Rigorous mapping of data to qualitative properties of parameter values and dynamics: a case study on a two-variable Lotka-Volterra system. (English) Zbl 1519.92191
Summary: In this work, we describe mostly analytical work related to a novel approach to parameter identification for a two-variable Lotka-Volterra (LV) system. Specifically, this approach is qualitative, in that we aim not to determine precise values of model parameters but rather to establish relationships among these parameter values and properties of the trajectories that they generate, based on a small number of available data points. In this vein, we prove a variety of results about the existence, uniqueness, and signs of model parameters for which the trajectory of the system passes exactly through a set of three given data points, representing the smallest possible data set needed for identification of model parameter values. We find that in most situations such a data set determines these values uniquely; we also thoroughly investigate the alternative cases, which result in nonuniqueness or even nonexistence of model parameter values that fit the data. In addition to results about identifiability, our analysis provides information about the long-term dynamics of solutions of the LV system directly from the data without the necessity of estimating specific parameter values.
MSC:
| 92D25 | Population dynamics (general) |
| 34A55 | Inverse problems involving ordinary differential equations |
Keywords:
inverse problem; parameter identification; data fitting; linear-in-parameters; predator-preySoftware:
EnKFReferences:
| [1] | Aster, RC; Borchers, B.; Thurber, CH, Parameter estimation and inverse problems (2018), Amsterdam: Elsevier, Amsterdam |
| [2] | Broomhead, DS; King, GP, Extracting qualitative dynamics from experimental data, Physica D, 20, 2-3, 217-236 (1986) · Zbl 0603.58040 · doi:10.1016/0167-2789(86)90031-X |
| [3] | Calvetti D, and Somersalo E (2007) An introduction to bayesian scientific computing: ten lectures on subjective computing (Vol. 2). Springer Science & Business Media · Zbl 1137.65010 |
| [4] | Calvetti, D.; Somersalo, E., Inverse problems: from regularization to Bayesian inference, Wiley Interdiscip Rev Comput Stat, 10, 3 (2018) · Zbl 07910815 · doi:10.1002/wics.1427 |
| [5] | Cao, J.; Wang, L.; Xu, J., Robust estimation for ordinary differential equation models, Biometrics, 67, 4, 1305-1313 (2011) · Zbl 1274.62739 · doi:10.1111/j.1541-0420.2011.01577.x |
| [6] | Dalgaard P, and Larsen M (1990) Fitting numerical solutions of differential equations to experimental data: a case study and some general remarks. Biometrics, 1097-1109 |
| [7] | Duan, X.; Rubin, J.; Swigon, D., Identification of affine dynamical systems from a single trajectory, Inverse Prob, 36, 8 (2020) · Zbl 1451.34021 · doi:10.1088/1361-6420/ab958e |
| [8] | Duan X, Rubin JE, Swigon D (2023) Qualitative inverse problems: mapping data to the features of trajectories and parameter values of an ODE model. Inverse Prob, 39:075002 · Zbl 07691091 |
| [9] | Evensen G (2009) Data assimilation: the ensemble kalman filter. Springer Science & Business Media |
| [10] | Fort, H., On predicting species yields in multispecies communities: quantifying the accuracy of the linear Lotka-Volterra generalized model, Ecol Modell, 387, 154-162 (2018) · doi:10.1016/j.ecolmodel.2018.09.009 |
| [11] | Khan, T.; Chaudhary, H., Estimation and identifiability of parameters for generalized Lotka-Volterra biological systems using adaptive controlled combination difference anti-synchronization, Differ Eq Dyn Syst, 28, 3, 515-526 (2020) · Zbl 1472.92181 · doi:10.1007/s12591-020-00534-8 |
| [12] | Kloppers, P.; Greeff, J., Lotka-Volterra model parameter estimation using experiential data, App Math Comput, 224, 817-825 (2013) · Zbl 1334.92348 · doi:10.1016/j.amc.2013.08.093 |
| [13] | Kunze, H.; Hicken, J.; Vrscay, E., Inverse problems for odes using contraction maps and suboptimality of the? Collage method?, Inverse Probl, 20, 3, 977 (2004) · Zbl 1067.34010 · doi:10.1088/0266-5611/20/3/019 |
| [14] | Lazzus, JA; Vega-Jorquera, P.; Lopez-Caraballo, CH; Palma-Chilla, L.; Salfate, I., Parameter estimation of a generalized Lotka-Volterra system using a modified PSO algorithm, Appl Soft Comput, 96 (2020) · doi:10.1016/j.asoc.2020.106606 |
| [15] | MacKay, RS; Meiss, JD, Hamiltonian dynamical systems: a reprint selection (2020), Boca Raton, FL: CRC Press, Boca Raton, FL · doi:10.1201/9781003069515 |
| [16] | May, RM, Theoretical ecology: principles and applications (1976), Amsterdam: Elsevier - Health Sciences Division, Amsterdam |
| [17] | Meshkat, N.; Sullivant, S.; Eisenberg, M., Identifiability results for several classes of linear compartment models, Bull Math Biol, 77, 8, 1620-1651 (2015) · Zbl 1336.92004 · doi:10.1007/s11538-015-0098-0 |
| [18] | Packard, NH; Crutchfield, JP; Farmer, JD; Shaw, RS, Geometry from a time series Geometry from a time series, Phys Rev Lett, 45, 9, 712 (1980) · doi:10.1103/PhysRevLett.45.712 |
| [19] | Perko, L., Differential equations and dynamical systems (2013), New York, NY: Springer Science & Business Media, New York, NY |
| [20] | Ramsay, JO; Hooker, G.; Campbell, D.; Cao, J., Parameter estimation for differential equations: a generalized smoothing approach, J Royal Stat Soci Series B (Stat Methodol), 69, 5, 741-796 (2007) · Zbl 07555374 · doi:10.1111/j.1467-9868.2007.00610.x |
| [21] | Smith, RC, Uncertainty quantification: theory, implementation, and applications (2013), Philadelphia, PA: SIAM, Philadelphia, PA |
| [22] | Stanhope, S.; Rubin, J.; Swigon, D., Identifiability of linear and linear-in-parameters dynamical systems from a single trajectory, SIAM J Appl Dyn Syst, 13, 4, 1792-1815 (2014) · Zbl 1303.93066 · doi:10.1137/130937913 |
| [23] | Stanhope, S.; Rubin, J.; Swigon, D., Robustness of solutions of the inverse problem for linear dynamical systems with uncertain data, SIAM/ASA J Uncertain Quantif, 5, 1, 572-597 (2017) · Zbl 1391.93051 · doi:10.1137/16M1062466 |
| [24] | Stuart, AM, Inverse problems: a Bayesian perspective, Acta Numer, 19, 451-559 (2010) · Zbl 1242.65142 · doi:10.1017/S0962492910000061 |
| [25] | Swigon, D.; Stanhope, SR; Zenker, S.; Rubin, JE, On the importance of the Jacobian determinant in parameter inference for random parameter and random measurement error models, SIAM/ASA J Uncertain Quantif, 7, 3, 975-1006 (2019) · Zbl 1433.62091 · doi:10.1137/17M1114405 |
| [26] | Takens, F., Dynamical systems and turbulence, Warwick 1980, 366-381 (1981), Berlin: Heidelberg, Springer, Berlin · doi:10.1007/BFb0091924 |
| [27] | Tarantola A (2005) Inverse problem theory and methods for model parameter estimation. SIAM · Zbl 1074.65013 |
| [28] | Wangersky, PJ, Lotka-Volterra population models, Annu Rev Ecol Syst, 9, 189-218 (1978) · doi:10.1146/annurev.es.09.110178.001201 |
| [29] | Wu, L.; Wang, Y., Estimation the parameters of Lotka-Volterra model based on grey direct modelling method and its application, Expert Syst Appl, 38, 6, 6412-6416 (2011) · doi:10.1016/j.eswa.2010.09.013 |
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