Delay-difference models

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Delay-difference models

• Fish 458, Lecture 14

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Delay-difference models

• Provide an intermediate option between
  age-aggregated and full age-structured models.
• Are based on some key simplifying assumptions
  that allow age-structured dynamics to be
  simplified to a single equation involving total
  biomass or total numbers only.
• Do not require much information.
• Are useful when a realistic model is needed
  that doesnt require substantial data / computer
  memory.

3
A delay difference model for the total number of
animals in a population
Note s is natural survival if there is no
exploitation or survival from all causes of
mortality but then it is necessary to assume
that the exploitable part of the population is
the same as the mature part.
4
Some typical recruitment functions
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Moving from numbers to biomass-I

• The previous delay-difference model ignored
  growth because it dealt only with numbers.
• We will now extend the framework to include
• Growth
• Time-dependent mortality (due to, for example,
  harvesting).

6
Moving from numbers to biomass-II

• The simplest way to model changes in (mature)
  biomass is to assume that recruitment is in
  units of biomass and to take account of growth.
• g is the proportional change in mass from one
  year to the next, i.e.
• st is the survival rate during year t from
  all causes.
• This model is the lagged recruitment, survival
  and growth (LRSG) model. Lets derive it from
  first principles.

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Deriving the LRSG model - I

• is the biomass of animals aged L and older at
  the start of year t
• The survival rate during year t is the product of
  survival from natural causes and from
  exploitation (note that vulnerability is assumed
  to be 1 for all animals older than age L and zero
  below this age)
• Dont forget the standard age-structured dynamics
  equation

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Deriving the LRSG Model-II
9
Extending the biomass-based delay-difference
model-I

• One of the fundamental assumptions of the LRSG
  model is its growth model.
• However, for ggt1 this model implies that mass
  increases exponentially with age. In contrast,
  mass-at-age for most species exhibits asymptotic
  behavior.

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Starting with an Asymptotic Growth Curve

• The von Bertalanffy growth curve is probably the
  most common used in fisheries. Assuming that
  mass-at-age (rather than length-at-age) follows a
  von Bertalanffy growth curve, mass-at-age is
  governed by the recursive equation
• where ? is the Brody growth coefficient.

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Deriving the full Deriso-Schnute model
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More on the Deriso-Schnute model
Survival of biomass
Growth
Notes
Recruitment
- mass-at-recruitment
The original Deriso version corresponds to
The virgin (no fishing) equilibrium is given by
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Extensions

• Allowing for partial recruitment.
• Harvesting can be continuous or occur
  instantaneously in the middle of the year.
• A variety of functions are available to predict
  recruitment.
• Delay-difference models based on size can also be
  derived.

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Overview-I

• Provides an elegant link between the
  age-aggregated (e.g. logistic) models and more
  complicated (age-, size- and stage-structured)
  models.
• Rests on some key simplifying assumptions
• Mass-at-age follows a von Bertlanffy growth
  curve.
• Recruitment and maturity occur at the same time.
• Vulnerability is independent of age above the
  age-at-maturity.
• Natural mortality is independent of age above the
  age-at-maturity.

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Overview-II

• Some of the assumptions are highly restrictive,
  particularly
• Vulnerability independent of age
• Age-at-recruitment equals age-at-maturity
• Only one fishery.
• Not used much today because we have the computing
  resources to implement full age-structured models.

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Comparison of Dynamic Schaefer, age-structured
and Deriso-Schnute models (Cape hake)

• Age-structured model
• M assumed to be 0.4yr-1.
• Mass-at-age age-specific.
• Logistic vulnerability - inflection point at age
  3.
• Maturity at age 4.
• Deriso-Schnute model
• M assumed to be 0.4yr-1.
• Estimate ? from the mass-at-age data.
• Knife-edged vulnerability / maturity at age 3.
• Parameters B0, steepness (r and K), q and ?.
• Fitted to CPUE and survey data
• Stock assumed to be at B0 in 1917.

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Model Selection

• The three models are not nested but the data and
  likelihood functions are identical.
• We can compare these models using AIC (and AICc)

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Model Selection
Schafer model -LnL -95.00 Delay-difference
model -LnL -106.02 Age-structured model -LnL
-104.58
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Readings

• Hilborn and Walters (1992), Chapter 9
• Deriso, R.B. (1980). Can. J. Fish. Aquat. Sci.
  37 268-282.
• Schnute, J. (1985). Can. J. Fish. Aquat. Sci. 42
  414-429.
• Schnute, J. (1987). Can. J. Fish. Aquat. Sci..
  44 924-940.
• Fournier, D.A., Doonan, I.J. (1987). Can. J.
  Fish. Aquat. Sci. 44 422-437.