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## Embedding population dynamics models in inference

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Title: Embedding population dynamics models in inference

1
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)

2
AIM A generalized methodology for defining and
fitting matrix population models that
accommodates process variation (demographic and
environmental stochasticity), observation error
and model uncertainty
3
Hidden process models
• Special case
• state-space models
• (first-order Markov)

4
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).
5
States
Suppose we have m states at the start of year t.
Then numbers of animals by state are
NB These numbers are unknown!
6
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
7
Survival sub-process
Given nt
NB a model (involving hyperparameters) can be
specified for
or
can be modelled as a random effect
8
Survival sub-process

Survival
9
Ageing sub-process
No first-year animals left!
Given us,t
NB process is deterministic
10
Ageing sub-process
Age incrementation
11
Birth sub-process
Given ua,t
New first-year animals
NB a model may be specified for
12
Birth sub-process
Births
13
The BAS model
where
14
The BAS model
15
Leslie matrix
The product BAS is a Leslie projection matrix
16
Other processes
Growth
17
The BGS model with m2
18
Lefkovitch matrix
The product BGS is a Lefkovitch projection matrix
19
Sex assignment
New-born
20
Genotype assignment
21
Movement
e.g. two age groups in each of two locations
22
Movement BAVS model
23
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
24
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

25
Elements required for Bayesian inference
Prior for parameters
pdf (prior) for initial state
pdf for state at time t given earlier states
Observation pdf
26
Bayesian inference
Joint prior for and the
Likelihood
Posterior
27
Types of inference
Filtering
Smoothing
One step ahead prediction
28
Generalizing the framework
Model prior
Prior for parameters
pdf (prior) for initial state
pdf for state at time t given earlier states
Observation pdf
29
Generalizing the framework
• Replace

by
where
and is a possibly random operator
30
Example British grey seals
31
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

32
Estimated pup production
33
Questions
• What is the future population trajectory?
• What types of data will help address this
question?
• Biological interest in birth, survival and
movement rates

34
Empirical predictions
35
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)

36
Grey seal state model states
• 7 age classes
• pups (n0)
• age 1 age 5 females (n1-n5)
• age 6 females (n6) breeders
• 48 colonies aggregated into 4 regions

37
Grey seal state model processes
• a year starts just after the breeding season
• 4 sub-processes
• survival
• age incrementation
• movement of recruiting females
• breeding

38
Grey seal state model survival
• density-independent adult survival us,a,c,t
• density-dependent pup survival us,0,c,t
Binomial(n0,c,t-1, f juv,c,t) where f juv,c,t
f juv.max/(1ßcn0,c,t-1)

39
Grey seal state model age 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

40
Grey seal state model movement 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

41
Grey seal state model movement
• (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

42
Grey seal state model breeding
• density-independent
• ub,0,c,t Binomial(um,6,c,t , a)

43
Grey seal state model matrix formulation
• E(ntnt-1, T) B Mt A St nt-1

44
Grey seal state model matrix formulation
• E(ntnt-1, T) Pt nt-1

45
Grey seal observation model
• pup production estimates normally distributed,
with variance proportional to expectation
• y0,c,t Normal(n0,c,t , ?2n0,c,t)

46
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)

47
Grey seal model prior distributions
48
Posterior parameter estimates
49
Smoothed pup estimates
50
51
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

52
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.