1. Integrated Population Modeling a natural tool for population dynamics Integrated Population Modeling a natural tool for population dynamics Jean-Dominique LEBRETON CEFE, CNRS, Montpellier, France jean-dominique.lebreton@cefe.cnrs.fr

3. INTRODUCTION 1.Demog. & census info, the Greater snow goose 2.Model trajectory vs census STATE SPACE MODELS 3.State equation 4.Observation equation 5.State-space model FITTING STATE SPACE MODELS 6.The Kalman filter, Kalman smoother 7.« Integrated » likelihood 8.Normal approximation BACK TO THE GREATER SNOW GOOSE

4. 1.Demographic and census information Changes in numbers tell us something about population mechanisms Changes in numbers tell us something about population mechanisms Demographic information (e.g., CR data + statistical model) used to understand "what happened" Demographic information (e.g., CR data + statistical model) used to understand "what happened" More precisely Census or surveys can be used to estimate rate of population change Census or surveys can be used to estimate rate of population change A "dynamic model" is needed to translate demographic estimates into estimates of rate of population change A "dynamic model" is needed to translate demographic estimates into estimates of rate of population change The two types of estimates of rate of change are compared The 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”

5. The Greater Snow Goose Anser caerulescens atlantica : the eastern population of snow goose Anser caerulescens atlantica : the eastern population of snow goose Breeds eastern arctic Canada islands Breeds eastern arctic Canada islands Winters mostly near Cheasapeake bay Winters mostly near Cheasapeake bay Migration stopover in spring and autumn along Saint-Laurent

6. The GSG population growth

7. The GSG: a real world problem YearSpring greater snow goose’s population on St-Laurence River Breeding success (% of juveniles in the fall flock) 1993417 50047.8 1994596 0009.2 1995612 00016.6 1996669 00025.0 1997657 50041.6 1998835 00037.5 1999803 0002.0 2000814 00022.7 2001837 00027.5 YearFarmersSurfaceTotal Loss ($) Loss/ ha ($) 19923013 309466 589141 19931671 427211 514148 19943964 188534 891128 19954076 508904 043139 19963754 884844 213175 19974064 656537 280115 19984877 0031 264 397180 19994964 978978 513196 Total 36 9535 741 440155

8. The GSG: a real world problem YearSpring greater snow goose’s population on St-Laurence River Breeding success (% of juveniles in the fall flock) 1993417 50047.8 1994596 0009.2 1995612 00016.6 1996669 00025.0 1997657 50041.6 1998835 00037.5 1999803 0002.0 2000814 00022.7 2001837 00027.5 « Why we need go on spring hunting »

9. Leslie Matrix Model 0 0 ¤ ¤ ¤ 0 0 0 0 ¤ 0 0 0 0 ¤ ¤ Age-dep. breeding Prop. * fecund. * 1st year survival a 3 *f*s 1 f*s 1 Age-dep. breeding Prop. * fecund. * 1st year survival a 3 *f*s 1 f*s 1 Age - dependent survival Age - dependent survival PopulationProjectionsPopulationProjections

10. Leslie Matrix Model 0 0 ¤ ¤ ¤ 0 0 0 0 ¤ 0 0 0 0 ¤ ¤ Age-dep. breeding Prop. * fecund. * 1st year survival a 3 *f*s 1 f*s 1 Age-dep. breeding Prop. * fecund. * 1st year survival a 3 *f*s 1 f*s 1 Age - dependent survival Age - dependent survival PopulationProjectionsPopulationProjections HarvestHarvest

11. Capture-recapture analysis: Survival of GSG with winter harvest as a covariate High harvest  low survival  slow growth or decrease

12. 2. Model trajectory vs census survival driven by harvest (+ further time-dependent parameters) Year 73 74 … i i+1 … 2002 Harvest x 73 x 74 … x i x i+1... x 02 Survival  73  74...  i  i+1...  02 Matrix M 73 M 74... M i M i+1... M 02 Numbers obtained by a « time-varying matrix model » (using N 73 based on average stable age structure): N i+1 =M(  i )*N i N i+1 =M(  i )*N i

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

14. « tuning »:S’=1.07*S

15. What do we need ? Better integration of demographic information and survey information (rather than "ad hoc" comparison) Better integration of demographic information and survey information (rather than "ad hoc" comparison) Structured models (Matrix models in practice) should play a central role Structured models (Matrix models in practice) should play a central role Usually census less detailed than vector used in model Usually census less detailed than vector used in model Strong need for adequate methodology in relation with "integrated monitoring" (census + marked individuals) Strong need for adequate methodology in relation with "integrated monitoring" (census + marked individuals) Methodology must be probabilistic for estimation, model selection, etc… Methodology must be probabilistic for estimation, model selection, etc…

16. INTRODUCTION 1.Demog. & census info, the Greater snow goose 2.Model trajectory vs census STATE SPACE MODELS 3.State equation 4.Observation equation 5.State-space model FITTING STATE SPACE MODELS The Kalman filter, Kalman smoother « Integrated » likelihood« Integrated » likelihood Normal approximationNormal approximation BACK TO THE GREATER SNOW GOOSE

17. 3. State equation A linear (Markov) model for a state vector = Matrix model, Greater snow goose

18. General form (Harvey 1989, p. 101) First order Markov Process  t = T t  t-1 + c t + R t  t T t is a m x m matrix c t is a m x 1 matrix R t is a m x g matrix (optional)  t is a g x 1 matrix of random deviations, E(  t )=0 var(  t )=Q t cov(  t-1,  t )=0 E(  t )=0 var(  t )=Q t cov(  t-1,  t )=0

19. State equation: change over time  t = T t  t-1 + c t + R t  t T t, c t and R t are "system matrices" T t, c t and R t are "system matrices" They may change with time in a predetermined way only They 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

20. State equation In population applications:  t = T t  t-1 + c t + R t  t often reduces to  t = T t (  )  t-1 +  t i.e., c t =0 and R t = Id T t (  ) will be in a first step considered as known, i.e. the parameters are supposed to be known, i.e. one works conditional on the parameter values

21. Initial value The system starts with values provided for E(  0 ) = a 0 var(  0 )=P 0 Cf the calculation of N 73 for the GSG Usually easy and not critical

22.  t : what stochasticity? For survival : binomial For fecundity: Poisson i.e. Demographic stochasticity All can be easily approximated by Normal distributions, in particular for large population sizes

23. 4. Observation equation y(t) =  N i (t) +  t = (1 1 1 1) N(t)+  t In the case of the snow goose, one observes the total spring (pre-breeding population) More generally, Y t can be a vector (examples later)

24. General form Y t = Z t  t + d t +  t Z t is an N x m matrix (i.e. Y t can be a vector) d t is an N x 1 matrix  t is a N X 1 random matrix, E(  t )=0, var (  t )=H t As previously, the system matrices Z t and d t can only change over time in a predetermined way Often Z t is constant, and d t =0, i.e. the observation equation reduces to: Y t = Z  t +  t

25. 5. State-space model A state-space model is the combination of: A state equation (SE)  t = T t (  )  t-1 + c t + R t  t A state equation (SE)  t = T t (  )  t-1 + c t + R t  t An observation equation (OE) Y t = Z t  t + d t +  t An observation equation (OE) Y t = Z t  t + d t +  t Only Y T = (Y 1, Y 2,...,Y T ) is observed Only Y T = (Y 1, Y 2,...,Y T ) is observed Under the assumptions above, everything is linear Under the assumptions above, everything is linear Normal distributions for  t,  t,  0  Normal distributions for  t, Y t, t = 1,...,T (by linearity) Normal distributions for  t,  t,  0  Normal distributions for  t, Y t, t = 1,...,T (by linearity)

26. A classical example Satellite tracking State equation : dynamical model for the coordinates (x y z) of a satellite, based on Kepler's laws etc... Observation equation: distances to ground stations converted into coordinates measured with error Purpose: estimate (x y z) at any time based on all the information available (present and past + constraints induced by the movement model (SE))

27. y(t) =  N i (t) +  t = (1 1 1 1) N(t)+  t Observation equation for the pre-breeding census of the total population = = state equation (Leslie matrix) Greater Snow Goose How to combine the information brought by the survey and that brought by capture-recapture analyses on the parameters in the Leslie matrix ?

28. INTRODUCTION 1.Demog. & census info, the Greater snow goose 2.Model trajectory vs census STATE SPACE MODELS 3.State equation 4.Observation equation 5.State-space model FITTING STATE SPACE MODELS 6.The Kalman filter, Kalman smoother 7.« Integrated » likelihood 8.Normal approximation BACK TO THE GREATER SNOW GOOSE

29. For a state space model  t = T t (  )  t-1 + c t + R t  t Y t = Z t  t + d t +  t One may wish to estimate: One may wish to estimate: the state vector  t, based on Y t (filtering) the state vector  t, based on Y t (filtering) the states vectors  1,...,  T, based on Y T (smoothing) the states vectors  1,...,  T, based on Y T (smoothing) the parameters  (with little hope of estimating them all because of the usual loss of dimension from  t to Y t ) the parameters  (with little hope of estimating them all because of the usual loss of dimension from  t to Y t ) 7. The Kalman filter

30. for the state-space model  t = T t (  )  t-1 + c t + R t  t and Y t = Z t  t + d t +  t  t = T t (  )  t-1 + c t + R t  t and Y t = Z t  t + d t +  t with only Y T = (Y 1, Y 2,...,Y T ) observed with only Y T = (Y 1, Y 2,...,Y T ) observed The prob. density of Y T can be expressed as: The prob. density of Y T can be expressed as: f( Y T )= {  t=2,...T f(Y t | Y t-1,  0,  ) } f(Y 1 |  0,  )g(  0 ) By linearity, normal distributions for  t and  t lead to normal distributions for  t and Y t By linearity, normal distributions for  t and  t lead to normal distributions for  t and Y t We just need expectations and covariance matrices to be able to set up a likelihood We just need expectations and covariance matrices to be able to set up a likelihood The Kalman filter is a set of recurrence relationships for expectations and covariance matrices The Kalman filter is a set of recurrence relationships for expectations and covariance matrices

31. By linearity, normal distributions for  t and  t lead to normal distributions for  t and Y t By linearity, normal distributions for  t and  t lead to normal distributions for  t and Y t... thanks to the lemma:... thanks to the lemma: X | m  N(m,  X ) and m  N( ,V)  X  N( ,  X + V) X | m  N(m,  X ) and m  N( ,V)  X  N( ,  X + V) Non-normality implies tedious, often impractical, integrations Non-normality implies tedious, often impractical, integrations Monte-Carlo Markov Chain (MCMC) Bayesian algorithms can be viewed as stochastic integration algorithms Monte-Carlo Markov Chain (MCMC) Bayesian algorithms can be viewed as stochastic integration algorithms Linearity and normal distributions

32. Filtering Initial conditions:  0  N(a 0, P 0 ) Initial conditions:  0  N(a 0, P 0 ) Apply the SE, conditional on information at time 0:  1  N(a 1|0 + c 1, P 1|0 ) with a 1|0 = T 1 a 0 +c 1 and P 1|0 = T 1 P 0 T 1 ' + R 1 Q 1 R' 1 Apply the SE, conditional on information at time 0:  1  N(a 1|0 + c 1, P 1|0 ) with a 1|0 = T 1 a 0 +c 1 and P 1|0 = T 1 P 0 T 1 ' + R 1 Q 1 R' 1 Rearrange the SE and OE:  1 = a 1|0 + (  1-a 1|0 ) Rearrange the SE and OE:  1 = a 1|0 + (  1-a 1|0 ) Y 1 =Z 1 a 1|0 +d 1 +Z 1 (  1 -a 1|0 )+  1 Y 1 =Z 1 a 1|0 +d 1 +Z 1 (  1 -a 1|0 )+  1 for the state-space model  t = T t (  )  t-1 + c t + R t  t and Y t = Z t  t + d t +  t with covariance matrices var(  t )=Q t and var(  t )=H t  t = T t (  )  t-1 + c t + R t  t and Y t = Z t  t + d t +  t with covariance matrices var(  t )=Q t and var(  t )=H t Using var(AX)=A var(X) A' and other classical properties

33. Filtering From  1  N(a 1|0 + c 1, P 1|0 )  1 = a 1|0 + (  1 -a 1|0 ) From  1  N(a 1|0 + c 1, P 1|0 )  1 = a 1|0 + (  1 -a 1|0 ) Y 1 =Z 1 a 1|0 +d 1 +Z 1 (  1 -a 1|0 )+  1 Y 1 =Z 1 a 1|0 +d 1 +Z 1 (  1 -a 1|0 )+  1  1 a 1|0 P 1|0 P 1|0 Z 1 '  1 a 1|0 P 1|0 P 1|0 Z 1 '  N,  N, Y 1 Z 1 a 1|0 +d 1 Z 1 P 1|0 Z 1 P 1|0 Z 1 '+H 1 Y 1 Z 1 a 1|0 +d 1 Z 1 P 1|0 Z 1 P 1|0 Z 1 '+H 1 Get distribution of  1 conditional on Y 1 by multiple regression Get distribution of  1 conditional on Y 1 by multiple regression

34. X | Y = N(  X -  XY  YY -1 (y-  Y ),  XX -  XY  YY -1  YX ) E(X | Y) corresponds to D X|Y X  X  XX  XY Y  Y  YX  YY  N,

35. Lemma: multiple regression Lemma: multiple regression X  X  XX  XY X  X  XX  XY  N,  N, Y  Y  YX  YY Y  Y  YX  YY The distribution of X conditional on Y is: The distribution of X conditional on Y is: X | Y  N (  X +  XY  YY -1 (Y-  Y ),  XX -  XY  YY -1  YX ) X | Y  N (  X +  XY  YY -1 (Y-  Y ),  XX -  XY  YY -1  YX )

36. Filtering From From  1 a 1|0 P 1|0 P 1|0 Z 1 '  1 a 1|0 P 1|0 P 1|0 Z 1 '  N,  N, Y 1 Z 1 a 1|0 +d 1 Z 1 P 1|0 Z 1 P 1|0 Z 1 '+H 1 Y 1 Z 1 a 1|0 +d 1 Z 1 P 1|0 Z 1 P 1|0 Z 1 '+H 1  1 | Y 1  N ( a 1, P 1 ), with a 1 = a 1|0 + P 1|0 Z 1 ' F 1 -1 (Y 1 -Z 1 a 1|0 -d 1 ) a 1 = a 1|0 + P 1|0 Z 1 ' F 1 -1 (Y 1 -Z 1 a 1|0 -d 1 ) P 1 = P 1|0 - P 1|0 Z 1 ' F 1 -1 Z 1 'P 1|0 P 1 = P 1|0 - P 1|0 Z 1 ' F 1 -1 Z 1 'P 1|0 where F 1 = Z 1 P 1|0 Z 1 ' + H 1.... go on over time with the same recursion to get  t | Y t

37. Filtering E(  t | Y t ) is also the Minimum MSE Estimate of  t E(  t | Y t ) is also the Minimum MSE Estimate of  t In passing the distribution of Y t|t-1 is obtained (as was that of Y 1 conditional on the information at time 0) In passing the distribution of Y t|t-1 is obtained (as was that of Y 1 conditional on the information at time 0) Hence f(Y t | Y t-1, a 0,  ) and the likelihood can be produced Hence f(Y t | Y t-1, a 0,  ) and the likelihood can be produced E(  t | Y t ) is a prediction based on the past, well adapted to dynamical systems in real time (prediction of the coordinates of a satellite based on the info available at time t) E(  t | Y t ) is a prediction based on the past, well adapted to dynamical systems in real time (prediction of the coordinates of a satellite based on the info available at time t) E(  t | Y T ), based on all info before and after t, will be more relevant in general to population dynamics models E(  t | Y T ), based on all info before and after t, will be more relevant in general to population dynamics models

38. It will be also more precise than E(  t | Y t ), that may be useful for predictions one-step ahead It will be also more precise than E(  t | Y t ), that may be useful for predictions one-step ahead Estimation of  t by E(  t | Y T ) is called "smoothing" Estimation of  t by E(  t | Y T ) is called "smoothing" Estimation of a single  t is based "fixed-point smoothing" Estimation of a single  t is based "fixed-point smoothing" May be useful to estimate missing values May be useful to estimate missing values Estimation of all  t = "interval smoothing" Estimation of all  t = "interval smoothing" Based on backwards recursions starting from T Based on backwards recursions starting from T Smoothing

39. Start from a T|T = a T and P T|T = P T obtained from the KF Start from a T|T = a T and P T|T = P T obtained from the KF iterate iterate a t|T = a t +P t * (a t+1|T -T t+1 a t ) P t|T = P t + P t *(P t+1|T -P t+1|t )P t *' P t * = P t T t+1 ' P t+1|t -1 Proof involved Proof involved Produces Minimum MSE estimators, in particular of state vector Produces Minimum MSE estimators, in particular of state vector Smoothing

40. 7. « Integrated » Likelihood Traditional use of Log L C (Y,  ) conditional on  : Traditional use of Log L C (Y,  ) conditional on  : - Estimate state vectors  t Non traditional use (Morgan et al.): based on independence, combine with capture-recapture Log-Likelihood Log L R (CR-Data,  ) as: Non traditional use (Morgan et al.): based on independence, combine with capture-recapture Log-Likelihood Log L R (CR-Data,  ) as: Log L R (CR-Data,  ) + Log L C (Y,  ) Log L R (CR-Data,  ) + Log L C (Y,  ) - Estimate  using all info (CR + Census) - Estimate variance of census - Estimate state vectors  t - Estimate pop size using all info (CR + Census)

41. 8. Normal approximation Integration as Log L R (CR-Data,  ) + Log L C (Y,  ) is difficult in practice, because the first term is involved Integration as Log L R (CR-Data,  ) + Log L C (Y,  ) is difficult 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,  ): approximation based on MLEs asympt.distribution Simplify calculations of Log LR(CR-Data,  ): approximation based on MLEs asympt.distribution   N ( ,  ), replacing  by estimate S   N ( ,  ), replacing  by estimate S Log LR(CR-Data,  )  -(n log(2  )+log(det S) )/2-(  -  )'S -1 (  -  )/2 (approx. of deviance by paraboloid tangent at MLE) Log LR(CR-Data,  )  -(n log(2  )+log(det S) )/2-(  -  )'S -1 (  -  )/2 (approx. of deviance by paraboloid tangent at MLE)

42. INTRODUCTION 1.Demog. & census info, the Greater snow goose 2.Model trajectory vs census STATE SPACE MODELS 3.State equation 4.Observation equation 5.State-space model FITTING STATE SPACE MODELS 6.The Kalman filter, Kalman smoother 7.« Integrated » likelihood 8.Normal approximation BACK TO THE GREATER SNOW GOOSE

43. Kalman Filter Greater snow goose Observed survey (continuous line, with 95 % CI) Estimated pop size ( dotted line)

44. Kalman Filter Greater snow goose Numbers predicted in age classes 1, 2, 3, 4+...

45. Kalman Filter Greater snow goose...used to cross- validate the results by comparing the modelled and observed age structure in autumn

46. Kalman Filter Greater snow goose Adult survival: CR estimate Adult survival: Integrated modeling Estimate

47. Conclusions Greater snow goose A genuine exponential growth A genuine exponential growth Slightly modified by variation in reproductive output Slightly modified by variation in reproductive output Despite no compensation of hunting mortality Despite no compensation of hunting mortality Being used to forecast effects of spring hunting Being used to forecast effects of spring hunting

48. A tribute Patrick « George » LESLIE, whose famous 1945 paper launched the development of « matrix models » Hal CASWELL, whose 2001 book « matrix population models » is an up-to-date review of the subject and its recent developments

49. Integrating census and demographic information Byron Morgan, who, among many different contributions to Statistical Ecology, is developing the Kalman Filter methodology in Vertebrate population dynamics Gilles Gauthier, who is developing the Greater Snow Goose program at Université Laval, Québec

50. Further ideas / discussion Integrate more likelihoods (fecundity etc...) Integrate more likelihoods (fecundity etc...) Non linear algorithms (Density-dependence !) Non linear algorithms (Density-dependence !) Bayesian approaches Bayesian approaches

51. State Equation (A single age class) N f+S 0 0 N N M = M 0 0 N M +  t N H t+1 H 0 0 N H t Observation Equation : census + harvest 1 0 0 N 1 0 0 N Y t = 0 0 1 N M +  t N H N H State-space model for harvested population

52. White Stork : census of breeders State equation accounting for incomplete recruitment at age 3. State vector is (N 1, N 2, N_NB 3, N_B 3, N 4 ) Matrix: Matrix: 0 00f S 1 fS 1 0 00f S 1 fS 1 S 00 0 0 S 00 0 0 0(1-U 3 )S0 0 0 0 U 3 S0 0 0 0 0S S S 0 0S S S Observation equation: Y t = (0 0 0 1 1) N t + e t Y t = (0 0 0 1 1) N t + e t

53. Dispersal (Cormorant, BH Gull...) State equation : Multistate Leslie matrix model (with states breeder and non-breeder) (Cf Lebreton, TPB 1996), + demographic variability Observation equation: colony-specific numbers of breeders (censuses) Demographic information: multisite recruitment model (Cf Lebreton et al., Oikos 2003, Grosbois, Delon 2004) Objective: obtain robust estimates of dispersal (natal & breeding) incorporating the census information