Explicit Representation of Periodic Solutions and Parameter Estimation Problem of Predator-Prey System
DOI:
https://doi.org/10.29304/jqcm.2022.14.1.881Keywords:
Predator Prey model, parameter estimation problem, integral representation, periodic solutionsAbstract
It is known that some dynamical systems have not analytical solution which is impossible or sometime difficult to find. In fact, undertaking a manual simulation or using complicated methods (need to find derivatives such that Newton method) of such systems is a difficult task due to the complexity of the computations. Therefore, a computerized simulation is frequently required to find accurate results in fast execution time especially for solving biological problems like Predator Prey model. This paper aims to find integral representation solutions for Predator Prey model using the new method proposed by Mohammed, J. & Tyukin, I in [16]. These integral representation solutions are periodic and depend on parameters of the Predator Prey model explicitly, then, however, it requires to estimate these parameters. Application of the method to solve parameter estimation problem of predator–prey model is illustrated in details with constructing the solutions for state and parameter estimation.
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References
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[28] RASHEED, M., Rashid, A., Rashid, T., Abed Hamed, S., & Sabri, A. (2021). Analysis of Non-Linear Device by means of Numerical Algorithms.
Journal of Al-Qadisiyah for Computer Science and Mathematics, 13(3), Math Page 79-. https://doi.org/10.29304/jqcm.2021.13.3.845.
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Journal of Al-Qadisiyah for Computer Science and Mathematics, 11(2), math 46-53. https://doi.org/10.29304/jqcm.2019.11.2.548.
[30] RASHEED, M., SHIHAB, S., RASHID, T., & Enneffati, M. (2021). Two Numerical Algorithms for Solving Nonlinear Equation of Solar Cell.
Journal of Al-Qadisiyah for Computer Science and Mathematics, 13(1), Math Page 87-94. https://doi.org/10.29304/jqcm.2021.13.1.752
[2] Al-Ameri, J. M. (2018). Explicit Representations of Periodic Solutions of Nonlinearly Parameterized Ordinary Differential Equations And Their Applications To Inverse Problems (Doctoral dissertation, University of Leicester).
[3] Al Themairi, A., & Alqudah, M. A. (2021). Existence and Uniqueness of Caputo Fractional Predator-Prey Model of Holling-Type II with Numerical Simulations. Mathematical Problems in Engineering, 2021.
[4] Besançon, G. (2000). Remarks on nonlinear adaptive observer design. Systems & control letters, 41(4):271–280.
[5] Bock, H. G., Kostina, E., and Schlöder, J. P. (2007). Numerical methods for parameter estimation in nonlinear differential algebraic equations. GAMM Mitteilungen, 30(2):376–408.
[6] De Terán, F., & Dopico, F. (2011). Consistency and efficient solution of the Sylvester equation for*-congruence. The Electronic Journal of Linear Algebra, 22, 849-863.
[7] Hua, F., Sieving, K. E., Fletcher Jr, R. J., & Wright, C. A. (2014). Increased perception of predation risk to adults and offspring alters avian reproductive strategy and performance. Behavioral Ecology, 25(3), 509-519.
[8] Hammouri, H. and de Morales,L. (1990). Observer synthesis for state affine systems . In Decision and Control, 1990, Proceedings of the 29th IEEE Conferenceon, 784–785. IEEE.
[9] Khan, A., Alshehri, H. M., Gómez-Aguilar, J. F., Khan, Z. A., & Fernández-Anaya, G. (2021). A predator–prey model involving variable-order fractional differential equations with Mittag-Leffler kernel. Advances in Difference Equations, 2021(1), 1-18.
[10] Liu, B., Teng, Z., & Chen, L. (2006). Analysis of a predator–prey model with Holling II functional response concerning impulsive control strategy. Journal of Computational and Applied Mathematics, 193(1), 347-362.
[11] Ljung, L. (1999), system identification: Theory for the user.
[12] Luenberger, D. (1966). Observers for multivariable systems. IEEE Transactions on Automatic Control, 11(2):190–197.
[13] Luenberger, D. (1971). An introduction to observers. IEEE Transactions on automatic control, 16(6):596–602.
[14] Mao, Y., Tang, W., Liu, Y., & Kocarev, L. (2009). Identification of biological neurons using adaptive observers. Cognitive processing, 10(1), 41-53.
[15] Ma, T., & Cao, C. (2020). Estimation using L1 adaptive descriptor observer for multivariable systems with nonlinear uncertainties and measurement noises. European Journal of Control, 52, 11-18.
[16] Mohammed, J. A. A., & Tyukin, I. (2017). Explicit parameter-dependent representations of periodic solutions for a class of nonlinear systems. IFAC-PapersOnLine, 50(1), 4001-4007.
[17] Nelder, J. A. and Mead, R. (1965). A simplex method for function minimization. The computer journal, 7(4):308–313.
[18] Stechlinski, P., Khan, K. A., & Barton, P. I. (2018). Generalized sensitivity analysis of nonlinear programs. SIAM Journal on Optimization, 28(1), 272-301.
[19] Creel, S., & Christianson, D. (2008). Relationships between direct predation and risk effects. Trends in Ecology & Evolution, 23(4), 194-201.
[20] Johnson, T., & Tucker, W. (2008). Rigorous parameter reconstruction for differential equations with noisy data. Automatica, 44(9), 2422-2426.
[21] Tyukin, I. Y., Steur, E., Nijmeijer, H., & Van Leeuwen, C. (2013). Adaptive observers and parameter estimation for a class of systems nonlinear in the parameters. Automatica, 49(8), 2409-2423.
[22] Tyukin, I. Y., Gorban, A. N., Tyukina, T. A., Al-Ameri, J. M., & Korablev, Y. A. (2016). Fast sampling of evolving systems with periodic trajectories. Mathematical Modelling of Natural Phenomena, 11(4), 73-88.
[23] Yavuz, M., & Sene, N. (2020). Stability analysis and numerical computation of the fractional predator–prey model with the harvesting rate. Fractal and Fractional, 4(3), 35.
[24] Zhan, C., Situ, W., Yeung, L. F., Tsang, P. W. M., & Yang, G. (2014). A parameter estimation method for biological systems modelled by ode/dde models using spline approximation and differential evolution algorithm. IEEE/ACM transactions on computational biology and bioinformatics, 11(6), 1066-1076.
[25] Z.J. Jing, L.S. Chen (1984). The existence and uniqueness of limit cycles in general predator-prey differential equations. Chineese Sci. Bull., 9, 521–523.
[26] M. RASHEED, A. Rashid, A. Rashid, T. Rashid, S. Abed Hamed, and O. Abed AL-Farttoosi, “Application of Numerical Analysis for Solving
Nonlinear Equation”, JQCM, vol. 13, no. 3, pp. Math Page 70-78, Sep. 2021.
[27] G. Arif, S. Wuhaib, and M. Rashad, “Infected Intermediate Predator and Harvest in Food Chain”, JQCM, vol. 12, no. 1, pp. Math Page
120 -138, Apr. 2020.
[28] RASHEED, M., Rashid, A., Rashid, T., Abed Hamed, S., & Sabri, A. (2021). Analysis of Non-Linear Device by means of Numerical Algorithms.
Journal of Al-Qadisiyah for Computer Science and Mathematics, 13(3), Math Page 79-. https://doi.org/10.29304/jqcm.2021.13.3.845.
[29] Kareem Wanas, A., & Swamy, S. (2019). Differential Subordination Results for Holomorphic Functions Related to Differential Operator.
Journal of Al-Qadisiyah for Computer Science and Mathematics, 11(2), math 46-53. https://doi.org/10.29304/jqcm.2019.11.2.548.
[30] RASHEED, M., SHIHAB, S., RASHID, T., & Enneffati, M. (2021). Two Numerical Algorithms for Solving Nonlinear Equation of Solar Cell.
Journal of Al-Qadisiyah for Computer Science and Mathematics, 13(1), Math Page 87-94. https://doi.org/10.29304/jqcm.2021.13.1.752
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Published
2022-03-05
How to Cite
Khudhir, J. M. (2022). Explicit Representation of Periodic Solutions and Parameter Estimation Problem of Predator-Prey System. Journal of Al-Qadisiyah for Computer Science and Mathematics, 14(1), Math Page 1– 9. https://doi.org/10.29304/jqcm.2022.14.1.881
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