Age-structured models

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Age-structured models

• Fish 458, Lecture 3

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Why age-structured models?

• Advantages
• Populations have age-structure!
• More realistic - many basic population processes
  (birth rate, death rate, growth, movement) are
  age-specific.
• Much of the data we collect are structured by
  age.
• Easily to build in actual processes and highly
  flexible.

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Why age-structured models?

• Disadvantages
• Increased complexity.
• The data needed to apply age-structured models
  are often not available.
• Some of the questions being addressed do not
  require information on age-structure.
• Age-lumped models often perform as well as
  age-structured models.
• Still not that realistic (no predation,
  competition, size and spatial structure)!

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Some symbols
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State variables, Forcing Functions and Parameters

• State variables
• Numbers-at-age
• Fraction harvested
• Spawning biomass
• Forcing function catch
• Parameters
• Natural mortality, egg production-at-age,
  mass-at-age, vulnerability-at-age,
  survival-at-age, oldest age

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The Basic Age-Structured Model
Plus-group age
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The Stock-Recruitment Relationship

• The function g determines the number of offspring
  (age 0) as a function of the egg production.
  Typical examples
• Note that this model has no stochastic
  components, i.e. it is a deterministic model
  (sometimes called an age-structured production
  model).

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Some Assumptions of this Model

• The fishing occurs at the start of the year.
• No immigration and emigration.
• Fecundity, natural mortality, mass and
  vulnerability dont change over time.
• Vulnerability and mass dont change with fishing
  pressure (i.e. no density-dependence in these
  parameters).
• Age x is chosen so that fecundity, natural
  mortality, mass and vulnerability are the same
  for all ages above age x.

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Vulnerability, Selectivity and Availability

• Conventional definitions
• Selectivity The probability of catching an
  individual of a given age scaled to the maximum
  probability over all ages, given that all animals
  are available to be caught.
• Availability The relative probability, as a
  function of age, of being in the area in which
  catching occurs.
• Vulnerability The combination of selectivity and
  availability.

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The Basic Model Again-I
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The Basic Model Again-II(The steps in setting up
a model)

1. Specify the initial (year y1) age-structure.
2. Set ycy1.
3. Calculate the mortality (fishing and natural)
   during year yc.
4. Project ahead and hence compute the
   numbers-at-age for animals aged 1 and older at
   the start of year yc1.
5. Compute the egg production at the start of year
   yc1 and hence the number of 0-year-olds at the
   start of year yc1.
6. Increase yc by 1 and go to step 3.

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Building Age-Structured Models

• Be careful of timing. In the previous model
• Spawning start of the year
• Natural mortality throughout the year
• Exploitation start of the year
• Growth instantaneous at the start of year
• These are not the only possible assumptions.
• Southern hemisphere krill no growth in winter!
• The results may be sensitive to when population
  dynamic processes occur (especially if survival
  is low).

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An Alternative Model (northern cod-like)
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Assumptions of the alternative model

• The fishery occurs a fraction ? after the start
  of the year.
• Vulnerability is age and time-dependent.
• Natural survival is independent of age.
• Only animals aged 2 and older are considered in
  the model.
• No stock-recruitment relationship, i.e. this is a
  stochastic model.

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What about a population in equilibrium??

• Equilibrium implies
• Constant recruitment
• Time-invariant exploitation rate
• For the basic model therefore

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Calculating the plus-group
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Building an age-structured model-I

• There are two fisheries with different
  vulnerabilities.
• One fishery operates from January-June and the
  other from July-December.
• Animals younger than 5 are discarded (dead) by
  fishery 1.
• Recruitment (age 0) is relate to egg production
  according to a stochastic Ricker
  stock-recruitment relationship.
• Survival is independent of age.

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The Equations
Note This model implicitly discards the catch
of animals younger than 5 by not including then
in the landed catch.
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Readings

• Burgeman et al. (1994) Chapter 4
• Haddon (2001) Chapter 2
• Au and Smith (1997). Can. J. Fish. Aquat. Sci.
  54 415-420.