Embedding population dynamics models in inference S.T. Buckland, K.B. Newman, L. Thomas and J...
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Transcript of Embedding population dynamics models in inference S.T. Buckland, K.B. Newman, L. Thomas and J...
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)
AIMA generalized methodology for
defining and fitting matrix population models that
accommodates process variation (demographic and environmental stochasticity), observation error
and model uncertainty
Hidden process models
Special case:
state-space models
(first-order Markov)
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).
StatesSuppose we have m states at the start of year t. Thennumbers of animals by state are:
tm
t
t
t
t
n
n
n
n
,
,3
,2
,1
n
NB: These numbersare unknown!
Intermediate states
The process that updates nt to nt+1
can be split into ordered sub-processes.
1,,, ttbtatst nuuun
e.g. survival ageing births:
This makes model definition much simpler
Survival sub-process
tm
t
t
mtms
ts
ts
n
n
n
u
u
u
,
,2
,1
2
1
,,
,2,
,1,
00
00
00
)E(
)E(
)E(
Given nt:
mjnu jtjtjs ,,1),(binomial~ ,,,
NB a model (involving hyperparameters) can be specified for
or can be modelled as a random effect
Survival sub-process
Survival
1tn ,1 tsu ,1,
2tn ,2 tsu ,2,
mtmn , tmsu ,,
Ageing sub-process
Given us,t:
NB process is deterministic
tms
tms
ts
ts
tma
ta
ta
u
u
u
u
u
u
u
,,
,1,
,2,
,1,
,,
,3,
,2,
1100
0010
0001
No first-year animals left!
Ageing sub-process
Age incrementation
tsu ,1,
tsu ,2,
tau ,3,
tmau ,,
tau ,2,
tmsu ,1,
tmsu ,,
Birth sub-processGiven ua,t:
NB a model may be specified for
m
jjjtjat ppun
210,,1,1 ),,,(lmultinomiae.g.
i
jiji
ij ipp ,1with
tma
ta
tam
tm
t
t
t
u
u
u
n
n
n
nE
,,
,3,
,2,32
1,
1,3
1,2
1,1
100
010
001
)(
New first-yearanimals
Birth sub-process
Births
tau ,2,
tau ,3,
tmau ,,
2
3
m
tn ,1
tn ,2
tn ,3
tmn ,
The BAS model
where
100
010
00132
m
B
1100
0010
0001
A
m
00
00
00
2
1
S
ttt BASnθnn ),|(E 1
φ
λθ
The BAS model
Leslie matrix
The product BAS is a Leslie projection matrix:
mm
mmmm
1
2
1
13221
00
000
000
BAS
Other processes
Growth:
1000
0100
0000
0001
00001
1
12
2
21
1
m
mm
G
The BGS model with m=2
Lefkovitch matrix
The product BGS is a Lefkovitch projection matrix:
2
1
0
0
1
01
10
1
BGS
21
21)1(
Sex assignment
tb
tb
tb
tx
tx
tx
tx
u
u
u
u
u
uE
uE
,3,
,2,
,1,
,4,
,3,
,2,
,1,
100
010
001
00
)(
)(
New-born
Adult female
Adult male
tbtx
tbtx
txtbtx
tbtx
uu
uu
uuu
uu
,3,,4,
,2,,3,
,1,,1,,2,
,1,,1, ),(binomial~
Genotype assignment
Movemente.g. two age groups in each of two locations
1221
1221
1221
1221
100
010
010
001
V
Movement: BAVS model
Observation equation
ttttE nOθny ),|(
e.g. metapopulation with two sub-populations, each split into adults and young,unbiased estimates of total abundance of each sub-population available:
t
t
t
t
t
tt
n
n
n
n
yE
yEE
,12
,02
,11
,01
,2
,1
1100
0011
)(
)()(y
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
Elements required for Bayesian inference
)(θg Prior for parameters
)|( 00 θng pdf (prior) for initial state
),,...,|( 01 θnnn tttg pdf for state at time t given earlier states
),|( θny tttf Observation pdf
Bayesian inference
Joint prior for and the :
T
ttttggg
10100 ),,,|()|()( θnnnθnθ
θ tn
Likelihood:
T
ttttf
1
),|( θny
Posterior:
),,(
),|(),,,|()|()(),,|,,,(
1
10100
10T
T
ttttttt
TT f
fgggg
yy
θnyθnnnθnθyyθnn
Types of inference
Filtering:
Smoothing:
One step ahead prediction:
),,|,( 1 ttg yyθn
),,|,( 1 Ttg yyθn
),,|,( 11 ttg yyθn
Generalizing the framework
)|( Mθg Prior for parameters
),|( 00 Mθng pdf (prior) for initial state
),,,...,|( 01 Mθnnn tttg pdf for state at time t given earlier states
),,|( Mθny tttf Observation pdf
)(Mg Model prior
Generalizing the framework
Replace
by
where
tttt nPθnn ),|(E 1
)(1 ttt nPn
)))((()( ,1,1, tttKtKtt nPPPnP
and is a possibly random operator)(, tkP
Example: British grey sealsBritish grey seal breeding colonies
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
Estimated pup production
Year
Pu
p c
ou
nt
1960 1970 1980 1990 2000
05
00
01
00
00
15
00
0
orkney
Year
Pu
p c
ou
nt
1960 1970 1980 1990 2000
05
00
01
00
00
15
00
0
outer hebrides
Year
Pu
p c
ou
nt
1960 1970 1980 1990 2000
05
00
01
00
00
15
00
0
inner hebrides
Year
Pu
p c
ou
nt
1960 1970 1980 1990 2000
05
00
01
00
00
15
00
0
north sea
Questions
• What is the future population trajectory?
• What types of data will help address this question?
• Biological interest in birth, survival and movement rates
Empirical predictions
Year
Pu
p c
ou
nt
1960 1970 1980 1990 2000 2010
50
00
15
00
0
orkney
Year
Pu
p c
ou
nt
1960 1970 1980 1990 2000 2010
20
00
60
00
10
00
01
40
00
outer hebrides
Year
Pu
p c
ou
nt
1960 1970 1980 1990 2000 2010
15
00
25
00
35
00
inner hebrides
Year
Pu
p c
ou
nt
1960 1970 1980 1990 2000 2010
10
00
30
00
50
00
north sea
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)
• 7 age classes– pups (n0)
– age 1 – age 5 females (n1-n5)
– age 6+ females (n6+) = breeders
• 48 colonies – aggregated into 4 regions
Grey seal state model:states
Grey seal state model: processes
• a “year” starts just after the breeding season
• 4 sub-processes– survival– age incrementation– movement of recruiting females– breeding
uus,a,cs,a,c
,t,t
nna,c,t-a,c,t-
11
uui,a,ci,a,c
,t,t
uum,a,m,a,
c,tc,t
nna,c,a,c,
tt
breedinbreedingg
movemmovementent
ageagesurvivalsurvival
Grey seal state model: survival
• density-independent adult survival us,a,c,t ~ Binomial(na,c,t-1,φadult) a=1-6
• density-dependent pup survivalus,0,c,t ~ Binomial(n0,c,t-1, φ juv,c,t)where φ juv,c,t= φ juv.max/(1+βcn0,c,t-1)
Grey seal state model:age incrementation and sexing
• ui,1,c,t ~Binomial (us,0,c,t , 0.5)
• ui,a+1,c,t = us,a,c,t a=1-4
• ui,6+,c,t = us,5,c,t + us,6+,c,t
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
Grey seal state model:movement
• (um,5,c→1,t, ... , um,5,c→4,t) ~ Multinomial(ui,5,c,t, ρc→1,t, ... , ρc→4,t)
• ρc→i,t =θc→i,t / Σj θc→j,t
• θc→i,t =
– γsf when c=i
– γdd max([φjuv,i,t-φjuv,c,t],0)/exp(γdistdc,i) when c≠i
Grey seal state model:breeding
• density-independent
• ub,0,c,t ~ Binomial(um,6+,c,t , α)
Grey seal state model: matrix formulation
• E(nt|nt-1, Θ) ≈ B Mt A St nt-1
Grey seal state model:matrix formulation
• E(nt|nt-1, Θ) ≈ Pt nt-1
Grey seal observation model
• pup production estimates normally distributed, with variance proportional to expectation:
y0,c,t ~ Normal(n0,c,t , ψ2n0,c,t)
Grey seal model: parameters
• survival parameters: φa, φjuv.max, β1 ,..., βc
• breeding parameter: α
• movement parameters: γdd, γdist, γsf
• observation variance parameter: ψ
• total 7 + c (c is number of regions, 4 here)
Grey seal model: prior distributions
0.93 0.95 0.97
01
02
03
04
0
phi.adult 0.966
0.6 0.7 0.8 0.9
01
23
45
phi.juv.max 0.734
0.92 0.96
01
02
03
0
alpha 0.973
0.06 0.07 0.08 0.09
01
03
05
0
psi 0.07
2 4 6 8 10 14
0.0
0.1
00
.20
0.3
0
gamma.dd 3.32
0.5 1.5 2.5
0.0
0.4
0.8
gamma.dist 0.792
0.2 0.6 1.0 1.4
0.0
1.0
2.0
gamma.sf 0.355
0.0006 0.0010 0.0014
05
00
15
00
beta.ns 0.000906
0.0008 0.0014 0.0020
05
00
10
00
15
00
beta.ih 0.00127
0.0002 0.0004
02
00
04
00
06
00
0
beta.oh 0.000304
0.00010 0.00020 0.00030
04
00
08
00
0beta.ork 0.000183
Posterior parameter estimates
Smoothed pup estimates
Year
Pup
s
1985 1990 1995 2000
1500
3500
North Sea
Year
Pup
s
1985 1990 1995 2000
1500
3000
Inner Hebrides
Year
Pup
s
1985 1990 1995 2000
8000
1200
0
Outer Hebrides
Year
Pup
s
1985 1990 1995 2000
6000
1600
0
Orkneys
Predicted adults
Year
Adu
lts
2004 2008 2012
9000
1300
0North Sea
Year
Adu
lts
2004 2008 2012
7000
1000
0
Inner Hebrides
Year
Adu
lts
2004 2008 2012
2500
040
000
Outer Hebrides
Year
Adu
lts
2004 2008 2012
4000
060
000
Orkneys
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
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.