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 n t to n t+1 can be split into ordered sub-processes. e.g. survival ageing births: This makes model definition much simpler

7. Survival sub-process Given n t : NB a model (involving hyperparameters) can be specified for orcan be modelled as a random effect

8. Survival sub-process Survival

9. Ageing sub-process Given u s,t : NB process is deterministic No first-year animals left!

10. Ageing sub-process Age incrementation

11. Birth sub-process Given u a,t : NB a model may be specified for New first-year animals

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 m=2

18. Lefkovitch matrix The product BGS is a Lefkovitch projection matrix:

19. Sex assignment New-born Adult female Adult male

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 Prior for parameters pdf (prior) for initial state pdf for state at time t given earlier states Observation pdf Model prior

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. 7 age classes –pups (n 0 ) –age 1 – age 5 females (n 1 -n 5 ) –age 6+ females (n 6+ ) = breeders 48 colonies – aggregated into 4 regions Grey seal state model: states

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 u s,a,c,t n a,c,t-1 u i,a,c,t u m,a,c,t n a,c,t breedingmovementagesurvival

38. Grey seal state model: survival density-independent adult survival u s,a,c,t ~ Binomial(n a,c,t-1,φ adult ) a=1-6 density-dependent pup survival u s,0,c,t ~ Binomial(n 0,c,t-1, φ juv,c,t ) where φ juv,c,t = φ juv.max /(1+β c n 0,c,t-1 )

39. Grey seal state model: age incrementation and sexing u i,1,c,t ~Binomial (u s,0,c,t, 0.5) u i,a+1,c,t = u s,a,c,t a=1-4 u i,6+,c,t = u s,5,c,t + u s,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 (u m,5,c→1,t,..., u m,5,c→4,t ) ~ Multinomial(u i,5,c,t, ρ c→1,t,..., ρ c→4,t ) ρ c→i,t =θ c→i,t / Σ j θ c→j,t θ c→i,t = –γ sf when c=i –γ dd max([φ juv,i,t -φ juv,c,t ],0)/exp(γ dist d c,i ) when c≠i

42. Grey seal state model: breeding density-independent u b,0,c,t ~ Binomial(u m,6+,c,t, α)

43. Grey seal state model: matrix formulation E(n t |n t-1, Θ) ≈ B M t A S t n t-1

44. Grey seal state model: matrix formulation E(n t |n t-1, Θ) ≈ P t n t-1

45. Grey seal observation model pup production estimates normally distributed, with variance proportional to expectation: y 0,c,t ~ Normal(n 0,c,t, ψ 2 n 0,c,t )

46. Grey seal model: parameters survival parameters: φ a, φ juv.max, β 1,..., β c breeding parameter: α movement parameters: γ dd, γ dist, γ sf observation variance parameter: ψ 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. Predicted adults

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.