DYNAMICAL ANALYSIS OF FRACTIONAL-ORDER PREDATOR-PREY MODEL WITH PREY REFUGE AND ADDITIONAL FOOD FOR PREDATOR

A.S. Zulaikha, S. Anam

IJPAM: Volume 118, No. 3 (2018)

Title

DYNAMICAL ANALYSIS OF FRACTIONAL-ORDER
PREDATOR-PREY MODEL WITH PREY REFUGE
AND ADDITIONAL FOOD FOR PREDATOR

Authors

Agus Suryanto Zulaikha

, Syaiful Anam

$^{1,2}$ Departement of Mathematics
Brawijaya University
Jl. Veteran Malang 65145, INDONESIA

Abstract

In this work, a fractional-order predator-prey model with prey refuge and additional food for predator is presented. First, the non-negativity, boundedness, existence and uniqueness of the solution for the model are discussed. We also determine the equilibrium points and their existence conditions as well as their stability behaviour, both locally and globally. Some numerical simulations are performed to illustrate analytical results.

History

Received: 2017-12-07
Revised: 2018-02-05
Published: April 15, 2018

AMS Classification, Key Words

AMS Subject Classification: 34A08, 34D23, 92D25
Key Words and Phrases: fractional-order model, prey refuge, additional food, global stability, Lyapunov function, Grünwald-Letnikov method

Download Section

Download paper from here.
You will need Adobe Acrobat reader. For more information and free download of the reader, see the Adobe Acrobat website.

Bibliography

1: J. D. Murray, Mathematical Biology: An Introduction, 3rd Ed., Springer-Verlag, Inc. Berlin (2005), 507-508.
2: A. Sih, Prey refuges and predator-prey stability, Theoretical Population Biology 31 (1987), 1-12, DOI: https://doi.org/10.1016/0040-5809(87)90019-0.
3: Y. Huang, F. Chen and L. Zhong, Stability analysis of a prey-predator model with Holling type III response function incorporating a prey refuge, Applied Mathematics and Computation 182 (2006), 672-683, DOI: 10.1016/j.amc.2006.04.030.
4: J. P. Tripathi, S. Abbas and M. Thakur, Dynamical analysis of a prey-predator model with Beddington-DeAngelis type function response incorporating a prey refuge, Nonlinear Dynamics 80 (2015), 177-196, DOI: 10.1007/s11071-014-1859-2.
5: S. Samanta, R. Dhar, I. M. Elmojtaba and J. Chattopadhyay, The role of additional food in a predator-prey model with a prey refuge, Journal of Biological Systems 24(2) (2016), 1-21, DOI: 10.1142/S0218339016500182.
6: P. D. N. Srinivasu, B. S. R. V. Prasad and M. Venkatesulu, Biological control through provision of additional food to predators, Theoretical Population Biology 72 (2007), 111-120, DOI: https://doi.org/10.1016/j.tpb.2007.03.011.
7: B. Sahoo, Global Stability of predator prey system with alternative prey, Biotechnology (2012), Article ID: 898039, DOI: http://dx.doi.org/10.5402/2013/898039.
8: B. S. R. V. Prasad, M. Banerjee and P. D. N. Srinivasu, Dynamics of additional food provided predator-prey system with mutually interfering predators, Mathematical Biosciences 246(1) (2013), 176-190, DOI: https://doi.org/10.1016/j.mbs.2013.08.013.
9: M. Sen, P. D. N. Srinivasu and M. Banerjee, Global dynamics of an additional food provided predator-prey system with constant harvest in predators, Applied Mathematics and Computation 250 (2015), 193-211, DOI: https://doi.org/10.1016/j.amc.2014.10.085.
10: H. M. Ulfa, A. Suryanto and I. Darti, Dynamics of Leslie-Gower predator-prey model with additional food for predators, International Journal of Pure and Applied Mathematics (IJPAM) 115(2) (2017), 199-209, DOI: 10.12732/ijpam.v115i2.1.
11: J. Ghosh, B. Sahoo and S. Poria. Prey-predator dynamics with prey refuge providing additional food to predator, Chaos Solitons and Fractals 96 (2017), 110-119, DOI: https://doi.org/10.1016/j.chaos.2017.01.010.
12: E. Ahmed, A.M.A. El-Sayed and H.A.A. El-Saka, Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models, Journal of Mathematical Analysis and Applications 325 (2007), 542-553, DOI: https://doi.org/10.1016/j.jmaa.2006.01.087.
13: A. Suryanto, I. Darti and S. Anam, Stability analysis of a fractional order modified Leslie-Gower model with additive Allee Effect, International Journal of Mathematics and Mathematical Sciences (2017), Article ID: 8273430, DOI: https://doi.org/10.1155/2017/8273430.
14: K. Nugraheni, A. Suryanto and Trisilowati, Dynamics of a fractional order Eco-Epidemiological model, Journal of Tropical Life Science 7(3) (2017), 243-250, DOI: http://dx.doi.org/10.11594/jtls.07.03.09.
15: H. Li, L. Zhang, C. Hu, J. Yao-Lin and Z. Teng, Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge, Journal of Applied Mathematics and Computing 64 (2016), 435-449, DOI: 10.1007/s12190-016-1017-8.
16: Y. Li, Y. Chen and I. Podlubny, Stability of fractional-order nonlinear dynamics: Lyapunov direct method and generalized Mittag-Leffler stability, Computers & Mathematics with Applications 59 (2010), 1810-1821, DOI: 10.1016/j.camwa.2009.08.019.
17: I. Petras, Fractional-Order Nonlinear Systems, Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg (2011).
18: Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, vol. 1929 of Lecture Notes in Mathematics, Springer, Berlin, Germany, (2008).
19: Z. M. Odibat and N. T. Shawagfeh, Generalized Taylors formula, Applied Mathematics and Computation 186 (2007), 286-293, DOI: 10.1016/j.amc.2006.07.102.
20: D. Matignon, Stability results for fractional differential equations with applications to control processing, in Computational Engineering in Systems Applications (1996), 963-968.
21: C. V. De-León, Volterra-type lyapunov functions for fractional-order epidemic systems, Communications in Nonlinear Science and Numerical Simulation 24 (2015), 75-85, DOI: 10.1016/j.cnsns.2014.12.013.
22: J. Huo, H. Zhao and L. Zhu, The effect of vaccines on backward bifurcation in a fractional order HIV model, ISSN: 1468-1218 Nonlinear Analysis: Real World Applications 26 (2015), 289-305, DOI: 10.1016/j.nonrwa.2015.05.014.
23: A. Kilbas, H. Srivastava and J. Trujillo, Theory and Application of Fractional Differential Equations, Elsevier, New York (2006).
24: I. Darti and A. Suryanto, Stability preserving non-standard finite difference scheme for a harvesting Leslie–Gower predator–prey model, Journal of Difference Equations and Applications 21 (2015), 528-534, DOI: 10.1080/10236198.2015.1029922.
25: I. Darti and A. Suryanto, Dynamics preserving nonstandard finite difference method for the modified leslie-gower predator-prey model with holling-type II functional response, Far East Journal of Mathematical Sciences 99 (2016), 615 - 774, DOI: 10.17654/MS099050719.
26: T. Fayeldi, A. Suryanto and A. Widodo, Dynamical behaviors of a discrete SIR epidemic model with nonmonotone incidence rate, International Journal of Applied Mathematics and Statistics 47 (2013), 416-423.

How to Cite?

DOI: 10.12732/ijpam.v118i3.21 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2018
Volume: 118
Issue: 3
Pages: 751 - 765

Google Scholar; DOI (International DOI Foundation); WorldCAT.

CC BY This work is licensed under the Creative Commons Attribution International License (CC BY).