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.