1. MSY from age-structured models

2. 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”

3. 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

4. 5 MSY Bmsy.xlsx sheet “MSY Bmsy”

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

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

7. 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”

8. 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

9. Sustainable Fisheries Act 2007 (16 U. S. C
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.
Changed in 1996
i.e. BMSY

10. 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”

11. 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

12. 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

13. Common extensions
Splitting the sexes: especially when growth and vulnerability differ by sex
Splitting by space

14. 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

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

16. 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

17. 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”

18. Fitting models to data using sums of squares

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

20. 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

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

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

23. Data
Weight (kg)
Fork length (cm)

24. 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

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

26. 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”

27. 5 Length weight.xlsx
sheet “surface”

28. 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

29. 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

30. 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

31. 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

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

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

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

35. 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

36. 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

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

38. 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

39. 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

40. 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