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Estimating a Predator‐Prey Dynamical Model with the Parameter Cascades Method

Author

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  • Jiguo Cao
  • Gregor F. Fussmann
  • James O. Ramsay

Abstract

Summary Ordinary differential equations (ODEs) are widely used in ecology to describe the dynamical behavior of systems of interacting populations. However, systems of ODEs rarely provide quantitative solutions that are close to real field observations or experimental data because natural systems are subject to environmental and demographic noise and ecologists are often uncertain about the correct parameterization. In this article we introduce “parameter cascades” as an improved method to estimate ODE parameters such that the corresponding ODE solutions fit the real data well. This method is based on the modified penalized smoothing with the penalty defined by ODEs and a generalization of profiled estimation, which leads to fast estimation and good precision for ODE parameters from noisy data. This method is applied to a set of ODEs originally developed to describe an experimental predator–prey system that undergoes oscillatory dynamics. The new parameterization considerably improves the fit of the ODE model to the experimental data sets. At the same time, our method reveals that important structural assumptions that underlie the original ODE model are essentially correct. The mathematical formulations of the two nonlinear interaction terms (functional responses) that link the ODEs in the predator–prey model are validated by estimating the functional responses nonparametrically from the real data. We suggest two major applications of “parameter cascades” to ecological modeling: It can be used to estimate parameters when original data are noisy, missing, or when no reliable priori estimates are available; it can help to validate the structural soundness of the mathematical modeling approach.

Suggested Citation

  • Jiguo Cao & Gregor F. Fussmann & James O. Ramsay, 2008. "Estimating a Predator‐Prey Dynamical Model with the Parameter Cascades Method," Biometrics, The International Biometric Society, vol. 64(3), pages 959-967, September.
  • Handle: RePEc:bla:biomet:v:64:y:2008:i:3:p:959-967
    DOI: 10.1111/j.1541-0420.2007.00942.x
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    References listed on IDEAS

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    1. Jiguo Cao & James Ramsay, 2007. "Parameter cascades and profiling in functional data analysis," Computational Statistics, Springer, vol. 22(3), pages 335-351, September.
    2. J. O. Ramsay & G. Hooker & D. Campbell & J. Cao, 2007. "Parameter estimation for differential equations: a generalized smoothing approach," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(5), pages 741-796, November.
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    2. Jeon, Jong-ha & Kim, Pilwon, 2014. "Reconstruction of systems with impulses and delays from time series data," Chaos, Solitons & Fractals, Elsevier, vol. 69(C), pages 64-73.
    3. Liu, Baisen & Wang, Liangliang & Nie, Yunlong & Cao, Jiguo, 2019. "Bayesian inference of mixed-effects ordinary differential equations models using heavy-tailed distributions," Computational Statistics & Data Analysis, Elsevier, vol. 137(C), pages 233-246.
    4. Arnone, Eleonora & Azzimonti, Laura & Nobile, Fabio & Sangalli, Laura M., 2019. "Modeling spatially dependent functional data via regression with differential regularization," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 275-295.

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