Embedding population dynamics models in inference

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Embedding population dynamics models in inference

• S.T. Buckland, K.B. Newman, L. Thomas
• and J Harwood (University of St Andrews)
• Carmen Fernández
• (Oceanographic Institute, Vigo, Spain)

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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
This makes model definition much simpler
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Survival sub-process
Given nt
NB a model (involving hyperparameters) can be
specified for
or
can be modelled as a random effect
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Survival sub-process
       
Survival
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Ageing sub-process
No first-year animals left!
Given us,t
NB process is deterministic
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Ageing sub-process
Age incrementation
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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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Birth sub-process
Births
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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
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The BGS model with m2
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Lefkovitch matrix
The product BGS is a Lefkovitch projection matrix
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Sex assignment
New-born
Adult female
Adult male
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Genotype assignment
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Movement
e.g. two age groups in each of two locations
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Movement BAVS model
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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
Model prior
Prior for parameters
pdf (prior) for initial state
pdf for state at time t given earlier states
Observation pdf
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Generalizing the framework

• Replace

by
where
and is a possibly random operator
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Example British grey seals
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British grey seals

• Hard to survey outside of breeding season 80 of
  time at sea, 90 of this time underwater
• Aerial surveys of breeding colonies since 1960s
  used to estimate pup production
• (Other data intensive studies, radio tracking,
  genetic, counts at haul-outs)
• 6 per year overall increase in pup production

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Estimated pup production
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Questions

• What is the future population trajectory?
• What types of data will help address this
  question?
• Biological interest in birth, survival and
  movement rates

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Empirical predictions
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Population dynamics model

• Predictions constrained to be biologically
  realistic
• Fitting to data allows inferences about
  population parameters
• Can be used for decision support
• Framework for hypothesis testing (e.g. density
  dependence operating on different processes)

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Grey seal state modelstates

• 7 age classes
• pups (n0)
• age 1 age 5 females (n1-n5)
• age 6 females (n6) breeders
• 48 colonies aggregated into 4 regions

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Grey seal state model processes

• a year starts just after the breeding season
• 4 sub-processes
• survival
• age incrementation
• movement of recruiting females
• breeding

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Grey seal state model survival

• density-independent adult survival us,a,c,t
  Binomial(na,c,t-1,fadult) a1-6
• density-dependent pup survivalus,0,c,t
  Binomial(n0,c,t-1, f juv,c,t)where f juv,c,t
  f juv.max/(1ßcn0,c,t-1)

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Grey seal state modelage incrementation and
sexing

• ui,1,c,t Binomial (us,0,c,t , 0.5)
• ui,a1,c,t us,a,c,t a1-4
• ui,6,c,t us,5,c,t us,6,c,t

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Grey seal state modelmovement of recruiting
females

• females only move just before breeding for the
  first time
• movement is fitness dependent
• females move if expected survival of offspring is
  higher elsewhere
• expected proportion moving proportional to
• difference in juvenile survival rates
• inverse of distance between colonies
• inverse of site faithfulness

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Grey seal state modelmovement

• (um,5,c?1,t, ... , um,5,c?4,t)
  Multinomial(ui,5,c,t, ?c?1,t, ... , ?c?4,t)
• ?c?i,t ?c?i,t / Sj ?c?j,t
• ?c?i,t
• ?sf when ci
• ?dd max(fjuv,i,t-fjuv,c,t,0)/exp(?distdc,i)
  when c?i

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Grey seal state modelbreeding

• density-independent
• ub,0,c,t Binomial(um,6,c,t , a)

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Grey seal state model matrix formulation

• E(ntnt-1, T) B Mt A St nt-1

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Grey seal state modelmatrix formulation

• E(ntnt-1, T) Pt nt-1

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Grey seal observation model

• pup production estimates normally distributed,
  with variance proportional to expectation
• y0,c,t Normal(n0,c,t , ?2n0,c,t)

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Grey seal model parameters

• survival parameters fa, fjuv.max, ß1 ,..., ßc
• breeding parameter a
• movement parameters ?dd, ?dist, ?sf
• observation variance parameter ?
• total 7 c (c is number of regions, 4 here)

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Grey seal model prior distributions
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Posterior parameter estimates
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Smoothed pup estimates
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Predicted adults
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Seal model

• Other state process models
• More realistic movement models
• Density-dependent fecundity
• Other forms for density dependence
• Fit model at the colony level
• Include observation model for pup counts
• Investigate effect of including additional data
• data on vital rates (survival, fecundity)
• data on movement (genetic, radio tagging)
• less frequent pup counts?
• index of condition
• Simpler state models

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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. 2006. Hidden
process models for animal population dynamics.
Ecological Applications 16, 74-86.
Buckland, S.T., Newman, K.B., Fernández, C.,
Thomas, L. and Harwood, J. Embedding population
dynamics models in inference. Submitted to
Statistical Science.