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)

AIM A 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).

States Suppose we have m states at the start of year t. Then numbers of animals by state are: NB: These numbers are unknown!

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

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

Survival sub-process Survival

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

Ageing sub-process Age incrementation

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

Birth sub-process Births

The BAS model where

The BAS model

Leslie matrix The product BAS is a Leslie projection matrix:

Other processes Growth:

The BGS model with m=2

Lefkovitch matrix The product BGS is a Lefkovitch projection matrix:

Sex assignment New-born Adult female Adult male

Genotype assignment

Movement e.g. two age groups in each of two locations

Movement: BAVS model

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:

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

Bayesian inference Joint prior for and the : Likelihood: Posterior:

Types of inference Filtering: Smoothing: One step ahead prediction:

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

Generalizing the framework Replace by where and is a possibly random operator

Example: British grey seals

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

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

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 (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

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

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 )

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

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 (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

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

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

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

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 )

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)

Grey seal model: prior distributions

Posterior parameter estimates

Smoothed pup estimates

Predicted adults

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 State-space models for the dynamics of wild animal populations. Ecological Modelling 171, Thomas, L., Buckland, S.T., Newman, K.B. and Harwood, J A unified framework for modelling wildlife population dynamics. Australian and New Zealand Journal of Statistics 47, Newman, K.B., Buckland, S.T., Lindley, S.T., Thomas, L. and Fernández, C Hidden process models for animal population dynamics. Ecological Applications 16, Buckland, S.T., Newman, K.B., Fernández, C., Thomas, L. and Harwood, J. Embedding population dynamics models in inference. Submitted to Statistical Science.