Statistical Science

Embedding Population Dynamics Models in Inference

Stephen T. Buckland, Ken B. Newman, Carmen Fernández, Len Thomas, and John Harwood

Full-text: Open access


Increasing pressures on the environment are generating an ever-increasing need to manage animal and plant populations sustainably, and to protect and rebuild endangered populations. Effective management requires reliable mathematical models, so that the effects of management action can be predicted, and the uncertainty in these predictions quantified. These models must be able to predict the response of populations to anthropogenic change, while handling the major sources of uncertainty. We describe a simple “building block” approach to formulating discrete-time models. We show how to estimate the parameters of such models from time series of data, and how to quantify uncertainty in those estimates and in numbers of individuals of different types in populations, using computer-intensive Bayesian methods. We also discuss advantages and pitfalls of the approach, and give an example using the British grey seal population.

Article information

Statist. Sci., Volume 22, Number 1 (2007), 44-58.

First available in Project Euclid: 1 August 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Hidden process models filtering Kalman filter matrix population models Markov chain Monte Carlo particle filter sequential importance sampling state-space models


Buckland, Stephen T.; Newman, Ken B.; Fernández, Carmen; Thomas, Len; Harwood, John. Embedding Population Dynamics Models in Inference. Statist. Sci. 22 (2007), no. 1, 44--58. doi:10.1214/088342306000000673.

Export citation


  • Berliner, L. M. (1996). Hierarchical Bayesian time series models. In Maximum Entropy and Bayesian Methods (K. Hanson and R. Silver, eds.) 15--22. Kluwer, Dordrecht.
  • Besbeas, P., Freeman, S. N. and Morgan, B. J. T. (2005). The potential of integrated population modelling. Aust. N. Z. J. Stat. 47 35--48.
  • Besbeas, P., Freeman, S. N., Morgan, B. J. T. and Catchpole, E. A. (2002). Integrating mark-recapture-recovery and census data to estimate animal abundance and demographic parameters. Biometrics 58 540--547.
  • Besbeas, P., Lebreton, J.-D. and Morgan, B. J. T. (2003). The efficient integration of abundance and demographic data. Appl. Statist. 52 95--102.
  • Buckland, S. T., Magurran, A. E., Green, R. E. and Fewster, R. M. (2005). Monitoring change in biodiversity through composite indices. Philos. Trans. R. Soc. Lond. Ser. B 360 243--254.
  • 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.
  • Burnham, K. P. and Anderson, D. R. (2002). Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, 2nd ed. Springer, New York.
  • Calder, C., Lavine, M., Müller, P. and Clark, J. S. (2003). Incorporating multiple sources of stochasticity into dynamic population models. Ecology 84 1395--1402.
  • Caswell, H. (2001). Matrix Population Models: Construction, Analysis and Interpretation, 2nd ed. Sinauer Associates, Sunderland, MA.
  • Catchpole, E. A. and Morgan, B. J. T. (1997). Detecting parameter redundancy. Biometrika 84 187--196.
  • Catchpole, E. A., Morgan, B. J. T. and Freeman, S. N. (1998). Estimation in parameter-redundant models. Biometrika 85 462--468.
  • Clark, J. S. and Bjørnstad, O. N. (2004). Population time series: Process variability, observation errors, missing values, lags and hidden states. Ecology 85 3140--3150.
  • Clark, J. S., Ferraz, G. A., Oguge, N., Hays, H. and DiCostanzo, J. (2005). Hierarchical Bayes for structured, variable populations: From recapture data to life-history prediction. Ecology 86 2232--2244.
  • Collie, J. S. and Sissenwine, M. P. (1983). Estimating population size from relative abundance data measured with error. Canadian J. Fisheries and Aquatic Sciences 40 1871--1879.
  • Cunningham, C. L., Reid, D. G., McAllister, M. K., Kirkwood, G. P. and Darby, C. D. (2007). Modelling the migration of multiple stocks: a Bayesian state-space model for north-east Atlantic mackerel. African J. Marine Science. To appear.
  • Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood estimation from incomplete data via the EM algorithm (with discussion). J. Roy. Statist. Soc. Ser. B 39 1--38.
  • Doucet, A., de Freitas, N. and Gordon, N., eds. (2001). Sequential Monte Carlo Methods in Practice. Springer, New York.
  • Dupuis, J. A. (1995). A Bayesian estimation of movement and survival probabilities from capture--recapture data. Biometrika 82 761--772.
  • Gilks, W. R., Richardson, S. and Spiegelhalter, D. J., eds. (1996). Markov Chain Monte Carlo in Practice. Chapman and Hall, London.
  • Gilks, W. R. and Roberts, G. O. (1996). Strategies for improving MCMC. In Markov Chain Monte Carlo in Practice (W. R. Gilks, S. Richardson and D. J. Spiegelhalter, eds.) 89--114. Chapman and Hall, London.
  • Gordon, N. J., Salmond, D. J. and Smith, A. F. M. (1993). Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proceedings-F 140 107--113.
  • Green, P. J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82 711--732.
  • Gudmundsson, G. (1987). Time series models of fishing mortality rates. Report RH-02-87, Raunvisindastofnun Haskolans, Univ. Iceland.
  • Gudmundsson, G. (1994). Time series analysis of catch-at-age observations. Appl. Statist. 43 117--126.
  • Gurney, W. S. C. and Nisbet, R. M. (1998). Ecological Dynamics. Oxford Univ. Press, New York.
  • Harvey, A. (1989). Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge Univ. Press.
  • Harwood, J. and Stokes, K. (2003). Coping with uncertainty in ecological advice: Lessons from fisheries. Trends in Ecology and Evolution 18 617--622.
  • Hilborn, R., Pikitch, E. K. and McAllister, M. K. (1994). A Bayesian estimation and decision analysis for an age-structured model using biomass survey data. Fisheries Research 19 17--30.
  • Hilborn, R. and Walters, C. J. (1992). Quantitative Fisheries Stock Assessment: Choice, Dynamics and Uncertainty. Chapman and Hall, New York.
  • Hooten, M. B., Wikle, C. K., Dorazio, R. M. and Royle, J. A. (2007). Hierarchical spatio-temporal matrix models for characterizing invasions. Biometrics 63 558--567.
  • Johnson, D. S. and Hoeting, J. A. (2003). Autoregressive models for capture--recapture data: A Bayesian approach. Biometrics 59 341--350.
  • Kalman, R. E. (1960). A new approach to linear filtering and prediction problems. Transactions of ASME-J. Basic Engineering 82 35--45.
  • King, R. and Brooks, S. P. (2002a). Model selection for integrated recovery/recapture data. Biometrics 58 841--851.
  • King, R. and Brooks, S. P. (2002b). Bayesian model discrimination for multiple strata capture--recapture data. Biometrika 89 785--806.
  • Lavine, M., Beckage, B. and Clark, J. S. (2002). Statistical modeling of seedling mortality. J. Agric. Biol. Environ. Stat. 7 21--41.
  • Lebreton, J. D. (1973). Introduction aux modèles mathématiques de la dynamique des populations. Informatique et Biosphère 77--116.
  • Lebreton, J. D. and Isenmann, P. (1976). Dynamique de la population camarguaise de mouettes rieuses Larus ridibundus L.: un modèle mathématique. La Terre et la vie 30 529--549.
  • Lefkovitch, L. P. (1965). The study of population growth in organisms grouped by stages. Biometrics 21 1--18.
  • Lele, S. R. (2006). Sampling variability and estimates of density dependence: A composite likelihood approach. Ecology 87 189--202.
  • Leslie, P. H. (1945). On the use of matrices in certain population mathematics. Biometrika 33 183--212.
  • Leslie, P. H. (1948). Some further notes on the use of matrices in population mathematics. Biometrika 35 213--245.
  • Liu, J. S. (2001). Monte Carlo Strategies in Scientific Computing. Springer, New York.
  • Liu, J. S. and Chen, R. (1998). Sequential Monte Carlo methods for dynamic systems. J. Amer. Statist. Assoc. 93 1032--1044.
  • McAllister, M. K. and Ianelli, J. N. (1997). Bayesian stock assessment using catch-age data and the sampling-importance sampling algorithm. Canadian J. Fisheries and Aquatic Sciences 54 284--300.
  • McAllister, M. K., Pikitch, E. K., Punt, A. E. and Hilborn, R. (1994). A Bayesian approach to stock assessment and harvest decisions using the sampling/importance resampling algorithm. Canadian J. Fisheries and Aquatic Sciences 51 2673--2687.
  • McConnell, B. J., Fedak, M. A., Lovell, P. and Hammond, P. S. (1999). Movements and foraging areas of grey seals in the North Sea. J. Applied Ecology 36 573--590.
  • Mendelssohn, R. (1988). Some problems in estimating population sizes from catch-at-age data. Fishery Bulletin 86 617--630.
  • Meyer, R. and Millar, R. B. (1999). Bayesian stock assessment using a state-space implementation of the delay difference model. Canadian J. Fisheries and Aquatic Sciences 56 37--52.
  • Millar, R. B. and Meyer, R. (2000). Non-linear state-space modelling of fisheries biomass dynamics by using Metropolis--Hastings within-Gibbs sampling. Appl. Statist. 49 327--342.
  • Myers, R. H. (1990). Classical and Modern Regression with Applications, 2nd ed. PWS-Kent, Boston.
  • Newman, K. B. (1998). State-space modeling of animal movement and mortality with application to salmon. Biometrics 54 1290--1314.
  • Newman, K. B. (2000). Hierarchic modeling of salmon harvest and migration. J. Agric. Biol. Environ. Stat. 5 430--455.
  • 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.
  • Newman, K. B., Fernández, C., Buckland, S. T. and Thomas, L. (2007). Monte Carlo inference for state-space models of wild animal populations. To appear.
  • Pitt, M. K. and Shephard, N. (1999). Filtering via simulation: Auxiliary particle filters. J. Amer. Statist. Assoc. 94 590--599.
  • Poole, D. and Raftery, A. E. (2000). Inference for deterministic simulation models: The Bayesian melding approach. J. Amer. Statist. Assoc. 95 1244--1255.
  • Quinn, T. J. II and Deriso, R. B. (1999). Quantitative Fish Dynamics. Oxford Univ. Press, New York.
  • Raftery, A. E., Givens, G. H. and Zeh, J. E. (1995). Inference from a deterministic population dynamics model for bowhead whales (with discussion). J. Amer. Statist. Assoc. 90 402--430.
  • Rivot, E., Prévost, E., Parent, E. and Baglinière, J. L. (2004). A Bayesian state-space modelling framework for fitting a salmon stage-structured population dynamic model to multiple time series of field data. Ecological Modelling 179 463--485.
  • Rubin, D. B. (1988). Using the SIR algorithm to simulate posterior distributions. In Bayesian Statistics 3 (J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds.) 395--402. Clarendon Press, Oxford.
  • Schnute, J. T. (1994). A general framework for developing sequential fisheries models. Canadian J. Fisheries and Aquatic Sciences 51 1676--1688.
  • Sullivan, P. (1992). A Kalman filter approach to catch-at-length analysis. Biometrics 48 237--257.
  • Thomas, L., Buckland, S. T., Newman, K. B. and Harwood, J. (2005). A unified framework for modelling wildlife population dynamics. Aust. N. Z. J. Stat. 47 19--34.
  • Thompson, D., Hammond, P. S., Nicholas, K. S. and Fedak, M. A. (1991). Movements, diving and foraging behaviour of grey seals (Halichoerus grypus). J. Zoology 224 223--232.
  • Trenkel, V. M., Elston, D. A. and Buckland, S. T. (2000). Calibrating population dynamics models to count and cull data using sequential importance sampling. J. Amer. Statist. Assoc. 95 363--374.
  • Tuljapurkar, S. (1997). Stochastic matrix models. In Structured-Population Models in Marine, Terrestrial and Freshwater Systems (S. Tuljapurkar and H. Caswell, eds.) 59--87. Chapman and Hall, New York.
  • Walters, C. (2002). Adaptive Management of Renewable Resources. Blackburn, Caldwell, NJ.
  • West, M. (1993a). Approximating posterior distributions by mixtures. J. Roy. Statist. Soc. Ser. B 55 409--422.
  • West, M. (1993b). Mixture models, Monte Carlo, Bayesian updating and dynamic models. In Computing Science and Statistics: Proc. 24th Symposium on the Interface 325--333. Interface Foundation of North America, Fairfax Station, VA.
  • West, M. and Harrison, J. (1997). Bayesian Forecasting and Dynamic Models, 2nd ed. Springer, New York.
  • Wikle, C. K. (2003). Hierarchical Bayesian models for predicting the spread of ecological processes. Ecology 84 1382--1394.
  • Wikle, C. K., Berliner, L. M. and Cressie, N. (1998). Hierarchical Bayesian space--time models. Environmental and Ecological Statistics 5 117--154.
  • Wolpert, R. L. (1995). Comment on ``Inference from a deterministic population dynamics model for bowhead whales,'' by A. E. Raftery, G. H. Givens and J. E. Zeh. J. Amer. Statist. Assoc. 90 426--427.