Models of total numbers or biomass

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Models of total numbers or biomass

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Numbers next year numbers this year births -
deaths immigrants - emigrants

• Nt1NtB-DI-E
• This is a tautological framework, we then flesh
  it out with functional responses
• BbNt
• DdNt
• I0
• E0
• exponential_model.xls

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A differential equation version

• dN/dt (b-d)N
• NtN0exp(b-dt)

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Model assumptions

• Population will grow or decline exponentially for
  an indefinite period
• Births and deaths are independent and there is
  no impact of age structure or sex ratio
• Birth and death rates are constant in time
• There is no environmental variability

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Adding stochasticity and building an individual
based model

• 1. For each time step from 1 to nt, do steps 2
  to 7.
• 2. Let N(t1) take the value of the current
  population N(t)
• 3. For each animal from 1 to N(t), do steps 4 to
  7
• 4. Choose a uniform random number U1
• 5. Choose a uniform random number U2
• 6. If U1 is less than d, then decrease N(t1) by
  1.
• 7. If U2 is less than b, then increase N(t1) by
  1.

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Show simulation IBM_with_stochasticity.xls
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Simplifying the calculations

• The expected number of births will be bNt
• The expected number of deaths will be dNt
• The number of births and deaths will be
  binomially distributed (remember QSCI 381!)
• With large sample size the binomial approaches
  the normal distribution. The variance of a
  binomial is p(1-p)N.

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Births

• Will be normally distributed with
• mean b Nt
• standard deviation sqrt(b(1-b)Nt)
• In EXCEL we can generate normal deviates using
  TOOLS
• With small sample size we can generate binomial
  deviates

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Steps

• Calculate expected standard deviation time t (st)
• Generate random deviates (measured in standard
  deviations), call this rt
• Number of births will be
• Ntbrtst
• problems occur at low numbers ... births may be
  negative and deaths positive

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Why we do this

• We dont have to calculate the probability of
  each individual living or giving birth
• This is very helpful with populations in the
  thousands or millions!
• Remember both models assume that birth and death
  for each individual are independent, there are
  no environmental effects!

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Types of stochasticity

• Phenotypic - not all individuals are alike
• Demographic - random births and deaths
• Environmental - some years are better than
  others, El Nino, hurricanes, deep freeze etc
• Spatial - not all places are alike

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Only demographic stochasticity was included in
previous models

• We often allow for a more general model of
  stochasticity
• Nt1f(Nt,p,ut) exp(wt)
• where wt is normally distributed with a mean zero
  and standard deviation ?
• this says -- numbers next year depend upon
  numbers this year, the parameters (b and d), any
  controls u (such as harvesting) and random
  environmental conditions

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A little deeper

• If w is normally distributed with mean zero
• when w0 exp(w) 1 -- this is the average year
• when wgt0 then exp(w) gt 1 these are good years
• when wlt0 then exp(w) lt 1 these are bad years
• because exp(w) is not symmetric, the expected
  value is not 1
• therefore we use a correction factor exp(w- ? 2/2)

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Demo lognormal distributions

• Then stochastic and stochastic 2 from
  exponential_model

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Adding habitat limitation
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This simply says that the rate of increase
declines as density approaches k
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Why compensation

• At high densities there will be a shortage of
  food, refuge from predators or some critical
  requirement
• Birth rates may decline, or mortality rates will
  increase
• This is called compensation and can be quantified
  as the difference between the rate of increase
  when resources are abundant and a rate of
  increase of 0.

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Anticipating depensation

• Rates of increase may decline at low densities
• This is known as depensation
• It makes local or total extinction much more
  likely
• Will be discussed later in course

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Things to do later in course

• Explore what happens if environmental conditions
  are not independent in time, but tend to come in
  runs of good and bad years more often than would
  be expected by chance.
• This is called serial autocorrelation