Estimating a Predator-Prey Dynamical Model with the Parameter Cascades Method
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 subjec...
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| Published in | Biometrics Vol. 64; no. 3; pp. 959 - 967 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Malden, USA
Blackwell Publishing Inc
01.09.2008
Blackwell Publishing Blackwell Publishing Ltd |
| Subjects | |
| Online Access | Get full text |
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| Abstract | 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. |
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| AbstractList | 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. 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.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. 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. [PUBLICATION 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. 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. /// Les équations différentielles ordinaires (EDO) sont largement utilisées pour décrire le comportement dynamique des populations en interaction, Cependant les systèmes EDO donnent rarement des solutions quantitatives proches des observations réelles de terrain ou des résultats expérimentaux, parce que les systèmes naturels sont soumis à un bruit environnemental ou démographique et que les écologistes sont dans l'incertitude pour paramétrer correctement leurs données. Dans cet article nous introduisons les "cascades de paramètres" pour améliorer l'estimation des paramètres EDO de manière à bien ajuster les solutions aux données réelles. Cette méthode est basée sur le lissage pénalisé modifié en définissant la pénalité par les EDO et en généralisant l'estimation profilée pour aboutir à une estimation rapide et à une bonne précision des paramètres EDO sur des données affectées de bruit. Cette méthode est appliquée à un ensemble d'EDO prévues à l'origine pour décrire un système prédateur-proie expérimental soumis à des oscillations. Le nouveau paramétrage améliore considérablement l'ajustement du modèle EDO aux données expérimentales. La méthode révèle en outre que d'importantes hypothèses structurales sous-jacentes au modèle EDO originel sont essentiellement correctes. La formulation mathématique des deux termes d'interaction non linéaire (réponses fonctionnelles) qui lient les EDO dans le modèle prédateur-proie est validée par un estimateur non paramétrique des réponses fonctionnelles à partir des données réelles. Nous suggérons deux applications importantes des "cascades de paramètres" à la modélisation écologique. La méthode peut servir à estimer des paramètres quand les données sont soumises à des bruits ou manquantes, ou quand aucune estimée a priori fiable n'est disponible. La méthode peut par ailleurs aider à valider la qualité structurelle du modèle mathématique. 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. |
| Author | Ramsay, James O. Cao, Jiguo Fussmann, Gregor F. |
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| BackLink | https://www.ncbi.nlm.nih.gov/pubmed/18047526$$D View this record in MEDLINE/PubMed |
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| References_xml | – reference: Bock, H. G. (1983). Recent advances in parameter identification techniques for ordinary differential equations. In Numerical Treatment of Inverse Problems in Differential and Integral Equations, P. Deuflhard and E. Harrier (eds), 95-121. Basel , Switzerland : Birkhäuser. – reference: Turchin, P. (2003). Complex Population Dynamics. Princeton , NJ : Princeton University Press. – reference: Vos, M., Kooi, B. W., DeAngelis, D. L., and Mooij, W. M. (2004). Inducible defences and the paradox of enrichment. Oikos 105, 471-480. – reference: Hassell, M. P., Lawton, J. H., and Beddington, J. R. (1977). Sigmoid functional responses by invertebrate predators and parasitoids. Journal of Animal Ecology 46, 249-262. – reference: Fussmann, G. F. and Blasius, B. (2005). Community response to enrichment is highly sensitive to model structure. Biology Letters 1, 9-12. – reference: Murdoch, W., Briggs, C., and Nisbet, R. (2003). Consumer-Resource Dynamics. New York : Princeton University Press. – reference: Gelman, A., Bois, F., and Jiang, J. (1996). Physiological pharmacokinetic analysis using population modeling and informative prior distributions. Journal of the American Statistical Association 91, 1400-1412. – reference: Kendall, B. E., Briggs, C. J., Murdoch, W. W., Turchin, P., Ellner, S. P., McCauley, E., Nisbet, R. M., and Wood, S. N. (1999). Why do populations cycle? A synthesis of statistical and mechanistic modeling approaches. Ecology 80, 1789-1805. – reference: Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis, 2nd ed. New York : Springer. – reference: Rosenzweig, M. L. and MacArthur, R. H. (1963). Graphical representation and stability conditions of predator-prey interactions. American Naturalist 97, 209. – reference: Wood, S. N. (2001). Partially specified ecological models. Ecological Monographs 71, 1-25. – reference: Ellner, S. P., Seifu, Y., and Smith, R. H. (2002). Fitting population dynamic models to time-series data by gradient matching. Ecology 83, 2256-2270. – reference: Kondoh, M. (2003). Foraging adaptation and the relationship between food-web complexity and stability. Science 299, 1388-1391. – reference: Williams, R. J. and Martinez, N. D. (2004). Stabilization of chaotic and non-permanent food-web dynamics. European Physical Journal B 38, 297-303. – reference: Becks, L., Hilker, F. M., Malchow, H., Jürgens, K., and Arndt, H. (2005). Experimental demonstration of chaos in a microbial food web. Nature 435, 1226. – reference: Holling, C. S. (1959). The components of predation as revealed by a study of small-mammal predation of the European pine sawfly. The Canadian Entomologist 91, 293-320. – reference: Yoshida, T., Jones, L. E., Ellner, S. P., Fussmann, G. F., and Hairston, N. G. (2003). Rapid evolution drives ecological dynamics in a predator-prey system. Nature 424, 303-306. – reference: Himmelblau, D., Jones, C., and Bischoff, K. B. (1967). Determination of rate constants for complex kinetics models. Industrial Engineering Chemistry Fundamentals 6, 539. – reference: de Boor, C. and Swartz, B. (1973). Collocation at Gaussian points. SIAM Journal on Numerical Analysis 10, 582-606. – reference: Ellner, S. P., Kendall, B. E., Wood, S. N., McCauley, E., and Briggs, C. J. (1997). Inferring mechanism from time-series data: Delay-differential equations. Physics D 110, 182-194. – reference: Fussmann, G. F., Ellner, S. P., Hairston, N. G., Jones, L. E., Shertzer, K. W., and Yoshida, T. (2005). Ecological and evolutionary dynamics of experimental plankton communities. Advances in Ecological Research 37, 221-243. – reference: Jensen, C. X. J., Jeschke, J. M., and Ginzburg, L. R. (2007). A direct, experimental test of resource vs. consumer dependence: Comment. Ecology 88, 1600-1602. – reference: Ivlev, V. S. (1961). Experimental Ecology of the Feeding of Fishes. New Haven , CT : Yale University Press. – reference: Fussmann, G. F., Ellner, S. P., Shertzer, K. W., and Hairston, N. G. J. (2000). Crossing the Hopf bifurcation in a live predator-prey system. Science 290, 1358-1360. – reference: Cao, J. and Ramsay, J. O. (2007). Parameter cascades and profiling in functional data analysis. Computational Statistics 22, 335-351. – reference: Ionides, E. L., Breto, C., and King, A. A. (2006). Inference for nonlinear dynamical systems. Proceedings of the National Academy of Sciences 103, 18438-18443. – reference: Fussmann, G. F., Weithoff, G., and Yoshida, T. (2005). A direct, experimental test of resource vs. consumer dependence. Ecology 86, 2924-2930. – reference: Ramsay, J. O., Hooker, G., Campbell, D., and Cao, J. (2007). Parameter estimation for differential equations: A generalized smoothing approach (with discussion). Journal of the Royal Statistical Society, Series B 69, 741-746. – reference: Shertzer, K. W., Ellner, S. P., Fussmann, G. F., and Hairston, N. G. (2002). Predator-prey cycles in an aquatic microcosm: Testing hypotheses of mechanism. Journal of Animal Ecology 71, 802-815. – volume: 103 start-page: 18438 year: 2006 end-page: 18443 article-title: Inference for nonlinear dynamical systems publication-title: Proceedings of the National Academy of Sciences – volume: 22 start-page: 335 year: 2007 end-page: 351 article-title: Parameter cascades and profiling in functional data analysis publication-title: Computational Statistics – year: 2005 – year: 2003 – volume: 46 start-page: 249 year: 1977 end-page: 262 article-title: Sigmoid functional responses by invertebrate predators and parasitoids publication-title: Journal of Animal Ecology – volume: 88 start-page: 1600 year: 2007 end-page: 1602 article-title: A direct, experimental test of resource vs. consumer dependence: Comment publication-title: Ecology – volume: 424 start-page: 303 year: 2003 end-page: 306 article-title: Rapid evolution drives ecological dynamics in a predator‐prey system publication-title: Nature – volume: 71 start-page: 1 year: 2001 end-page: 25 article-title: Partially specified ecological models publication-title: Ecological Monographs – volume: 267 start-page: 1611 year: 2000 end-page: 1620 – volume: 69 start-page: 741 year: 2007 end-page: 746 article-title: Parameter estimation for differential equations: A generalized smoothing approach (with discussion) publication-title: Journal of the Royal Statistical Society, Series B – volume: 1 start-page: 9 year: 2005 end-page: 12 article-title: Community response to enrichment is highly sensitive to model structure publication-title: Biology Letters – volume: 38 start-page: 297 year: 2004 end-page: 303 article-title: Stabilization of chaotic and non‐permanent food‐web dynamics publication-title: European Physical Journal B – year: 1961 – volume: 290 start-page: 1358 year: 2000 end-page: 1360 article-title: Crossing the Hopf bifurcation in a live predator‐prey system publication-title: Science – volume: 110 start-page: 182 year: 1997 end-page: 194 article-title: Inferring mechanism from time‐series data: Delay‐differential equations publication-title: Physics D – volume: 83 start-page: 2256 year: 2002 end-page: 2270 article-title: Fitting population dynamic models to time‐series data by gradient matching publication-title: Ecology – volume: 10 start-page: 582 year: 1973 end-page: 606 article-title: Collocation at Gaussian points publication-title: SIAM Journal on Numerical Analysis – volume: 37 start-page: 221 year: 2005 end-page: 243 article-title: Ecological and evolutionary dynamics of experimental plankton communities publication-title: Advances in Ecological Research – start-page: 95 year: 1983 end-page: 121 – volume: 80 start-page: 1789 year: 1999 end-page: 1805 article-title: Why do populations cycle? A synthesis of statistical and mechanistic modeling approaches publication-title: Ecology – volume: 6 start-page: 539 year: 1967 article-title: Determination of rate constants for complex kinetics models publication-title: Industrial Engineering Chemistry Fundamentals – volume: 91 start-page: 1400 year: 1996 end-page: 1412 article-title: Physiological pharmacokinetic analysis using population modeling and informative prior distributions publication-title: Journal of the American Statistical Association – volume: 299 start-page: 1388 year: 2003 end-page: 1391 article-title: Foraging adaptation and the relationship between food‐web complexity and stability publication-title: Science – volume: 435 start-page: 1226 year: 2005 article-title: Experimental demonstration of chaos in a microbial food web publication-title: Nature – volume: 91 start-page: 293 year: 1959 end-page: 320 article-title: The components of predation as revealed by a study of small‐mammal predation of the European pine sawfly publication-title: The Canadian Entomologist – volume: 71 start-page: 802 year: 2002 end-page: 815 article-title: Predator‐prey cycles in an aquatic microcosm: Testing hypotheses of mechanism publication-title: Journal of Animal Ecology – volume: 86 start-page: 2924 year: 2005 end-page: 2930 article-title: A direct, experimental test of resource vs. consumer dependence publication-title: Ecology – volume: 97 start-page: 209 year: 1963 article-title: Graphical representation and stability conditions of predator‐prey interactions publication-title: American Naturalist – volume: 105 start-page: 471 year: 2004 end-page: 480 article-title: Inducible defences and the paradox of enrichment publication-title: Oikos – ident: e_1_2_10_2_1 doi: 10.1038/nature03627 – ident: e_1_2_10_7_1 doi: 10.1890/0012-9658(2002)083[2256:FPDMTT]2.0.CO;2 – ident: e_1_2_10_8_1 doi: 10.1098/rsbl.2004.0246 – ident: e_1_2_10_22_1 doi: 10.1515/9781400847259 – ident: e_1_2_10_12_1 doi: 10.1080/01621459.1996.10476708 – ident: e_1_2_10_10_1 doi: 10.1016/S0065-2504(04)37007-8 – ident: e_1_2_10_23_1 doi: 10.1002/0470013192.bsa239 – ident: e_1_2_10_13_1 doi: 10.2307/3959 – ident: e_1_2_10_15_1 doi: 10.4039/Ent91293-5 – ident: e_1_2_10_19_1 doi: 10.1098/rspb.2000.1186 – ident: e_1_2_10_9_1 doi: 10.1126/science.290.5495.1358 – ident: e_1_2_10_31_1 doi: 10.1038/nature01767 – volume-title: Experimental Ecology of the Feeding of Fishes year: 1961 ident: e_1_2_10_17_1 – ident: e_1_2_10_4_1 doi: 10.1007/s00180-007-0044-1 – ident: e_1_2_10_29_1 doi: 10.1140/epjb/e2004-00122-1 – ident: e_1_2_10_11_1 doi: 10.1890/04-1107 – ident: e_1_2_10_16_1 doi: 10.1073/pnas.0603181103 – ident: e_1_2_10_26_1 doi: 10.1046/j.1365-2656.2002.00645.x – volume-title: Complex Population Dynamics year: 2003 ident: e_1_2_10_27_1 – ident: e_1_2_10_14_1 doi: 10.1021/i160024a008 – ident: e_1_2_10_20_1 doi: 10.1890/0012-9658(1999)080[1789:WDPCAS]2.0.CO;2 – ident: e_1_2_10_18_1 doi: 10.1890/05-1945 – start-page: 95 volume-title: Recent advances in parameter identification techniques for ordinary differential equations year: 1983 ident: e_1_2_10_3_1 – ident: e_1_2_10_28_1 doi: 10.1111/j.0030-1299.2004.12930.x – ident: e_1_2_10_21_1 doi: 10.1126/science.1079154 – ident: e_1_2_10_30_1 doi: 10.1890/0012-9615(2001)071[0001:PSEM]2.0.CO;2 – ident: e_1_2_10_5_1 doi: 10.1137/0710052 – ident: e_1_2_10_25_1 doi: 10.1086/282272 – ident: e_1_2_10_6_1 doi: 10.1016/S0167-2789(97)00123-1 – ident: e_1_2_10_24_1 doi: 10.1111/j.1467-9868.2007.00610.x |
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| Snippet | Ordinary differential equations (ODEs) are widely used in ecology to describe the dynamical behavior of systems of interacting populations. However, systems of... Summary Ordinary differential equations (ODEs) are widely used in ecology to describe the dynamical behavior of systems of interacting populations. However,... Summary Ordinary differential equations (ODEs) are widely used in ecology to describe the dynamical behavior of systems of interacting populations. However,... |
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| SubjectTerms | 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 |
| Title | Estimating a Predator-Prey Dynamical Model with the Parameter Cascades Method |
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