Count based PVA

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Count based PVA

• Incorporating Density Dependence, Demographic
  Stochasticity, Correlated Environments,
  Catastrophes and Bonanzas

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Assumptions of the diffusion appoximation

• Population growth
• Is unaffected by population density
• Its only source of variability is environmental
  stochasticity
• No trends in its mean and the variance
• Its values are not correlated in successive years
• Moderate variability
• No observation error

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But..

• Incorporating these effects into PVA models
  require
• more and better data
• more mathematically complex models

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Negative density dependence

• The simplest way to incorporate negative density
  dependence is introduce a population ceiling to
  the density-independent population growth model

?tNt if Nt lt K
Nt1
K if Nt gt K
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The ceiling model
Program algo2 (prepared by Matt 10,50,.55,.45,60)
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Mean time to extinction
Where cµ/s2, dlog(Nc/Nx), and klog(K/Nx)
If NcK and Nx 1 then
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Extinction risk predicted by the Ceiling Model
µ 0.1
Program tbarpedro
µ 0.001
µ -0.1
s2 µ
s2 2µ
s2 4µ
s2 8µ
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The theta logistic model

• A gradually changing growth rate

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The theta logistic model
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The Bay checkerspot butterflyEuphydryas editha
bayensis
front
Harrison et al., 1991 JRC population
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The negative association remains after removing
the outlier in the right
back
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Density Dependent model

• Find the best model Fit three models to the data
  using nonlinear least-squares regression of
  log(Nt1/Nt) against Nt
• Models to be tested
• Density independent model log(Nt1/Nt)r
• The Ricker model
  log(Nt1/Nt)r(1-Nt/K)
• The theta logistic model
  log(Nt1/Nt)r1-(Nt/K)T

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Estimate the parameters of each model
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Model maximum likelihood of a model assuming
normally distributed deviations is

• ln(Lmax) -(q/2)ln(2?Vr) 1)
• Vr residual variance
• q Sample number

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Maximum log likelihood

• The probability of obtaining the observed data
  given a particular set of parameter values for a
  particular model
• Information criterion statistics combine the
  maximum log likelihood for a model with the
  number of parameters it include to provide a
  measure of support

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Support is higher for

• models with higher likelihoods, and
• models with fewer parameters
• More complex models are penalized because more
  parameters will always lead to a better fit to
  the data, but at the cost of less precision in
  the estimate of each parameter and incorporation
  of spurious patterns from the data into future
  populations

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Akaike Information Criteria

• To identify the best model
• AICc -2 ln(Lmax ) (2pq)/ (q-p-1)
• p Number of estimated parameters (including the
  residual variance)
• q sampling number

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Akaike weights
exp-0.5(AICc,i-AICc,best)
Wi
? exp-0.5(AICc,i-AICbest)
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Compute the maximum log likelihood and Akaike
weights for each model
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Simulate the model to predict population
viability
qVr
s2
q-1
Program extprobpedro
Program theta_logistic
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Simulate the model to predict population
viability
Program extprobpedro
Program theta_denindeppedro
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Allee effects

• We can simply set the quasi-extinction threshold
  at or above the population size at which Alee
  effects become important
• Explicitly include Alee effects in the population
  model

Nt2
? r-ßNt
Nt1
ANt
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The parameters
maximum
Value at A
The potential offspring
Fraction of potential reproduction that is
actually achieved
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A discrete-time model with Alee effects
generated by mate-finding problems
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A discrete-time model with Alee effects
generated by mate-finding problems
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Combined effects of Demographic Environmental
stochasticity
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Combined effects of Demographic and Environmental
stochasticity
r0.1,K15, T1, b.1
r0.1,K15, T1, b1.5
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Correlation of deviations
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Environmental correlation

• When the environmental effects on the population
  growth rate are correlated, the effective
  environmental variance in the log population
  growth rate is (Foley 1994)
• (1?)/(1-?)s2

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(1?)/(1-?)s2
Variance without correlation
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Generate the correlated environmental variation

• ? ? ?t-1 vs2v (1-?2)zt
• ? correlation coefficient
• zt random number drawn from a normal
  distribution with mean 0 and variance 1
• ?t-1 is the sum of a term due to correlation
  with the previous environment deviation and a new
  random term, scaled by a factor to assure that
  the long string of ? is s2

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Extinction risk and correlation
r0.8
r1.4
Nt1
Nt
Nt
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r0.8
r1.4
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Catastrophes and Bonanzas