Advances in Analysis
A Fractional-Order Model for the Spread of Pests in Tea Plants
- Moustafa El-Shahed*
Department of Mathematics, Faculty of Arts and Sciences, Qassim University, P.O. Box 3771, Qassim, Unizah 51911, Saudi Arabia.
- A. M. Ahmed
Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City, P.O. Box: 11884, Cairo, Egypt
- Ibrahim. M. E. Abdelstar
Quantitative Methods Unit, Faculty of Business and Economics, Qassim University, P.O. Box 6633, Qassim, Buraidah 51452, Saudi Arabia
 E. Ahmed, A. M. A. El-Sayed, E. M. El-Mesiry and H. A. A. El-Saka; Numerical solution for the fractional replicator equation, IJMPC, 16 (2005), 1.9.
 E. Ahmed, A. M. A. El-Sayed, H. A. A. El-Saka; On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rossler, Chua and Chen systems, Physics Letters A, 358 (2006), 1.4.
 E. Ahmed, A. M. A. El-Sayed, H. A. A. El-Saka; Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models, J. Math. Anal. Appl, 325 (2007), 542.553.
 M. S. Abd-Elouahab, N. E. Hamri, and J. Wang, Chaos control of a fractional-order financial system, Mathematical Problems in Engineering, 2010 (2010), 18 pages.
 R. M. Anderson; R. M. May; Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991.
 R. S. Barbosa, J. A. T. MacHado, B. M. Vinagre, and A. J. Calderón, Analysis of the van der Pol oscillator containing derivatives of fractional order, Journal of Vibration and Control, 13 (2007), 1291.1301.
 D. Cafagna and G. Grassi, Fractional-order Chua.s circuit: time-domain analysis, bifurcation, chaotic behavior and test for chaos, International Journal of Bifurcation and Chaos, 18 (2008), 615.639.
 S. C. Das and K. C. Bar Ua, Scope of bio-control of pests and diseases in tea plantations, Tea Research Association, Tocklai Experimental Station, Jorhat- 785008, Assam.
 L. Debnath; Recent applications of fractional calculus to science and engineering, International Journal of Mathematics and Mathematical Sciences, 54 (2003), 3413. 3442.
 K. Diethelm, N. J. Ford; Analysis of fractional differential equations, J Math Anal Appl, 256 (2002), 229.248.
 K. Diethelm, N. J. Ford, A.D. Freed; A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn, 29 (2002), 3.22.
 M. Elshahed and A. Alsaedi; The Fractional SIRC model and Influenza A, Mathematical Problems in Engineering, Article ID 480378 (2011), 1.9.
 M. Elshahed, F. Abd El-Naby, Fractional calculus model for childhood diseases and vaccine Applied Mathematical Sciences, 8 (2014), 4859 - 4866.
 R. Garrappa, Trapezoidal methods for fractional differential equations: Theoretical and computational aspects, Mathematics and Computers in Simulation, 110 (2015), 96.112.
 B. Han, Chen. Z., Behavioral and electrophysiological responses of natural enemies to synonymous from tea shoots and kairomones from tea aphids, Toxoptera aurantii, J Chem Ecol. 28 (2002), 2203-19.
 H. Jonathan Crane and Carlos F. Balerdi, Tea Growing in the Florida Home Land-scape, University of Florida. Original publication, 2005.
 E. Kaslik and S. Sivasundaram, Nonlinear dynamics and chaos in fractional-order neural networks, Neural Networks, 32 (2012), 245.256.
 E. Kaslik and S. Sivasundaram, Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions, Nonlinear Analysis: Real World Applications, 13 (2012), 1489.1497 .
 A. A. Kilbas.; H. M. Srivastava.; and J. J. Trujillo.; Theory and Applications of Fractional Differential Equations, Elsevier Science, Amsterdam, The Netherlands, 204 (2006).
 S. Kyu Choi, B. Kang, and N. Koo, Stability for Caputo Fractional Differential Systems, Hindawi Publishing Corporation, Abstract and Applied Analysis, 2014 (2014), 6 pages.
 C. Li, C. Tao; On the Fractional Adams method, Computers and Mathematics with Applications, 58 (2009), 1573.1588.
 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.
 D. Matignon; Stability results for fractional differential equations with applications to control processing, Computational Engineering in Systems and Applications, Multi-conference, 2 (1996), 963-968.
 A. E. Matouk, A. A. Elsadany, E. Ahmed, H. N. Agiza, Dynamical behavior of fractional-order Hastings.Powell food chain model and its discretization, Communications in Nonlinear Science and Numerical Simulation 27 (2015), 153-167.
 A. E. Matouk, A. A. Elsadany, Dynamical behaviors of fractional-order Lotka. Volterra predator-prey model and its discretization, Journal of Applied Mathematics and Computing 49 (2015), 269-283.
 A. E. Matouk, A. A. Elsadany, Dynamical analysis, stabilization and discretization of a chaotic fractional-order GLV model, http://link.springer.com/article/10.1007/s11071-016-2781-6., 2015
 M.S.A. Mamun and M. Ahmed, Prospect of indigenous plant extracts in tea pest management, Int. J. Agril. Res. Innov. & Tech. 1, 1&2 (2011): 16-23,
 I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA (1999).
 M. S. Tavazoei and M. Haeri, A proof for nonexistence of periodic solutions in time invariant fractional order systems, Automatica, 45 (2009), 1886.1890.
 C. Vargas-De-León, Volterra-type Lyapunov functions for fractional-order epidemic systems, Commun Nonlinear Sci Numer Simulat, 24 (2015), 75.85