Lumped population dynamics models

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Lumped population dynamics models

• Fish 458 Lecture 2

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Revision Nomenclature

• Which are the state variables, forcing functions
  and parameters in the following model
• population size at the start of year t,
• catch during year t,
• growth rate, and
• annual recruitment

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The Simplest Model-I

• Assumptions of the exponential model
• No emigration and immigration.
• The birth and death rates are independent of each
  other, time, age and space.
• The environment is deterministic.
• is the initial population size, and
• is the intrinsic rate of growth(b-d).
• Population size can be in any units (numbers,
  biomass, species, females).

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The Simplest Model - II

• Discrete version
• The exponential model predicts that the
  population will eventually be infinite (for rgt0)
  or zero (for rlt0).
• Use of the exponential model is unrealistic for
  long-term predictions but may be appropriate for
  populations at low population size.
• The census data for many species can be
  adequately represented by the exponential model.

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Fit of the exponential model to the bowhead
abundance data
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Extrapolating the exponential model
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Extending the exponential model(Extinction risk
estimation)

• Allow for inter-annual variability in growth
  rate
• This formulation can form the basis for
  estimating estimation risk
• ( - quasi-extinction level, time period,
  critical probability)

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Calculating Extinction Risk for the Exponential
Model

• The Monte Carlo simulation
• Set N0, r and ?
• Generate the normal random variates
• Project the model from time 0 to time tmax and
  find the lowest population size over this period
• Repeat steps 2 and 3 many (1000s) times.
• Count the fraction of simulations in which the
  value computed at step 3 is less than ?.
• This approach can be extended in all sorts of
  ways (e.g. temporally correlated variates).

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Numerical Hint(Generating a N(x,y2) random
variate)

• Use the NormInv function in EXCEL combined with a
  number drawn from the uniform distribution on 0,
  1 to generate a random number from N(0,12),
  i.e.
• Then compute

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The Logistic Model-I

• No population can realistically grow without
  bound (food / space limitation, predation,
  competition).
• We therefore introduce the notation of a
  carrying capacity to which a population will
  gravitate in the absence of harvesting.
• This is modeled by multiplying the intrinsic rate
  of growth by the difference between the current
  population size and the carrying capacity.

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The Logistic Model - II

• where K is the carrying capacity.
• The differential equation can be integrated to
  give

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Logistic vs exponential model(Bowhead whales)
Which model fits the census data better?
Which is more Realistic??
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The Logistic Model-III
r0.1 K1000
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Assumptions and caveats

• Stable age / size structure
• Ignores spatial, ecosystem considerations /
  environmental variability
• Has one more parameter than the exponential
  model.
• The discrete time version of the model can
  exhibit oscillatory behavior.
• The response of the population is instantaneous.
• Referred to as the Schaefer model in fisheries.

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The Discrete Logistic Model
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Some common extensions to the Logistic Model

• Time-lags (e.g. the lag between birth and
  maturity is x)
• Stochastic dynamics
• Harvesting
• where is the catch during year t.

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Surplus Production

• The logistic model is an example of a surplus
  production model, i.e.
• A variety of surplus production functions exist
• the Fox model
• the Pella-Tomlinson
  model
• Exercise show that Fox model is the limit p-gt0.

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Variants of the Pella-Tomlinson model
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Some Harvesting Theory

• Consider a population in dynamic equilibrium
• To find the Maximum Sustainable Yield
• For the Schaefer / logistic model

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Additional Harvesting Theory

• Find for the Pella-Tomlinson model

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Readings Lecture 2

• Burgman Chapters 2 and 3.
• Haddon Chapter 2