MSY from age-structured models

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”

MSY reference points loop over different values of u Equilibrium recruits at fishing rate u 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) uMSY is the exploitation rate u producing MSY SSBMSY is the spawning stock biomass at uMSY

5 MSY Bmsy.xlsx sheet “MSY Bmsy”

Equilibrium exploitation vs. catch MSY Sustainable yield Unsustainable Exploitation rate (u) uMSY 5 MSY Bmsy.xlsx sheet “MSY Bmsy”

Spawning output vs. catch MSY SSBMSY at 26% of SSB0 Sustainable yield SSB0 Spawning output (eggs) SSBMSY 5 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) 5 MSY Bmsy.xlsx sheet “MSY Bmsy”

Reference point: BMSY BMSY—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 Sustainable Fisheries Act 2007 (16 U.S.C. 1802 104-297, 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. Changed in 1996 i.e. BMSY

MSY most affected by steepness and natural mortality Natural survival = 0.9 Natural survival = 0.5 Sustainable yield (C) Exploitation rate (u) 5 MSY Bmsy.xlsx sheet “MSY by h and s”

Key characteristics of “basic” age-structured models Time-invariant production relationship Completely stable, if you stop fishing at any level the population recovers Basic models have higher rates of increase at lower population 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) These models are 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 Splitting by space

What to know What are the terms B40%, Fmax, R0, SSB0, MSY, BMSY, uMSY 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, BMSY

Brief note on discrete fishing vs. continuous fishing Refer to handout on discrete vs. continuous fishing 5 Discrete vs continuous handout.pdf

Discrete fishing and continuous fishing We have assumed discrete fishing at a point in time (after births, before natural mortality) Exploitation rate at age and year (ua,t) is assumed to be “separable”: separated into age-specific va and year-specific ut. i.e. ua,t = vaut Furthermore, va ≤ 1 and ut ≤ 1 (we can’t catch more fish than there are) This implies it is impossible to catch age groups with vulnerability less than 1: ua,t < 1 even if ut = 1

Continuous fishing (Baranov equation) Discrete fishing ut is bounded [0,1] Partially continuous fishing Ft is bounded [0,∞) Continuous fishing (Baranov equation) Ft is bounded [0,∞) See “5 Discrete vs. continuous handout.pdf”

Fitting models to data using sums of squares

Background reading Chapter 5, Hilborn R & Mangel M (1997) The ecological detective: confronting models with data. Princeton University Press, Princeton, New Jersey.

Why fit models to data Models are hypotheses about nature We use the data to see how much support there is for competing hypotheses Does your model fit the data? Only valid when comparing to other hypotheses—there is no real absolute measure of acceptable fit

What is needed Competing models Data Goodness-of-fit criterion Algorithm to maximize goodness of fit

Length-weight model Parameters to estimate Weight Length ln of both sides turns it into a linear model of the form Y = a + bX

Data Weight (kg) Fork length (cm)

Goodness of fit Every value of a and b makes predictions about the weight of each individual fish i The “best” parameter values are those that minimize the sum of squares

The “best” fit Weight (kg) Fork length (cm) 5 Length weight.xlsx, sheet “fit”

Sum of squares surface the shading comes from conditional formatting b = 3.4 b = 2.4 a = 1×10-5 Green = good fit, red = poor fit, = best fit a = 3×10-5 5 Length weight.xlsx, sheet “Surface”

5 Length weight.xlsx sheet “surface”

How to find the minimum sum of squares Direct search (for 1–3 parameters), e.g. SSQ surface in the previous slide Algebra (linear regression and linear models) Non-linear gradient searches Non-linear function minimization

What are the competing hypotheses? Different values of slope or intercept We could expand our analysis to include shapes other than a power function and ask if the data support them

Schaefer model with index of abundance Constant: index assumed linearly proportional to biomass Observed data are Ct and It Unfished biomass = K Observed index value Predicted index value Schaefer MB 1954 Some aspects of the dynamics of the population important to the management of the commercial marine fisheries. Inter-American Tropical Tuna Commission Bulletin 1(2):25-56

Why lnSSQ and not SSQ? So that predicted 10 vs. observed 20 has same weight as predicted 1 vs. observed 2 The underlying assumption is multiplicative error

Simulated lobster CPUE data a one-way trip of catch-per-unit-effort data Index Year 5 Lobster simulation.xlsx, sheet “Simulate”

Index Index Index Index Index Year r = 0.05, K = 40271, SSQ = 0.902 5 Lobster simulation.xlsx, sheet “Estimate”

Corresponding harvest rates Index Index Harvest rate Index Index Year 5 Lobster simulation.xlsx, sheet “Estimate”

Lobster model fits All the models fit the index data very well But estimated harvest rates (u) range from 0.19 to 0.65 in the final year We need auxiliary information about harvest rates Expert knowledge: from length data it is clear than almost all lobster are caught, thus harvest rates are around 0.7 in recent years. Solution: add a term (u2005 – 0.7)2 to the SSQ

Multiple sources of information We have two types of information The index series (CPUE) The knowledge (from length frequency analysis) that exploitation rates are high Leading to two problems How to weight multiple data sources How to make probabilistic statements about the results

Unequal weighting 5 Lobster simulation.xlsx, sheet “u=0.7”

Lessons learned A one-way trip is not very informative Harvest and growth are confounded We can “force” the model to have higher exploitation rates by adding a second term to the SSQ with a particular weight But this involves arbitrary weighting of CPUE index vs. final exploitation rate

Sum of squares summary Make the predicted close to the observed! A simple approach that can be applied to simple or very complex models Find the hypothesis that comes closest to the data Find competing hypotheses that fit data nearly as well Decide how to deal with alternative hypotheses that are almost-as-good

Next steps Models often try to explain many sources of data Move to likelihood that provides a logical and statistically valid method of weighting alternative data sources Likelihood also lets us make more probabilistic statements about competing fits