A Nonstandard Finite Difference Scheme for a Mathematical Model Presenting the Climate Change on the Oxygen-plankton System

Yıl 2024, Cilt: 13 Sayı: 3, 798 - 807, 26.09.2024

Zahraa Al Jammali , İlkem Turhan Çetinkaya

https://doi.org/10.17798/bitlisfen.1492437

Öz

This paper presents a mathematical model describing climate change in the oxygen-plankton system. The model consists of a system of non-linear ordinary differential equations. The Nonstandard Finite Difference (NSFD) method is applied to discretize the non-linear system. The stability of the continuous and discrete model is presented for the given parameters in the literature. The compatibility of the results has been seen. Moreover, the model is solved by both the NSFD method and the Runge–Kutta–Fehlberg (RKF45) method. The numerical results are compared. Furthermore, the efficiency of the NSFD method compared to classical methods such as the Euler method and the fourth order Runge-Kutta (RK4) method for the bigger step size is shown in tabular form.

Anahtar Kelimeler

Nonstandard Finite Difference Method, Oxygen-plankton System, Stability Analysis, Schur-Cohn criterion

Kaynakça

• [1] Y. Sekerci, and S. Petrovskii, “Mathematical modeling of plankton–oxygen dynamics under the climate change,” Bulletin of Mathematical Biology, vol. 77, pp. 2325-2353, 2015.
• [2] Y. Sekerci, and S. Petrovskii, “Mathematical modelling of spatiotemporal dynamics of oxygen in a plankton system,” Mathematical Modelling of Natural Phenomena, vol. 20, no.2, pp. 96-114, 2015.
• [3] P. Priyadarshini, and P. Veeresha, “Analysis of models describing thermocline depth-ocean temperature and dissolved oxygen concentration in the ocean-plankton community,” Waves in Random and Complex Media, pp. 1-25, 2023.
• [4] S. Mondal, G. Samanta, and M. De la Sen, “Dynamics of oxygen-plankton model with variable zooplankton search rate in deterministic and fluctuating environments,” Mathematics, vol. 10, no. 10, 1641, 2022.
• [5] Y. Sekerci, and R. Ozarslan, “Marine system dynamical response to a changing climate in frame of power law, exponential decay, and Mittag‐Leffler kernel,” Mathematical Methods in the Applied Sciences, vol. 43, no.8, pp. 5480-5506, 2020.
• [6] Y. Sekerci, and R. Ozarslan, “Oxygen-plankton model under the effect of global warming with nonsingular fractional order,” Chaos, Solitons & Fractals, vol. 132, 109532,2020
• [7] Y. Sekerci, and R. Ozarslan, “Fractional order oxygen–plankton system under climate change,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 30, no. 3, 2020.
• [8] C. Xu, Y. Zhao, J. Lin, Y. Pang, Z. Liu, J. Shen,..., and S. Ahmad, “Mathematical exploration on control of bifurcation for a plankton–oxygen dynamical model owning delay, ”Journal of Mathematical Chemistry, pp. 1-31, 2023.
• [9] A. Gökçe, “A mathematical study for chaotic dynamics of dissolved oxygen-phytoplankton interactions under environmental driving factors and time lag.” Chaos, Solitons & Fractals, vol. 151, 111268, 2021.
• [10] P. R. Chowdhury, M. Banerjee, and S. Petrovskii, “A two-timescale model of plankton-oxygen dynamics predicts formation of Oxygen Minimum Zones and global anoxia.” arXiv preprint arXiv:2309.15447.,2023
• [11] R. E. Mickens, Difference Equations Theory and Applications, Atlanta, Ga, USA: Chapman & Hall, 1990.
• [12] R. E. Mickens, Nonstandard finite difference models of differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1994.
• [13] R. E. Mickens, “Nonstandard Finite Difference Schemes for Differential Equations,” Journal of Difference Equations and Applications, vol. 8, no. 9, pp. 823-847, 2002.
• [14] R. E. Mickens, Advances in the applications of Nonstandard Finite Difference Schemes, Singapore: Wiley-Interscience, 2005.
• [15] R. E. Mickens, “Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition,” Numerical Methods for Partial Differential Equations, vol. 23, no. 3, pp. 672-691, 2006
• [16] K. C. Patidar, “On the use of nonstandard finite difference methods,” Journal of Difference Equations and Applications, vol. 11, no. 8, pp. 735-758. 2005.
• [17] K. C. Patidar, “Nonstandard finite difference methods: recent trends and further developments,” Journal of Difference Equations and Applications, vol. 22, no. 6, pp. 817-849, 2016.
• [18] I. U. Khan, A. Hussain, S. Li, and A. Shokri, “Modeling the transmission dynamics of coronavirus using nonstandard finite difference scheme,” Fractal and Fractional, vol. 7, no. 6, p. 451, 2023.
• [19] I. Zhang, S. Gao, and Q. Zou, “A non-standard finite difference scheme of a multiple infected compartments model for waterborne disease,” Differential Equations and Dynamical Systems, vol. 28, no. 1, pp. 59-73, 2020.
• [20] Y. Yang, J. Zhou, X. Ma, and T. Zhang, “Nonstandard finite difference scheme for a diffusive within-host virus dynamics model with both virus-to-cell and cell-to-cell transmissions,” Computers & Mathematics with Applications, vol. 72, no. 4, pp. 1013-1020, 2016.
• [21] M. Kocabıyık, N. Özdoğan, and M. Y. Ongun, “ Nonstandard Finite Difference Scheme for a Computer Virus Model,” Journal of Innovative Science and Engineering, vol. 4, no. 2, pp. 96-108, 2020.
• [22] Q. A. Dang, and M. T. Hoang, “Numerical dynamics of nonstandard finite difference schemes for a computer virus propagation model,” International Journal of Dynamics and Control, vol. 8, no. 3, pp. 772-778, 2020.
• [23] T. M. Hoang, A. Q. Dang, L. Q. Dang, “Nonstandard finite difference schemes for solving a modified epidemiological model for computer viruses,” Journal of Computer Science and Cybernetics, vol. 34, no. 2, pp. 171-185, 2018.
• [24] M. Yakıt Ongun, and İ. Turhan, “A numerical comparison for a discrete HIV infection of CD4+ T-Cell model derived from nonstandard numerical scheme,” Journal of Applied Mathematics, vol. 2013, 2013.
• [25] İ. T. Çetinkaya, M. Kocabıyık, and M.Y. Ongun, “Stability analysis of discretized model of glucose–insulin homeostasis,” Celal Bayar University Journal of Science, vol. 17, no. 4, pp. 369-377, 2021.
• [26] İ. T. Çetinkaya, “An Application of Nonstandard Finite Difference Method to a Model Describing Diabetes Mellitus and Its Complications,” Journal of New Theory, vol. 45, pp. 105-119, 2023.
• [27] M. Kocabıyık, M. Y. Ongun, “Discretization and Stability Analysis for a Generalized Type Nonlinear Pharmacokinetic Models,” Gazi University Journal of Science, vol. 36, no. 4, pp.1675-1691, 2023.
• [28] N. Özdoğan, M. Y. Ongun, “Dynamical Behaviours of a discretized model with Michaelis-Menten Harvesting Rate,” Journal of Universal Mathematics, vol. 5, no. 2, pp. 159-176, 2022.
• [29] S. Li, I. Bukhsh, I. U. Khan, M. I. Asjad, S. M. Eldin, M. Abd El-Rahman, and D. Baleanu, “The impact of standard and nonstandard finite difference schemes on HIV nonlinear dynamical model,” Chaos, Solitons & Fractals, vol. 173, 113755, 2023.
• [30] J. Calatayud, and M. Jornet, “An improvement of two nonstandard finite difference schemes for two population mathematical models,” Journal of Difference Equations and Applications, vol. 27, no. 3, pp. 422-430, 2021.
• [31] M. Z. Ndii, N. Anggriani, and A. K. Supriatna, “Comparison of the differential transformation method and non standard finite difference scheme for solving plant disease mathematical model,” Communication in Biomathematical sciences, vol. 1, no. 2, 2018.
• [32] M. Mehdizadeh Khalsaraei, A. Shokri, S. Noeiaghdam, and M. Molayi, “Nonstandard finite difference schemes for an SIR epidemic model,” Mathematics, vol. 9, no. 23, 3082, 2021.
• [33] A. Zeb, and A. Alzahrani, “Non-standard finite difference scheme and analysis of smoking model with reversion class,” Results in Physics, 21, 103785, 2021.
• [34] M. T. Hoang, and O. F. Egbelowo, “Nonstandard finite difference schemes for solving an SIS epidemic model with standard incidence,” Rendiconti del Circolo Matematico di Palermo Series 2, vol. 69, pp. 753-769, 2020.
• [35] M. T. Hoang, and J. C. Valverde, “A generalized model for the population dynamics of a two stage species with recruitment and capture using a nonstandard finite difference scheme,” Computational and Applied Mathematics, vol. 43, no. 1, pp. 1-27, 2024.
• [36] X. L. Liu, C. C. Zhu, “A non-standard finite difference scheme for a diffusive HIV-1 infection model with immune response and intracellular delay,” Axioms, vol. 11, no. 3, 129, 2022.
• [37] K. Nonlaopon, M. Mehdizadeh Khalsaraei, A. Shokri, and M. Molayi, “ Approximate solutions for a class of predator–prey systems with nonstandard finite difference schemes,” Symmetry, vol. 14, no. 8, 1660, 2022.
• [38] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, second edition, Springer, New York.
• [39] S. Elaydi, An Introduction to Difference Equations, third edition, Springer, New York, 2005.
• [40] R. Ozarslan, and Y. Sekerci, “Fractional order oxygen–plankton system under climate change,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 30, no. 3, 2020.