Embedding population dynamics models in inference

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

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

8. Survival sub-process Survival

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

10. Ageing sub-process Age incrementation

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

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

29. Generalizing the framework by Replace 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. Grey seal state model:states • 7 age classes • pups (n0) • age 1 – age 5 females (n1-n5) • age 6+ females (n6+) = breeders • 48 colonies – aggregated into 4 regions

37. survival age movement breeding na,c,t-1 us,a,c,t ui,a,c,t um,a,c,t na,c,t Grey seal state model: processes • a “year” starts just after the breeding season • 4 sub-processes • survival • age incrementation • movement of recruiting females • breeding

38. Grey seal state model: survival • density-independent adult survivalus,a,c,t ~ Binomial(na,c,t-1,φadult) a=1-6 • density-dependent pup survivalus,0,c,t ~ Binomial(n0,c,t-1, φjuv,c,t)where φjuv,c,t= φjuv.max/(1+βcn0,c,t-1)

39. 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

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 • (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 / Σjθc→j,t • θc→i,t = • γsf when c=i • γddmax([φjuv,i,t-φjuv,c,t],0)/exp(γdistdc,i) when c≠i

42. Grey seal state model:breeding • density-independent • ub,0,c,t ~ Binomial(um,6+,c,t , α)

43. Grey seal state model: matrix formulation • E(nt|nt-1, Θ) ≈ BMtAStnt-1

44. Grey seal state model:matrix formulation • E(nt|nt-1, Θ) ≈ Ptnt-1

45. Grey seal observation model • pup production estimates normally distributed, with variance proportional to expectation: y0,c,t~ Normal(n0,c,t , ψ2n0,c,t)

46. 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)

47. Grey seal model: prior distributions

48. Posterior parameter estimates

49. Smoothed pup estimates

50. Predicted adults