Population Ecology

, Volume 56, Issue 2, pp 427–434 | Cite as

Parameter estimation for reaction-diffusion models of biological invasions

  • Samuel Soubeyrand
  • Lionel Roques
Notes and Comments


In this note, we discuss parameter estimation for population models based on partial differential equations (PDEs). Parametric estimation is first considered in the perspective of inverse problems (i.e., when the observation of the solution of a PDE is exactly observed or noise-free). Then, adopting the point of view of statistics, we turn to parametric estimation for PDEs using more realistic noisy measurements. The approach that we describe uses mechanistic-statistical models which combine (1) a PDE-based submodel describing the dynamic under study and (2) a stochastic submodel describing the observation process. This Note is expected to contribute to bridge the gap between modelers using PDEs and population ecologists collecting and analyzing spatio-temporal data.


Inverse problem Mechanistic-statistical model Parametric estimation Partial differential equation Population spread Statistical inference 

Supplementary material

10144_2013_415_MOESM1_ESM.pdf (762 kb)
PDF (761 KB)

AVI (82 KB)

AVI (117 KB)


  1. Berliner LM (2003) Physical-statistical modeling in geophysics. J Geophys Res 108:8776Google Scholar
  2. Berliner LM, Cressie N, Jezek K, Kim Y, Lam CQ, van der Veen CJ (2008) Equilibrium dynamics of ice streams: a bayesian statistical analysis. Stat Method Appl 17:145–165CrossRefGoogle Scholar
  3. Buckland ST, Newman KB, Thomas L, Koesters NB (2004) State-space models for the dynamics of wild animal populations. Ecol Model 171:157–175CrossRefGoogle Scholar
  4. Campbell EP (2004) An introduction to physical-statistical modelling using Bayesian methods. Tech Rep 49, CSIRO Mathematical and Information Sciences, AustraliaGoogle Scholar
  5. Casella G (1985) An introduction to empirical Bayes data analysis. Am Stat 39:83–87CrossRefGoogle Scholar
  6. Choulli M (2009) Une introduction aux probl‘emes inverses elliptiques et paraboliques. Springer, Paris (in French)Google Scholar
  7. Cressie N, Wikle CK (2011) Statistics for spatio-temporal data. Wiley, HobokenGoogle Scholar
  8. Dacunha-Castelle D, Duflo M (1982) Probabilités et Statistiques. Probèlmes à Temps Mobile, vol 2. Masson, Paris (in French)Google Scholar
  9. Efron B (2013) Bayes’ theorem in the 21st century. Science 340:1177–1178PubMedCrossRefGoogle Scholar
  10. Evans LC (1998) Partial differential equations. University of California, BerkeleyGoogle Scholar
  11. Evensen G (2009) Data assimilation: the ensemble Kalman filter, 2nd edn. Springer-Verlag, BerlinCrossRefGoogle Scholar
  12. Hadamard J (1923) Lectures on Cauchy’s problem in linear partial differential equations. Dover Publications, New YorkGoogle Scholar
  13. Harmon R, Challenor P (1997) A Markov chain Monte Carlo method for estimation and assimilation into models. Ecol Model 101:41–59CrossRefGoogle Scholar
  14. Ionides EL, Breto C, King AA (2006) Inference for nonlinear dynamical systems. Proc Natl Acad Sci USA 103:18438–18443PubMedCentralPubMedCrossRefGoogle Scholar
  15. Isakov V (1990) Inverse source problems. American Mathematical Society, ProvidenceGoogle Scholar
  16. Jones E, Parslow J, Murray L (2010) A Bayesian approach to state and parameter estimation in a Phytoplankton–Zooplankton model. Aust Meteorol Ocean 59:7–16Google Scholar
  17. Marin J-M, Robert C (2007) Bayesian core: a practical approach to computational Bayesian statistics. Springer-Verlag, New YorkGoogle Scholar
  18. Murray JD (2002) Mathematical biology, 3rd edn. Springer-Verlag, BerlinGoogle Scholar
  19. Okubo A, Levin SA (2002) Diffusion and ecological problems—modern perspectives, 2nd edn. Springer-Verlag, New YorkGoogle Scholar
  20. Ramsay JO, Hooker G, Campbell D, Cao J (2007) Parameter estimation for differential equations: a generalized smoothing approach. J R Stat Soc B 69:741–796CrossRefGoogle Scholar
  21. Rivot E, Prévost E, Parent E, Baglinière JL (2004) A Bayesian state-space modelling framework for fitting a salmon stage-structured population dynamic model to multiple time series of field data. Ecol Model 179:463–485CrossRefGoogle Scholar
  22. Robert CP, Casella G (1999) Monte Carlo statistical methods. Springer, New YorkCrossRefGoogle Scholar
  23. Roques L, Cristofol M (2010) On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation. Nonlinearity 23:675–686CrossRefGoogle Scholar
  24. Roques L, Soubeyrand S, Rousselet J (2011) A statistical-reaction-diffusion approach for analyzing expansion processes. J Theor Biol 274:43–51PubMedCrossRefGoogle Scholar
  25. Ruppert D, Wand MP, Carroll RJ (2003) Semiparametric regression. Cambridge University Press, New YorkCrossRefGoogle Scholar
  26. Serfling RJ (2002) Approximation theorems of mathematical statistics. Wiley, New YorkGoogle Scholar
  27. Shigesada N, Kawasaki K (1997) Biological invasions: theory and practice. Oxford Series in Ecology and Evolution. Oxford University Press. OxfordGoogle Scholar
  28. Soubeyrand S, Laine A, Hanski I, Penttinen A (2009a) Spatio-temporal structure of host-pathogen interactions in a metapop- ulation. Am Nat 174:308–320PubMedCrossRefGoogle Scholar
  29. Soubeyrand S, Neuvonen S, Penttinen A (2009b) Mechanical-statistical modelling in ecology: from outbreak detections to pest dynamics. Bull Math Biol 71:318–338PubMedCrossRefGoogle Scholar
  30. Tong H (1990) Non-linear time series: a dynamical system approach. Oxford University Press, OxfordGoogle Scholar
  31. Turchin P (1998) Quantitative analysis of movement: measuring and modeling population redistribution in animals and plants. Sinauer, SunderlandGoogle Scholar
  32. Wikle CK (2003a) Hierarchical Bayesian models for predicting the spread of ecological processes. Ecology 84:1382–1394CrossRefGoogle Scholar
  33. Wikle CK (2003b) Hierarchical models in environmental science. Int Stat Rev 71:181–199CrossRefGoogle Scholar
  34. Wikle CK, Berliner LM (2007) A Bayesian tutorial for data assimilation. Physica D 230:1–16CrossRefGoogle Scholar

Copyright information

© The Society of Population Ecology and Springer Japan 2013

Authors and Affiliations

  1. 1.INRA, UR546 Biostatistics and Spatial ProcessesAvignonFrance

Personalised recommendations