Age structured models

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Age structured models

• Fish 458

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Key Readings

• Hilborn, R., and Walters, C. J. 1992.
  Quantitative fisheries stock assessment choice,
  dynamics and uncertainty. Chapman and Hall, New
  York.
• Quinn, T. J., Jr., and Deriso, R. B. 1999.
  Quantitative Fish Dynamics. Oxford University
  Press, New York.
• Lawson, T. A., and Hilborn, R. 1985. Equilibrium
  yields and yield isopleths from a general
  age-structured model of harvested populations.
  Can. J. Fish. Aquat. Sci. 42 1766-1771.

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Basic population processes

• Births
• Deaths
• Somatic Growth
• Movement (immigration and emigration)

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Possible Model Complexity

• Single species models
• Total numbers or biomass
• Age or size structure
• Adding spatial structure
• Multi-species models
• Including predators and prey
• Including competitors
• Full ecosystem models

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Define sequence of events

• For example that follows
• Begin year
• Spawning
• Fishing mortality
• Natural mortality

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Basic age-structured modela one sex model
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Definitions
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Assumptions

• no immigration or emigration
• parameters such as v, s, w and f dont change
  over time
• vulnerability and size not affected by fishing
• v, s, w and f the same for all ages above n-1

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Starting conditions
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The plus group initial conditions
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Vulnerability / selectivity
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Recruitment

• Beverton-Holt shaped recruitment

R0
0.9R0
Recruitment
0.5R0
S0
0.2S0
Spawning biomass
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Converting from steepness
depends on R0, steepness (h) and SpR
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Yield and spawning biomass-per-recruit
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Yield and SBPR calculations

• Look at yield and SBPR as function of u
• Traditionally (but rarely now) look at
  vulnerability schedule as affected by mesh-size
• Can add economic yield rather than biological

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Biomass of a cohortB.C. sablefish s0.9 u0
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Full calculations B.C. sablefishs0.9 u0.1
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Eggs and yield-per-recruit
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Common uses of yield and SBPR

• Currently commonly used in reference points on
  eggs
• For many species where there is little concern
  about recruitment overfishing, yield per recruit
  dominates

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Comments on recruitment

• Note that spawning biomass per recruit SpR is the
  same as the total E in the yield per recruit
  calculations when there is no harvest
• So that we take the recruitment in the unfished
  condition (R0) and multiply times the spawning
  biomass per recruit to obtain the unfished
  spawning biomass.

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Calculating MSY and BMSY

• 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

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Step 1 loop over different values of u Step
2 calculate SBPR(u), catch per recruit(u)
Step 3 calculate R(u) and C(u) Step 4 end
loop over values of u MSY is maximum C(u) BMSY is
the spawning stock biomass at the u that produces
maximum C(u)
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Common reference points

• BMSY biomass that produces maximum sustained
  yield used to be a target and now is often
  treated as a lower limit
• Fx fishing mortality rate that produces x of
  spawning biomass per recruit e.g. F40 common
  target exploitation rate reference point

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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

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What age structured models cant do

• models are very general framework, and almost any
  desired feature can be added
• for instance, depensation, density dependent
  growth and survival, environmental effects on
  recruitment and survival
• it is best to think of these models as a general
  framework in which to imbed specific recruitment,
  growth and survival hypotheses

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Why do people use them?

• Flexibility
• Realism

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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
• Explicit partition in space