Integrated Population Modeling a natural tool for population dynamics Integrated Population Modeling...

Integrated Population Integrated Population ModelingModeling a natural toola natural tool for for population population dynamics dynamics

Jean-Dominique LEBRETONJean-Dominique LEBRETONCEFE, CNRS, Montpellier, FranceCEFE, CNRS, Montpellier, France

[email protected]@cefe.cnrs.fr

INTRODUCTIONINTRODUCTION

1.1. Demog. & census info, the Greater snow gooseDemog. & census info, the Greater snow goose2.2. Model trajectory vs censusModel trajectory vs census

STATE SPACE MODELSSTATE SPACE MODELS

3.3. State equationState equation4.4. Observation equationObservation equation5.5. State-space modelState-space model

FITTING STATE SPACE MODELSFITTING STATE SPACE MODELS

6.6. The Kalman filter, Kalman smootherThe Kalman filter, Kalman smoother7.7. « Integrated » likelihood« Integrated » likelihood8.8. Normal approximationNormal approximation

BACK TO THE GREATER SNOW GOOSEBACK TO THE GREATER SNOW GOOSE

1.Demographic and census 1.Demographic and census informationinformation

• Changes in numbers tell us something about population mechanismsChanges in numbers tell us something about population mechanisms

• Demographic information (e.g., CR data + statistical model) Demographic information (e.g., CR data + statistical model) used to understand "what happened" used to understand "what happened"

More preciselyMore precisely

• Census or surveys can be used to estimate rate of population changeCensus or surveys can be used to estimate rate of population change

• A "dynamic model" is needed to translate demographic estimatesA "dynamic model" is needed to translate demographic estimates into estimates of rate of population change into estimates of rate of population change

• The two types of estimates of rate of change are comparedThe two types of estimates of rate of change are compared

• The model can be modified and parameter estimates “tuned” The model can be modified and parameter estimates “tuned”

The Greater Snow GooseThe Greater Snow Goose Anser caerulescens atlantica Anser caerulescens atlantica : the eastern population of snow goose: the eastern population of snow goose

Breeds eastern arctic Canada islandsBreeds eastern arctic Canada islands

Winters mostly near Cheasapeake bayWinters mostly near Cheasapeake bay

Migration stopover in Migration stopover in spring and autumn spring and autumn along Saint-Laurentalong Saint-Laurent

The GSG population growthThe GSG population growth

1970 1975 1980 1985 1990 1995 2000 2005

400 000

800 000

The GSG: a real world problemThe GSG: a real world problem

Year Spring greater snow goose’s population on St-Laurence River

Breeding success (% of juveniles in the fall flock)

1993 417 500 47.8

1994 596 000 9.2

1995 612 000 16.6

1996 669 000 25.0

1997 657 500 41.6

1998 835 000 37.5

1999 803 000 2.0

2000 814 000 22.7

2001 837 000 27.5

Year Farmers Surface Total Loss ($)

Loss/ ha ($)

1992 301 3 309 466 589 141

1993 167 1 427 211 514 148

1994 396 4 188 534 891 128

1995 407 6 508 904 043 139

1996 375 4 884 844 213 175

1997 406 4 656 537 280 115

1998 487 7 003 1 264 397 180

1999 496 4 978 978 513 196

Total 36 953 5 741 440 155

The GSG: a real world problemThe GSG: a real world problem

Year Spring greater snow goose’s population on St-Laurence River

Breeding success (% of juveniles in the fall flock)

1993 417 500 47.8

1994 596 000 9.2

1995 612 000 16.6

1996 669 000 25.0

1997 657 500 41.6

1998 835 000 37.5

1999 803 000 2.0

2000 814 000 22.7

2001 837 000 27.5

« « Why we need go on spring huntingWhy we need go on spring hunting » »

Leslie Matrix ModelLeslie Matrix Model

0 0 ¤ ¤ 0 0 ¤ ¤ ¤ 0 0 0 ¤ 0 0 0 0 ¤ 0 0 0 ¤ 0 0 0 0 ¤ ¤ 0 0 ¤ ¤

Age-dep. breeding Prop. * fecund. * 1st year survivalAge-dep. breeding Prop. * fecund. * 1st year survivalaa33*f*s*f*s1 1 f*s f*s11


Age - Age - dependent dependent survivalsurvival


PopulationPopulationProjectionsProjectionsPopulationPopulationProjectionsProjections

Leslie Matrix ModelLeslie Matrix Model

0 0 ¤ ¤ 0 0 ¤ ¤ ¤ 0 0 0 ¤ 0 0 0 0 ¤ 0 0 0 ¤ 0 0 0 0 ¤ ¤ 0 0 ¤ ¤





PopulationPopulationProjectionsProjectionsPopulationPopulationProjectionsProjections

HarvestHarvestHarvestHarvest

Capture-recapture analysis: Survival of GSG Capture-recapture analysis: Survival of GSG

with winter harvest as a covariatewith winter harvest as a covariate

1975 1980 1985 1990 1995 20000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

High harvest High harvest low survival low survival slow growth or decrease slow growth or decrease

2. Model trajectory vs census2. Model trajectory vs census survival driven by harvest (+ further time-dependent survival driven by harvest (+ further time-dependent

parameters)parameters)

Year 73 74 … i i+1 … 2002Year 73 74 … i i+1 … 2002

Harvest xHarvest x7373 x x7474 … x … xii x xi+1i+1 ... x ... x0202

Survival Survival 7373 7474 ... ... ii i+1i+1 ... ... 0202

Matrix MMatrix M7373 M M7474 ... M ... Mii M Mi+1i+1 ... M ... M0202

Numbers obtained by a « time-varying matrix model » Numbers obtained by a « time-varying matrix model » (using N(using N7373 based on average stable age structure): based on average stable age structure):

NNi+1i+1=M(=M(ii)*N)*Nii

1975 1980 1985 1990 1995 2000 0

400 000

800 000



1975 1980 1985 1990 1995 2000 0

400 000

800 000

« tuning »:« tuning »:S’=1.07*SS’=1.07*S



What do we need ?What do we need ?• Better integration of demographic information Better integration of demographic information

and survey information (rather than "ad hoc" and survey information (rather than "ad hoc" comparison)comparison)

• Structured models (Matrix models in practice) Structured models (Matrix models in practice) should play a central roleshould play a central role

• Usually census less detailed than vector used in Usually census less detailed than vector used in model model

• Strong need for adequate methodology in Strong need for adequate methodology in relation with "integrated monitoring" (census + relation with "integrated monitoring" (census + marked individuals)marked individuals)

• Methodology must be probabilistic for Methodology must be probabilistic for estimation, model selection, etc…estimation, model selection, etc…






The Kalman filter, Kalman smootherThe Kalman filter, Kalman smoother• « Integrated » likelihood« Integrated » likelihood• Normal approximationNormal approximation


3. State equation3. State equationA linear (Markov) model for a A linear (Markov) model for a state vectorstate vector = Matrix model, Greater snow goose= Matrix model, Greater snow goose

tttt

ttAA

A

A

t

t NG

N

N

N

N

SS

S

S

fSfSfS

N

N

N

N

N

)(

00

000

000

0

4

3

2

1413121

14

3

2

1

1

General form General form (Harvey 1989, p. 101)(Harvey 1989, p. 101)

First order Markov ProcessFirst order Markov Process

tt = T = Ttt t-1t-1 + c + ctt + R + Rtttt

TTtt is a m x m matrix is a m x m matrix

cctt is a m x 1 matrix is a m x 1 matrix

RRtt is a m x g matrix (optional) is a m x g matrix (optional)

tt is a g x 1 matrix of random deviations, is a g x 1 matrix of random deviations,

E(E(tt)=0 var()=0 var(tt)=Q)=Qtt cov( cov(t-1t-1, , tt)=0)=0

State equation: change over timeState equation: change over time

tt = T = Ttt t-1t-1 + c + ctt + R + Rt t tt

• TTtt, c, ctt and R and Rtt are "system matrices" are "system matrices"

• They may change with time in a predetermined way onlyThey may change with time in a predetermined way only

• Only quantities in greek symbols are random variables Only quantities in greek symbols are random variables

State equationState equation

In population applications:In population applications:

tt = T = Ttt t-1t-1 + c + ctt + R + Rt t tt

often reduces tooften reduces to

tt = T = Ttt(() ) t-1t-1 + + tt

i.e., ci.e., ctt =0 and R =0 and Rtt = Id = Id

TTtt(() will be in a first step considered as known,) will be in a first step considered as known,

i.e. the parameters are supposed to be known,i.e. the parameters are supposed to be known,

i.e. one works i.e. one works conditional on the parameter valuesconditional on the parameter values

Initial valueInitial value

The system starts with values provided forThe system starts with values provided for

E(E(00) = a) = a00

var(var(00)=P)=P00

Cf the calculation of NCf the calculation of N7373 for the GSG for the GSG

Usually easy and not criticalUsually easy and not critical

tt : what stochasticity? : what stochasticity?

For survival : binomialFor survival : binomialFor fecundity: PoissonFor fecundity: Poisson

i.e. Demographic stochasticityi.e. Demographic stochasticity

All can be easily approximated by All can be easily approximated by Normal distributionsNormal distributions,,in particular for large population sizes in particular for large population sizes

4. Observation equation4. Observation equation

y(t) = Ni(t) + t = (1 1 1 1) N(t)+ t

In the case of the snow goose, one observes the total In the case of the snow goose, one observes the total spring (pre-breeding population)spring (pre-breeding population)

More generally, YMore generally, Ytt can be a vector (examples later) can be a vector (examples later)

General formGeneral form

YYtt = Z = Ztt tt + d + dtt + + tt

ZZtt is an N x m matrix (i.e. Y is an N x m matrix (i.e. Ytt can be a vector) can be a vector)

ddtt is an N x 1 matrix is an N x 1 matrix

tt is a N X 1 random matrix, E( is a N X 1 random matrix, E(tt)=0, var ()=0, var (tt)=H)=Htt

As previously, the system matrices ZAs previously, the system matrices Z tt and d and dtt can only can only

change over time in a predetermined waychange over time in a predetermined way

Often ZOften Ztt is constant, and d is constant, and dtt=0, i.e. the observation =0, i.e. the observation

equation reduces to:equation reduces to:

YYtt = Z = Z tt + + tt

5. State-space model5. State-space model

A state-space model is the combination of:A state-space model is the combination of:

• A state equation (SE) A state equation (SE) tt = T = Ttt(() ) t-1t-1 + c + ctt + R + Rtttt

• An observation equation (OE) An observation equation (OE) YYtt = Z = Ztt tt + d + dtt + + tt

• Only Only YYTT= (Y= (Y11, Y, Y2 2 , ...,Y, ...,YTT) is observed) is observed

• Under the assumptions above, everything is linearUnder the assumptions above, everything is linear• Normal distributions for Normal distributions for tt, , tt, , 00

Normal distributions for Normal distributions for tt, Y, Ytt, t = 1,...,T (by linearity), t = 1,...,T (by linearity)

A classical exampleA classical exampleSatellite trackingSatellite tracking

State equation : dynamical model for the coordinates State equation : dynamical model for the coordinates (x y z) of a satellite, based on Kepler's laws etc...(x y z) of a satellite, based on Kepler's laws etc...

Observation equation: distances to ground stations Observation equation: distances to ground stations converted into coordinates measured with errorconverted into coordinates measured with error

Purpose: estimate (x y z) at any time based on all the Purpose: estimate (x y z) at any time based on all the information available (present and past + constraints information available (present and past + constraints induced by the movement model (SE))induced by the movement model (SE))

tttt

ttAA

A

A

t

t NG

N

N

N

N

SS

S

S

fSfSfS

N

N

N

N

N

)(

00

000

000

0

4

3

2

1413121

14

3

2

1

1

y(t) = Ni(t) + t = (1 1 1 1) N(t)+ t

Observation equation for the pre-breeding censusObservation equation for the pre-breeding census of the total population = of the total population =

= state equation = state equation (Leslie matrix) (Leslie matrix)

Greater Snow GooseGreater Snow Goose

How to combine the information brought by the surveyHow to combine the information brought by the surveyand that brought by capture-recapture analyses on the and that brought by capture-recapture analyses on the parameters in the Leslie matrix ?parameters in the Leslie matrix ?

For a state space modelFor a state space modeltt = T = Ttt(() ) t-1t-1 + c + ctt + R + Rt t tt

YYtt = Z = Ztt tt + d + dtt + + tt

One may wish to estimate:One may wish to estimate:

• the state vector the state vector tt, based on , based on YYtt (filtering) (filtering)

• the states vectors the states vectors 11,..., ,..., TT, based on , based on YYTT (smoothing) (smoothing)

• the parameters the parameters (with little hope of estimating them all (with little hope of estimating them all

because of the usual loss of dimension from because of the usual loss of dimension from tt to Y to Ytt))

7. The Kalman filter7. The Kalman filter

for the state-space modelfor the state-space model

tt = T = Ttt(() ) t-1t-1 + c + ctt + R + Rt t tt and Y and Ytt = Z = Ztt tt + d + dtt + + tt

with only with only YYTT= (Y= (Y11, Y, Y2 2 , ...,Y, ...,YTT) observed) observed

• The prob. density of The prob. density of YYTT can be expressed as: can be expressed as:

f(f(YYTT)=)={{ t=2,...Tt=2,...T f(Y f(Ytt | | YYt-1t-1, , 00, , ) ) }} f(Y f(Y11 | | 00, , )g()g(00))

• By linearity, normal distributions for By linearity, normal distributions for tt and and tt lead to lead to

normal distributions for normal distributions for tt and Y and Yt t

• We just need expectations and covariance matricesWe just need expectations and covariance matrices

to be able to set up a likelihood to be able to set up a likelihood• The Kalman filter is a set of recurrence relationships The Kalman filter is a set of recurrence relationships

for expectations and covariance matrices for expectations and covariance matrices

• By linearity, normal distributions for By linearity, normal distributions for tt and and tt lead to lead to

normal distributions for normal distributions for tt and Y and Ytt

• ... thanks to the lemma:... thanks to the lemma:

X | X | mm N( N(mm,,XX) and ) and mm N( N(,V) ,V) X X N( N(, , XX + V) + V)

• Non-normality implies tedious, often impractical, integrationsNon-normality implies tedious, often impractical, integrations

• Monte-Carlo Markov Chain (MCMC) Bayesian algorithmsMonte-Carlo Markov Chain (MCMC) Bayesian algorithms can be viewed as stochastic integration algorithms can be viewed as stochastic integration algorithms

Linearity and normal distributionsLinearity and normal distributions

FilteringFiltering

• Initial conditions:Initial conditions: 00 N(a N(a00, P, P00))

• Apply the SE, conditional on information at time 0:Apply the SE, conditional on information at time 0:

11 N(a N(a 1|01|0 + c + c11, P, P1|01|0) )

with a with a 1|01|0 = T = T11aa00+c+c11 and P and P1|01|0 = T = T11PP00TT11' + R' + R11QQ11R'R'11

• Rearrange the SE and OE:Rearrange the SE and OE: 1 1 = a= a1|01|0 + ( + (1-a1-a1|01|0))

YY11=Z=Z11aa1|01|0+d+d11+Z+Z11((11-a-a1|01|0)+)+11

for the state-space modelfor the state-space model

tt = T = Ttt(() ) t-1t-1 + c + ctt + R + Rt t t t and Yand Ytt = Z = Ztt tt + d + dtt + + tt

with covariance matrices var( with covariance matrices var(tt)=Q)=Qtt and var( and var(tt)=H)=Htt

Using var(AX)=A var(X) A' and other classical properties Using var(AX)=A var(X) A' and other classical properties

FilteringFiltering

From From 11 N(a N(a 1|01|0 + c + c11, P, P1|01|0) )

1 1 = a= a1|01|0 + ( + (11-a-a1|01|0))

YY11=Z=Z11aa1|01|0+d+d11+Z+Z11((11-a-a1|01|0)+)+11

11 aa1|01|0 P P1|01|0 P P1|0 1|0 ZZ11''

N N ,, YY11 ZZ11aa1|01|0+d+d1 1 ZZ11PP1|01|0 ZZ11PP1|0 1|0 ZZ11'+H'+H11

Get distribution of Get distribution of 11 conditional on Y conditional on Y11 by multiple regression by multiple regression

X | Y = N( X | Y = N( XX--XYXYYYYY-1-1(y-(y-YY),),

XXXX--XYXYYYYY-1-1YXYX))

E(X | Y) corresponds to DE(X | Y) corresponds to DX|YX|Y

XX XX XXXX XYXY

YY YY YXYX YYYY

N ,N ,

• Lemma: multiple regressionLemma: multiple regression

XX XX XXXX XYXY

N N ,, YY Y Y YXYX YYYY

• The distribution of X conditional on Y is:The distribution of X conditional on Y is:

X | YX | Y N N ((X X + + XY XY YYYY-1 -1 (Y-(Y-YY) , ) , XX XX - - XY XY YYYY

-1 -1 YXYX))

FilteringFiltering

From From 11 aa1|01|0 P P1|01|0 P P1|0 1|0 ZZ11''

N N ,, YY11 ZZ11aa1|01|0+d+d1 1 ZZ11PP1|01|0 ZZ11PP1|0 1|0 ZZ11'+H'+H11

11 | Y | Y11 NN ( a ( a11, P, P11) , with) , with

aa11 = a = a1|01|0 + P + P1|01|0ZZ11' F' F11-1 -1 (Y(Y11-Z-Z11aa1|01|0-d-d11))

PP11 = P = P1|01|0 - P - P1|01|0ZZ11' F' F11-1 -1 ZZ11'P'P1|01|0

where Fwhere F11 = Z = Z11PP1|01|0 Z Z11' + H' + H11

.... go on over time with the same recursion to get.... go on over time with the same recursion to get tt | | YYtt

FilteringFiltering

• E(E(tt | | YYtt) is also the Minimum MSE Estimate of ) is also the Minimum MSE Estimate of tt

• In passing the distribution of YIn passing the distribution of Y t|t-1t|t-1 is obtained is obtained

(as was that of Y (as was that of Y11 conditional on the information at time 0) conditional on the information at time 0)

• Hence f(YHence f(Ytt | | YYt-1t-1, a, a00, , ) and the likelihood can be produced ) and the likelihood can be produced

• E(E(tt | | YYt t ) is a prediction based on the past, well adapted to ) is a prediction based on the past, well adapted to

dynamical systems in real time (prediction of the coordinates dynamical systems in real time (prediction of the coordinates

of a satellite based on the info available at time t)of a satellite based on the info available at time t)

• E(E(tt | | YYTT ), based on all info before and after t, will be more ), based on all info before and after t, will be more

relevant in general to population dynamics modelsrelevant in general to population dynamics models

• E(E(tt | | YYTT ), based on all info before and after t, will be more ), based on all info before and after t, will be more

relevant in general to population dynamics modelsrelevant in general to population dynamics models

• It will be also more precise than E(It will be also more precise than E(tt | | YYtt ), that may be ), that may be

useful for predictions one-step aheaduseful for predictions one-step ahead

• Estimation of Estimation of tt by E( by E(tt | | YYTT ) is called "smoothing" ) is called "smoothing"

• Estimation of a single Estimation of a single tt is based "fixed-point smoothing" is based "fixed-point smoothing"

• May be useful to estimate missing valuesMay be useful to estimate missing values

• Estimation of all Estimation of all tt = "interval smoothing" = "interval smoothing"

• Based on backwards recursions starting from TBased on backwards recursions starting from T

SmoothingSmoothing

• Start from aStart from aT|TT|T = a = aTT and P and PT|TT|T = P = PTT obtained from the KF obtained from the KF

• iterate iterate

aa t|T t|T = a = att+P+Ptt* (a* (at+1|Tt+1|T-T-Tt+1t+1aatt))

PPt|Tt|T = P = Ptt + P + Ptt*(P*(Pt+1|Tt+1|T-P-Pt+1|tt+1|t)P)Ptt*' *'

PPtt* = P* = PttTTt+1t+1' P' Pt+1|tt+1|t-1-1

• Proof involvedProof involved• Produces Minimum MSE estimators, Produces Minimum MSE estimators, in particular of state vector in particular of state vector

SmoothingSmoothing

7. « Integrated » 7. « Integrated » LikelihoodLikelihood

• Traditional use of Log LTraditional use of Log LCC(Y,(Y,) conditional on ) conditional on : :- Estimate state vectors Estimate state vectors tt

• Non traditional use (Morgan et al.): based on Non traditional use (Morgan et al.): based on independence, combine with capture-recapture independence, combine with capture-recapture Log-Likelihood Log L Log-Likelihood Log LRR(CR-Data, (CR-Data, ) as:) as:

Log LLog LRR(CR-Data, (CR-Data, ) + Log L) + Log LCC(Y,(Y,))

- Estimate Estimate using all info (CR + Census) using all info (CR + Census)- Estimate variance of censusEstimate variance of census- Estimate state vectors Estimate state vectors tt

- Estimate pop size using all info (CR + Census)Estimate pop size using all info (CR + Census)

8. Normal approximation8. Normal approximation

• Integration as Log LIntegration as Log LRR(CR-Data, (CR-Data, ) + Log L) + Log LCC(Y,(Y,) is ) is

difficult in practice, because the first term is involveddifficult in practice, because the first term is involved

• Specific Matlab code (Besbeas et al.)Specific Matlab code (Besbeas et al.)

• Specific program (integrated M-SURGE ?)Specific program (integrated M-SURGE ?)

• Simplify calculations of Log LR(CR-Data, Simplify calculations of Log LR(CR-Data, ): ): approximation based on MLEs asympt.distribution approximation based on MLEs asympt.distribution

NN ( (, , ), replacing ), replacing by estimate S by estimate S • Log LR(CR-Data, Log LR(CR-Data, ) ) -( -(n log(2n log(2)+log(det S) )/2-()+log(det S) )/2-( - - )'S)'S-1-1(( - - )/2)/2 (approx. of deviance by paraboloid tangent at MLE) (approx. of deviance by paraboloid tangent at MLE)

Kalman FilterKalman FilterGreater snow gooseGreater snow goose

Observed surveyObserved survey(continuous line, with 95 % CI)(continuous line, with 95 % CI)

Estimated pop sizeEstimated pop size((dotted line)dotted line)


Numbers predicted in Numbers predicted in age classes 1, 2, 3, age classes 1, 2, 3, 4+ ...4+ ...

1985 1990 1995 2000 2005 0

100 000

300 000

year

N1

1985 1990 1995 2000 2005 0

100 000

200 000

year

N2

1985 1990 1995 2000 2005 0

100 000

200 000

year

N3

1985 1990 1995 2000 2005100 000

500 000

year

N4


...used to cross-...used to cross-validate the results validate the results by comparing the by comparing the modelled and modelled and observed age observed age structure in autumnstructure in autumn


0.7 0.75 0.8 0.85 0.9 0.95 10.7

0.75

0.8

0.85

0.9

0.95

1

Y=X

Y=1.05 X

Adult survival: CR estimateAdult survival: CR estimate

Adult survival: Adult survival: Integrated modelingIntegrated modelingEstimateEstimate

ConclusionsConclusionsGreater snow gooseGreater snow goose

• A genuine exponential growthA genuine exponential growth• Slightly modified by variation inSlightly modified by variation in reproductive output reproductive output• Despite no compensation ofDespite no compensation of hunting mortality hunting mortality• Being used to forecast effectsBeing used to forecast effects of spring hunting of spring hunting

A tributeA tribute

Patrick « George » LESLIE,Patrick « George » LESLIE,whose famous 1945 paperwhose famous 1945 paperlaunched the development launched the development of « matrix models »of « matrix models »

Hal CASWELL,Hal CASWELL,whose 2001 bookwhose 2001 book« matrix population models »« matrix population models »is an up-to-date review of is an up-to-date review of the subject and its recent the subject and its recent developmentsdevelopments

Integrating census and Integrating census and demographic demographic informationinformation

Byron Morgan, who, among manyByron Morgan, who, among manydifferent contributions to Statistical different contributions to Statistical Ecology, is developing the Ecology, is developing the Kalman Filter methodology in Kalman Filter methodology in Vertebrate population dynamicsVertebrate population dynamics

Gilles Gauthier, who is developingGilles Gauthier, who is developingthe Greater Snow Goose program the Greater Snow Goose program at Université Laval, Québecat Université Laval, Québec

Further ideas / Further ideas / discussiondiscussion

• Integrate more likelihoods (fecundity etc...)Integrate more likelihoods (fecundity etc...)• Non linear algorithms (Density-dependence !)Non linear algorithms (Density-dependence !)• Bayesian approachesBayesian approaches

State Equation (A single age class) State Equation (A single age class)

NN f+S f+S 0 0 0 N0 N

NNMM = M 0 0 N = M 0 0 NMM + + tt

NNHH t+1 H 0 t+1 H 0 0 N 0 NHH t t

Observation Equation : census + harvest Observation Equation : census + harvest

1 0 0 N1 0 0 N

Y Y tt = 0 0 1 N = 0 0 1 NMM + + tt

NNHH

State-space model for State-space model for harvested populationharvested population

White Stork : census of breedersWhite Stork : census of breeders

State equation accounting for incomplete recruitment State equation accounting for incomplete recruitment at age 3. State vector is (Nat age 3. State vector is (N11, N, N22, N_NB, N_NB33, N_B, N_B33, N, N44))

Matrix:Matrix: 00 0 0 00 f Sf S11 fSfS11

SS 0 0 00 0 0 0 000 (1-U(1-U33)S)S 00 0 0 0 0

00 U U33SS 00 0 0 0 0

00 0 0 SS S S S S Observation equation: Observation equation:

YYtt = (0 0 0 1 1) N = (0 0 0 1 1) Nt t + e+ ett

Dispersal (Cormorant, BH Gull...)Dispersal (Cormorant, BH Gull...)

State equation : Multistate Leslie matrix model (with State equation : Multistate Leslie matrix model (with states breeder and non-breeder) (Cf Lebreton, TPB states breeder and non-breeder) (Cf Lebreton, TPB 1996), + demographic variability1996), + demographic variability

Observation equation: colony-specific numbers of Observation equation: colony-specific numbers of breeders (censuses)breeders (censuses)

Demographic information: multisite recruitment model Demographic information: multisite recruitment model (Cf Lebreton et al., Oikos 2003, Grosbois, Delon 2004)(Cf Lebreton et al., Oikos 2003, Grosbois, Delon 2004)

Objective: obtain robust estimates of dispersal (natal & Objective: obtain robust estimates of dispersal (natal & breeding) incorporating the census informationbreeding) incorporating the census information

Integrated Population Modeling a natural tool for population dynamics Integrated Population Modeling...

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