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MATHEMATICAL AND STABILITY ANALYSIS OF FRACTIONAL ORDER MODEL FOR SPREAD OF PESTS IN TEA PLANTS

ZAIN UL ABADIN ZAFAR

Faculty of Information Technology, University of Central Punjab, Lahore, Pakistan

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ZAHIR SHAH

https://orcid.org/0000-0002-5539-4225

Department of Mathematics, University of Lakki Marwat, Lakki Marwat 28420, Khyber Pakhtunkhwa, Pakistan

E-mail Address: zahir@ulm.edu.pk

Corresponding author.

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NIGAR ALI

Department of Mathematics, University of Malakand, Pakistan

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EBRAHEEM O. ALZAHRANI

Department of Mathematics, Faculty of Science, King Abdulaziz, University, P. O. Box 80203, Jeddah 21589, Saudi Arabia

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, and

MESHAL SHUTAYWI

Department of Mathematics, College of Science & Arts, King Abdulaziz University, P. O. Box 344, Rabigh 21911, Saudi Arabia

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https://doi.org/10.1142/S0218348X21500080Cited by:17 (Source: Crossref)

Abstract

In this paper, the fractional order model for the binge of pests in tea plants is studied numerically. This model consists of three compartments such as tea plant, pest and predator. The local and non-local stability investigation of the system is also deliberated. The Grunwald–Letnikov (GL coefficients) method and generalized Euler method (GEM) are used to elucidate and simulate the proposed system. We have obtained stability conditions for equilibrium points, provided a numerical example, and proved our results. The results illustrate the concentrations of tea plants, pests, and predators all reach their equilibrium values as time passes. An imperative feature of this model is that it controls the motion at which the solution to equilibrium is grasped.

Keywords:

References

1. B. Han and Z. Chen , Behavioral and electrophysiological responses of natural enemies to synonymous from tea shoots and kairomones from tea aphids, Toxoptera aurantii, J. Chem. Ecol. 28 (2002) 2203–2219. Crossref, Web of Science, Google Scholar
2. M. El-shahed and I. Elsony , A fractional-order model for the spread of pests in tea plants, Adv. Anal. 1(2) (2016) 68–79. Google Scholar
3. L. Debnath , Recent applications of fractional calculus to science and engineering, Int. J. Math. Math. Sci. 54 (2003) 3413–3442. Crossref, Google Scholar
4. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo , Theory and Applications of Fractional Differential Equations, Vol. 204 (Elsevier Science, Amsterdam, The Netherlands, 2006). Google Scholar
5. I. Podlubny , Fractional Differential Equations (Academic Press, New York, NY, USA, 1999). Google Scholar
6. S. Roy, G. Handique, N. Muraleedharan, K. Dashora, S. M. Roy, A. Mukhopadhyay and A. Babu, Use of plant extracts for tea pest management in India, Appl. Microbiol. Biotechnol. doi:10.1007/s00253-016-7522-8. Google Scholar
7. A. Maiti, A. K. Pal and G. P. Samanta , Usefulness of biocontrol of pests in tea: a mathematical model, Math. Model. Nat. Phenom. 3 (2008) 96–113. Crossref, Web of Science, Google Scholar
8. E. Leah , Mathematical Models in Biology, Society for Industrial and Applied Mathematics (Random House, New York, 2005). Google Scholar
9. S. Zibaei and M. Namjoo , A non-standard finite difference scheme for solving a fractional-order model of HIV-1 Infection of CD4 $^{+}$ $^{+}$ T-Cells, Int. J. Mob. Commun. 6(2) (2015) 169–184. Google Scholar
10. R. E. Mickens , Advances in the Applications of Nonstandard Finite Difference Schemes (Wiley-Interscience, Singapore, 2005). Link, Google Scholar
11. R. E. Mickens , Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition, Numer. Methods D E 23(3) (2007) 672–691. Crossref, Web of Science, Google Scholar
12. S. Bushnaq, S. Ali, K. Shah and M. Arif , Approximate solutions to nonlinear fractional order partial differential equations arising in ion-acoustic waves, AIMS Math. 4(3) (2019) 721–739. Crossref, Web of Science, Google Scholar
13. H. Khan, M. Arif, S. T. Mohyud-Din and S. Bushnaq , Numerical solutions to systems of fractional Voltera Integro differential equations, using Chebyshev wavelet method, J. Taibah Univ. Sci. 12(5) (2018) 584–591. Crossref, Web of Science, Google Scholar
14. S. Ali, S. Bushnaq, K. Shah and M. Arif , Numerical treatment of fractional order Cauchy reaction diffusion equations, Chaos, Soliton Fractals 103 (2017) 578–587. Crossref, Web of Science, Google Scholar
15. E. Bonyah, A. Atangana and A. A. Elsadany , A fractional model for predator-prey with omnivore, Chaos 29, 013136 (2019), https://doi.org/10.1063/1.5079512. Crossref, Web of Science, Google Scholar
16. L. Ali, S. Islam, T. Gul, M. A. Khan and E. Bonyah , Solutions of nonlinear real world problems by a new analytical technique, Heliyon 4(11) (2018) e00913. Crossref, Web of Science, Google Scholar
17. S. C. Das and K. C. Barua , Scope of bio-control of pests and diseases in tea plantations, in Proceedings of the Eighth International Conference, Tea Research Global Perspective, Calcutta, 11–12 January 1990, pp. 119–125. Google Scholar
18. M. Mochizuki , Effectiveness and pesticide susceptibility of the pyrethroid-resistant predatory mite Amblyseius womersleyi in the integrated pest management of tea pests, BioControl 48(2) (2003) 207–221. Crossref, Web of Science, Google Scholar
19. A. Takafuji, A. Ozawa, H. Nernoto and T. Gotoh , Spider mites of Japan: Their biology and control, Exp. Appl. Carol. 24(5–6) (2000) 319–335. Crossref, Web of Science, Google Scholar
20. F. Takahashi, M. Inoue and A. Takafuji , Management of the spider-mite population in a vinylhouse vinery by releasing Phytoseiulus persimilis Athias-Henriot onto the ground cover, Jpn. J. Appl. Entomol. Zool. 42(2) (1998) 71–76. Crossref, Web of Science, Google Scholar
21. X. Wang, J. Yang and X. Li , Dynamics of a Heroin Epidemic Model with very Population, Appl. Math. 2 (2011) 732–738. Crossref, Google Scholar
22. R. E. Mickens , Numerical Integration of population models satisfying conservation laws: NSFD methods, J. Biol. Dyn. 1(4) (2007) 427–436. Crossref, Google Scholar
23. K. Shah, H. Khalil and R. A. Khan , Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations, Chaos, Soliton Fractals 77 (2015) 240–246. Crossref, Web of Science, Google Scholar
24. Z. U. A. Zafar, K. Rehan, M. Mushtaq and M. Rafiq , Numerical modelling for nonlinear biochemical reaction networks, Iran. J. Math. Chem. 8(4) (2017) 413–423. Web of Science, Google Scholar
25. Z. U. A. Zafar, K. Rehan and M. Mushtaq , Fractional-order scheme for bovine babesiosis disease and tick population, Adv. Differ. Equ. (NY) 86 (2017). Google Scholar
26. Z. U. A. Zafar, M. Mushtaq and K. Rehan , A non-integer order dengue internal transmission model, Adv. Differ. Equ. (NY) 23 (2018). Google Scholar
27. D. Matignon , Stability results for fractional differential equations with applications to control processing, Computational Engineering in Systems and Applications Multi-Conference, Lille, France, Vol. 2 (1996), pp. 963–968. Google Scholar
28. S. Kyu Choi, B. Kang and N. Koo , Stability for Caputo Fractional Differential Systems, Abstract and Applied Analysis (Hindawi Publishing Corporation, 2014), Article ID: 631419. Google Scholar
29. S. Ahmad and M. R. M. Rao , Theory of Ordinary Differential Equations, with Applications in Biology and Engineering (East West Press, New Delhi, 1999). Google Scholar
30. F. Keshtkar, H. Kheiri and G. H. Erjaee , Fixed points classification of nonlinear fractional differential equations as a dynamical system, J. Fract. Calc. Appl. 5(2) (2014) 59–70. Google Scholar
31. Z. U. A. Zafar, N. Ali, Z. Shah, W. Deebani, P. Roy and G. Zaman , Hopf Bifurcation and Global Dynamics of Time Delayed Dengue Model, Comput. Methods Prog. Biol. 195 (2020) 105530. Crossref, Web of Science, Google Scholar
32. Z. Odibat and S. Moamni , An algorithm for the numerical solution of differential equations of fractional order, J. Appl. Math. Inform. 26 (2008) 15–27. Google Scholar
33. Z. Odibat and N. Shawagfeh , Generalized Taylor’s formula, Appl. Math. Comput. 186 (2007) 286–293. Crossref, Web of Science, Google Scholar