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

• Published in: Biometrics Vol. 64; no. 3; pp. 959 - 967
• Main Authors: Cao, Jiguo, Fussmann, Gregor F., Ramsay, James O.
• Format: Journal Article
• Language: English
• Published: Malden, USA Blackwell Publishing Inc 01.09.2008; Blackwell Publishing; Blackwell Publishing Ltd
• Subjects: Animals; Biometric Practice; biometry; Biometry - methods; Chlorella vulgaris - growth & development; Chlorella vulgaris - physiology; data collection; Data smoothing; Datasets; Differential equations; dynamic models; Ecological modeling; ecologists; Ecology; Ecosystem; equations; Estimating techniques; Estimation methods; Food Chain; Functional responses; Inverse problem; Mathematical independent variables; mathematical models; Mathematical vectors; Models, Biological; Models, Statistical; Nitrogen; Nonlinear Dynamics; Nuisance parameters; Odes; Ordinary differential equation; Ordinary differential equations; Penalized smoothing; Predation; Predator-prey system; Profiling method; Rotifera - growth & development; Rotifera - physiology; Statistical analysis; System identification
• ISSN: 0006-341X; 1541-0420; 1541-0420
• DOI: 10.1111/j.1541-0420.2007.00942.x

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.