Density dependent count PVA

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Density dependent count PVA

• Jan. 28 2008

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Density dependence extinction time

• Ultimate extinction becomes certain
• Ceiling model like density independent but
  growth stops above K
• Mean extinction time given by
• (initial population size is K)
• How much benefit do we get from increasing
  carrying capacity?

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Extinction time carrying capacity
Nx 1 NC K
s2 0.1
m 0.1
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Example Bay checkerspot butterfly

• Endangered species restricted to serpentine
  outcrops in SF Bay area
• Extensively studied for 30 yr

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More general density dependence

• For other models, must use simulation to estimate
  mean extinction time
• For all models, must use simulation to estimate
  CDF
• Data analysis issues
• Which model should we use?
• What parameter values do we use?
• Reverse order must have best fit parameter
  values for each model before comparing them

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Parameter estimation

• Define
• f is the density dependence function
• P is the parameter(s) of the model
• Use nonlinear minimization to find value of theta
  that minimizes sum of squared residuals
• In Excel, use solver but often better to use a
  real stats program
• Can sometimes use linear regression (Ricker)

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Maximum likelihood of model

• Maximum log-likelihood given by
• Where
• SSE is minimum value from previous step
• Assumptions
• No measurement error
• Environmental stochasticity in r normally
  distributed

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Choosing a model

• Perform previous 2 steps for each model
• Want model with largest log-likelihood, but need
  to control for number of parameters, p (includes
  residual variance)
• Calculate corrected Akaike Information Criterion
• Choose model with smallest corrected AIC
• Differences lt 2 not significant
• Most stats programs report AIC

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Parameter estimates for JRC population
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Simulating the model

• Recursion equation is
• is the maximum likelihood parameter estimate
• is normally distributed random number with
  mean 0 and variance , where
• is the variance of the residuals

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JRC extinction risk
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Environmental autocorrelation

• Environmental autocorrelation more likely to be
  positive than negative
• In general, positive autocorrelation increases
  extinction risk
• For DI model, effective variance is
• For DD model, need to model autocorrelation in
  the residuals after fitting the density
  dependence function

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Another source of autocorrelation

• Autocorrelated residuals may indicate that you
  are using the wrong model
• Example population exhibits strongly
  overcompensating density dependence, but we fit
  density independent model
• Population density will tend to fluctuate
  regularly high low high low
• After fitting with a density independent model,
  residuals will be strongly negatively
  autocorrelated

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Simulating autocorrelation

• Estimate correlation between adjacent residuals
  of best model (r)
• In the simulation model, use the following for
  the noise term
• zt is normal random number with mean zero and
  variance one

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Catastrophes bonanzas

• The concept
• Draw a random number to determine whether
  simulation year is normal or extreme
• Generate a random r with mean and variance
  appropriate to year type
• The problem
• C B are rare we dont observe many
• Poor estimate of frequency
• Lousy estimate of distribution of effects

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Catastrophes bonanzas solutions

• Simulate with range of plausible values for
  frequency and effect
• How sensitive are results to unknown parameters?
• See text for details
• Look for relationship with known climate
  fluctuations
• E.g., bonanza in El Niño year
• Use frequency magnitude of climate fluctuation
  to estimate parameters

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Bootstrapping

• Use when distribution of residuals is not normal
• In simulations, uses the observed residuals to
  generate random numbers
• Fit DD model (if appropriate) and calculate
  residuals
• In simulation, generate by picking an
  observed residual at random
• If there is autocorrelation, estimate the zt and
  bootstrap these

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Further reading

• Cawthorne, R.A., and J.H. Marchant. 1980. The
  effects of the 1978/79 winter on British bird
  populations. Bird Study 27 163-172.
• Hilborn, R., and M. Mangel. 1997. The Ecological
  Detective Confronting Models with Data
  (Princeton).
• Lande, R., S. Engen, and B.-E. Saether. 2003.
  Stochastic Population Dynamics in Ecology and
  Conservation (Oxford).