A Threshold Model for Heron Productivity

  • Panagiotis Besbeas
  • Byron J. T. Morgan


We demonstrate the potential of conditionally Gaussian state-space models in integrated population modeling, when certain model parameters may be functions of previous observations. The approach is applied to a heron census, and the data are best described by a model with three population-size thresholds which determine the population productivity. The model provides an explanation of how the population rebounds rapidly after major falls in size, which are characteristic of the data. By contrast, a simple logarithmic regression of productivity on population size was not significant. The results are of ecological interest, and suggest hypotheses for further investigation. Supplementary figures are available online.

Key Words

Conditionally Gaussian Density dependence Integrated population modeling Kalman filter Non-linear models P-splines State-space model 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Supplementary material

13253_2011_80_MOESM1_ESM.pdf (47 kb)
(PDF 46.6 kB)


  1. Abadi, F., Gimenez, O., Arlettaz, R., and Schaub, M. (2012), “Estimating the Strength of Density-Dependence in the Presence of Observation Errors Using Integrated Population Models”, submitted for publication. Google Scholar
  2. Besbeas, P., and Morgan, B. J. T. (2012), “Kalman Filter Initialization for Integrated Population Modelling”, to appear in Applied Statistics. Google Scholar
  3. 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. MathSciNetzbMATHCrossRefGoogle Scholar
  4. Besbeas, P., Lebreton, J.-D., and Morgan, B. J. T. (2003), “The Efficient Integration of Abundance and Demographic Data,” Applied Statistics, 52, 95–102. MathSciNetzbMATHCrossRefGoogle Scholar
  5. Besbeas, P., Freeman, S. N., and Morgan, B. J. T. (2005), “The Potential of Integrated Population Modelling,” Australian & New Zealand Journal of Statistics, 47, 35–48. MathSciNetzbMATHCrossRefGoogle Scholar
  6. Besbeas, P., Borysiewicz, R., and Morgan, B. J. T. (2009), “Completing the Ecological Jigsaw,” in Modelling Demographic Processes in Marked Populations. Environmental and Ecological Statistics Series, Vol. 3, eds. D. L. Thomson, E. G. Cooch, and M. J. Conroy, pp. 515–542. Google Scholar
  7. Breslow, N. (1974), “Covariance Analysis of Censored Survival Data,” Biometrics, 30, 89–99. CrossRefGoogle Scholar
  8. Brooks, S. P., King, R., and Morgan, B. J. T. (2004), “A Bayesian Approach to Combining Animal Abundance and Demographic Data,” Animal Biodiversity and Conservation, 27, 515–529. Google Scholar
  9. 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. CrossRefGoogle Scholar
  10. Crick, H. Q. P. (2004), “The Impact of Climate Change on Birds,” Ibis, 146 (Suppl. 1), 48–56. CrossRefGoogle Scholar
  11. Dennis, B., Ponciano, J. M., Lele, S. R., Taper, M. L., and Staples, D. F. (2006), “Estimating Density Dependence, Process Noise, and Observation Error,” Ecological Monographs, 76, 323–341. CrossRefGoogle Scholar
  12. de Valpine, P. (2002), “Review of Methods for Fitting Time-Series Models With Process and Observation Error and Likelihood Calculations for Nonlinear, Non-Gaussian State-Space Models,” Bulletin of Marine Science, 70, 455–471. Google Scholar
  13. de Valpine, P., and Hastings, A. (2002), “Fitting Population Models Incorporating Process Noise and Observation Error,” Ecological Monographs, 72, 57–76. CrossRefGoogle Scholar
  14. de Valpine, P., and Hilborn, R. (2005), “State-Space Likelihoods for Nonlinear Fisheries Time-Series,” Canadian Journal of Fisheries and Aquatic Sciences, 62, 1937–1952. CrossRefGoogle Scholar
  15. Ennola, K., Sarvala, J., and Devai, G. (1998), “Modelling Zooplankton Population Dynamics With the Extended Kalman Filtering Technique,” Ecological Modelling, 110, 135–149. CrossRefGoogle Scholar
  16. Freckleton, R. P., Watkinson, A. R., Green, R. E., and Sutherland, W. J. (2006), “Census Error and the Detection of Density Dependence,” Journal of Animal Ecology, 75, 837–851. CrossRefGoogle Scholar
  17. Freeman, S. N., and Morgan, B. J. T. (1992), “A Modelling Strategy for Recovery Data From Birds Ringed as Nestlings,” Biometrics, 48, 217–236. CrossRefGoogle Scholar
  18. Gimenez, O., Barbraud, C., Crainiceanu, C., Jenouvrier, S., and Morgan, B. J. T. (2006), “Semiparametric Regression in Capture-Recapture Modelling,” Biometrics, 62, 691–698. MathSciNetzbMATHCrossRefGoogle Scholar
  19. Grenfell, B. T., Wilson, K., Finkenstädt, B. F., Coulson, T. N., Murray, S., Albon, S. D., Pemberton, J. M., Clutton-Brock, T. H., and Crawley, M. J. (1998), “Noise and Determinism in Synchronised Sheep Dynamics,” Nature, 394, 675–677. CrossRefGoogle Scholar
  20. Grosbois, V., Harris, M. P., Anker-Nilssen, T., McCleery, R. H., Shaw, D. N., Morgan, B. J. T., and Gimenez, O. (2009), “Survival at Multi-Population Scales Using Mark-Recapture Data,” Ecology, 90, 2922–2932. CrossRefGoogle Scholar
  21. Harvey, A. C. (1989), Forecasting, Structural Time Series Models and the Kalman Filter, Cambridge: Cambridge University Press. Google Scholar
  22. Kitagawa, G. (1987), “Non-Gaussian State-Space Modeling of Nonstationary Time Series,” Journal of the American Statistical Association, 82, 1032–1063. MathSciNetzbMATHCrossRefGoogle Scholar
  23. Knape, J. (2008), “Estimability of Density Dependence in Models of Time Series Data,” Ecology, 89, 2994–3000. CrossRefGoogle Scholar
  24. Marchant, J. H., Freeman, S. N., Crick, H. P. Q., and Beaven, L. P. (2004), “The BTO Heronries Census of England and Wales 1928–2000: New Indices and a Comparison of Analytical Methods,” Ibis, 146, 323–334. CrossRefGoogle Scholar
  25. Meyer, R., and Millar, R. B. (1999), “BUGS in Bayesian Stock Assessments,” Canadian Journal of Fisheries and Aquatic Sciences, 56, 1078–1086. Google Scholar
  26. Millar, R. B., and Meyer, R. (2000), “Non-Linear State Space Modelling of Fisheries Biomass Dynamics by Using Metropolis–Hastings Within-Gibbs Sampling,” Applied Statistics, 49, 327–342. MathSciNetzbMATHCrossRefGoogle Scholar
  27. North, P. M., and Morgan, B. J. T. (1979), “Modelling Heron Survival Using Weather Data,” Biometrics, 35, 667–682. MathSciNetCrossRefGoogle Scholar
  28. Schnute, J. (1994), “A General Framework for Developing Sequential Fisheries Models,” Canadian Journal of Fisheries and Aquatic Sciences, 51, 1676–1688. CrossRefGoogle Scholar
  29. Stenseth, N. C., Chan, K.-S., Tavecchia, G., Coulson, T., Mysterud, A., Clutton-Brock, T., and Grenfell, B. (2004), “Modelling Non-Additive and Nonlinear Signals From Climatic Noise in Ecological Time Series: Soay Sheep as an Example,” Proceedings of the Royal Society of London. Series B, 271, 1985–1993. CrossRefGoogle Scholar
  30. Tavecchia, G., Besbeas, P., Coulson, T., Morgan, B. J. T., and Clutton-Brock, T. H. (2009), “Estimating Population Size and Hidden Demographic Parameters With State-Space Modelling,” The American Naturalist, 173, 722–733. CrossRefGoogle Scholar
  31. Thomas, L., Buckland, S. T., Newman, K. B., and Harwood, J. (2004), “A Unified Framework for Modelling Wildlife Population Dynamics,” Australian & New Zealand Journal of Statistics, 47, 19–34. MathSciNetGoogle Scholar

Copyright information

© International Biometric Society 2011

Authors and Affiliations

  1. 1.Department of StatisticsAthens University of Economics and BusinessAthensGreece
  2. 2.School of Mathematics, Statistics and Actuarial ScienceUniversity of KentCanterburyEngland, UK

Personalised recommendations