Handling uncertainty in models of population dynamics

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Handling uncertainty in models of population
dynamics

• S.T. Buckland, K.B. Newman,
• L. Thomas and J. Harwood
• University of St Andrews
• Carmen Fernández
• University of Lancaster and Institute of Marine
  Research, Vigo

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AIMA generalized methodology for defining and
fitting matrix population models that
accommodates process variation (demographic and
environmental stochasticity), observation error
and model uncertainty
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Hidden process models

• Special case
• state-space models
• (first-order Markov)

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States
We categorize animals by their state, and
represent the population as numbers of animals
by state.
Examples of factors that determine state age
sex size class genotype sub-population
(metapopulations) species (e.g. predator-prey
models, community models).
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States
Suppose we have m states at the start of year t.
Then numbers of animals by state are
NB These numbers are unknown!
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Intermediate states
The process that updates nt to nt1 can be split
into ordered sub-processes.
e.g. survival ageing births
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Survival sub-process
Given nt
NB a model (involving hyperparameters) can be
specified for
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Ageing sub-process
No first-year animals left!
Given us,t
NB process is deterministic
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Birth sub-process
Given ua,t
New first-year animals
NB a model may be specified for
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The BAS model
where
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The BAS model
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Leslie matrix
The product BAS is a Leslie projection matrix
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Other processes
Growth sex assignment genotype assignment
movement (metapopulations) competition predato
r-prey effects
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Observation equation
e.g. metapopulation with two sub-populations,
each split into adults and young, unbiased
estimates of total abundance of each
sub-population available
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Fitting models to time series of data

• Kalman filter
• Normal errors, linear models
• or linearizations of non-linear models
• Markov chain Monte Carlo
• Sequential Monte Carlo methods

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Elements required for Bayesian inference
Prior for parameters
pdf (prior) for initial state
pdf for state at time t given earlier states
Observation pdf
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Bayesian inference
Joint prior for and the
Likelihood
Posterior
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Types of inference
Filtering
Smoothing
One step ahead prediction
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Generalizing the framework

• Replace

by
where
and is a possibly random operator
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Incorporating model uncertainty

• MCMC use reversible jump MCMC

Sequential MC methods define a set of models
and associated prior, and use model averaging
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Example British grey seals
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Estimated pup production
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Population dynamics model

• Predictions constrained to be biologically
  realistic
• Can predict outcomes from different management
  actions
• Framework for exploring ecological processes
  (e.g. density dependence operating on different
  processes)

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Posterior parameter estimates
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Smoothed pup estimates
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Predicted adults
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References
Buckland, S.T., Newman, K.B., Thomas, L. and
Koesters, N.B. 2004. State-space models for the
dynamics of wild animal populations. Ecological
Modelling 171, 157-175.
Thomas, L., Buckland, S.T., Newman, K.B. and
Harwood, J. 2005. A unified framework for
modelling wildlife population dynamics.
Australian and New Zealand Journal of Statistics
47, 19-34.
Newman, K.B., Buckland, S.T., Lindley, S.T.,
Thomas, L. and Fernández, C. in press. Hidden
process models for animal population dynamics.
Ecological Applications.