2 yr 3 yr 4 yr 5 yr = F max 4 per recruit analysis.xlsx, sheet Fmax

Unfished spawning biomass In the absence of harvest, spawning biomass per recruit SBPR 0 is the same as the total egg production in the yield-per- recruit calculations Therefore unfished spawning biomass (common symbols are SSB 0, B 0, SB 0 or E 0 ) is SBPR 0 multiplied by recruitment in the unfished population (R 0 )

Equilibrium recruitment for exploitation rate u Beverton-Holt equation, recruits depend on spawners Spawners depend on recruits

Calculating MSY and B MSY Unlike YPR calculations of F max, 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 B MSY by using analytic formulae for the yield as a function of exploitation rate. MSY is the highest yield, B MSY is the stock size that produces the highest yield

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) SSB MSY is the spawning stock biomass at the u that produces MSY u MSY is the exploitation rate u producing MSY

4 MSY Bmsy.xlsx sheet “MSY Bmsy”

Equilibrium exploitation vs. catch u MSY MSY Unsustainable (u)(u) 4 MSY Bmsy.xlsx sheet “MSY Bmsy”

Spawning output vs. catch SSB MSY MSY 4 MSY Bmsy.xlsx sheet “MSY Bmsy” SSB MSY at 26% of SSB 0 SSB 0

Total biomass vs. catch B MSY (or TB MSY ) MSY 4 MSY Bmsy.xlsx sheet “MSY Bmsy” B0B0 TB MSY at 32% of TB 0

Reference point: B MSY B MSY —biomass that produces maximum sustained yield—used to be a target for fisheries management, but now often treated as a lower limit MSY—also known as the optimum yield

Sustainable Fisheries Act 2007 (16 U.S.C , updated from 1977 Magnuson-Stevens Act) (33) The term “optimum”, with respect to the yield from a fishery, means the amount of fish which— (A) will provide the greatest overall benefit to the Nation, particularly with respect to food production and recreational opportunities, and taking into account the protection of marine ecosystems; (B) is prescribed as such on the basis of the maximum sustainable yield from the fishery, as modified reduced by any relevant economic, social, or ecological factor; and (C) in the case of an overfished fishery, provides for rebuilding to a level consistent with producing the maximum sustainable yield in such fishery. i.e. B MSY Changed in 1996

MSY most affected by steepness and natural mortality Natural survival = 0.9Natural survival = 0.5 Exploitation rate (u) 4 MSY Bmsy.xlsx sheet “MSY by h and s”

Key characteristics of “basic” age-structured models Time-invariant production relationship All models totally stable, if you stop fishing at any level the population recovers All models show higher rates of increase at lower densities

What age-structured models can’t do Very little: almost any desired feature can be added to the basic framework (e.g. depensation, density-dependent growth and survival, environmental effects on recruitment and survival) Can think of these models as a general framework in which to embed specific recruitment, growth and survival hypotheses

Common extensions Splitting the sexes: especially when growth and vulnerability differ by sex Explicit partitioning between mature and immature, or vulnerable and not vulnerable individuals Splitting by space

What to know What are the terms B 40%, F max, R 0, SSB 0, MSY, B MSY, u MSY What determines shape of yield-per-recruit What determines shape of total yield curve (exploitation rate vs. yield) How to derive equilibrium recruitment How to estimate MSY, B MSY