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International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Angheluta-Bauer, Luiza / Chen, Xi / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi / Yang, Xu


IMPACT FACTOR 2017: 1.162

CiteScore 2018: 1.11

SCImago Journal Rank (SJR) 2018: 0.288
Source Normalized Impact per Paper (SNIP) 2018: 0.510

Mathematical Citation Quotient (MCQ) 2017: 0.12

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2191-0294
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Volume 20, Issue 2

Issues

Dynamical Behaviors of a Fractional-Order Predator–Prey Model with Holling Type IV Functional Response and Its Discretization

A. M. Yousef / S. Z. Rida / Y. Gh. Gouda / A. S. Zaki
Published Online: 2019-01-26 | DOI: https://doi.org/10.1515/ijnsns-2017-0152

Abstract

In this paper, we investigate the dynamical behaviors of a fractional-order predator–prey with Holling type IV functional response and its discretized counterpart. First, we seek the local stability of equilibria for the fractional-order model. Also, the necessary and sufficient conditions of the stability of the discretized model are achieved. Bifurcation types (include transcritical, flip and Neimark–Sacker) and chaos are discussed in the discretized system. Finally, numerical simulations are executed to assure the validity of the obtained theoretical results.

Keywords: predator–prey system; fractional calculus,discretization; functional response; bifurcations; chaos

MSC 2010: 26A33; 34C23; 37D45

References

  • [1]

    A. A. Berryman, The origins and evolution of predator–prey theory, Ecology. 73 (1992), 1530–1535.CrossrefGoogle Scholar

  • [2]

    A. M. A. El-Sayed, Fractional-order diffusion-wave equation, Int. J. Theor. Phys. 35 (1996), 311–322.CrossrefGoogle Scholar

  • [3]

    A. E. M. El-Misiery, E. Ahmed, On a fractional model for earthquakes, Appl. Math. Comput. 178 (2006), 207–211.

  • [4]

    F. C. Meral, T. J. Royston, R. Magin, Fractional calculus in viscoelasticity: An experimental study, Commun. Nonlinear. Sci. Numer. Simul. 15 (2010), 939–945.CrossrefGoogle Scholar

  • [5]

    D. A. Benson, M. M. Meerschaert, J. Revielle, Fractional calculus in hydrologic modeling: A numerical perspective, Adv. Water. Res. 51 (2013), 479–497.Crossref

  • [6]

    A. Sapora, P. Cornetti, A. Carpinteri, Wave propagation in nonlocal elastic continua modelled by a fractional calculus approach, Commun. Nonlinear. Sci. Numer. Simul. 18 (2013), 63–74.Crossref

  • [7]

    J. A. Tenreiro Machado, M. E. Mata, Pseudo phase plane and fractional calculus modeling of western global economic downturn, Commun. Nonlinear Sci. Numer. Simul. 22 (2015), 396–406.Crossref

  • [8]

    M. P. Aghababa, H. P. Aghababa, The rich dynamics of fractional-order gyros applying a fractional controller. Proc IMechE Part I, J. Syst. Control Eng. 227 (2013), 588–601.Crossref

  • [9]

    M. P. Aghababa, Fractional modeling and control of a complex nonlinear energy supply-demand system, Complexity 20 (2015), 74–86.Crossref

  • [10]

    M. P. Aghababa, M. Borjkhani, Chaotic fractional-order model for muscular blood vessel and its control via fractional control scheme, Complexity 20 (2014), 37–46.Crossref

  • [11]

    E. Ahmed, A. S. Elgazzar, On fractional order differential equations model for nonlocal epidemics, Phys A. 379 (2007), 607–614.Crossref

  • [12]

    R. L. Bagley, R. A. Calico, Fractional order state equations for the control of viscoelastically damped structures, J. Guid. Control Dyn. 14 (1991), 304–311.Crossref

  • [13]

    H. H. Sun, A. A. Abdelwahab, B. Onaral, Linear approximation of transfer function with a pole of fractional order. IEEE Trans. Autom. Control. 29 (1984), 441–444Crossref

  • [14]

    M. Ichise, Y. Nagayanagi, T. Kojima, An analog simulation of noninteger order transfer functions for analysis of electrode process, J. Electroanal. Chem. 33 (1971), 253–265.Crossref

  • [15]

    A. M. Yousef, S. M. Salman, Backward Bifurcation in a Fractional-Order SIRS Epidemic Model with a Nonlinear Incidence Rate, IJNSNS 17 (2016), 343–420.

  • [16]

    S. M. Salman, A. M. Yousef, On a fractional-order model for HBV infection with cure of infected cells, J. Egypt. Math. Soc. 25 (2017), 445–451.Crossref

  • [17]

    A. M. A. El-Sayed, A. E. M. El-Mesiry, H. A. A. El-Saka, On the fractional-order logistic equation. Appl. Math. Lett. 20 (2007), 817–823.Crossref

  • [18]

    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.Crossref

  • [19]

    O. Heaviside, Electromagnetic Theory, Chelsea, New York, 1971.

  • [20]

    D. Kusnezov, A. Bulgac, G. D. Dang, Quantum levy processes and fractional kinetics, Phys. Rev. Lett. 82 (1999), 1136–1139.Crossref

  • [21]

    M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geophys. J. R. Astron. Soc. 13 (1967), 529–539.Crossref

  • [22]

    F. Ben Adda, Geometric interpretation of the fractional derivative. J. Fract. Calc. 11 (1997), 21–52.

  • [23]

    I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal. 5 (2002), 367–386.

  • [24]

    E. N. Lorenz, Deterministic non-periodic flows, J. Atmos. Sci. 20 (1963), 130–141.Crossref

  • [25]

    D. Jana, Chaotic dynamics of a discrete predator-prey system with prey refuge, Appl. Math. Comput. 224 (2013), 848–865.

  • [26]

    H. N. Agiza, A. E. Matouk, Adaptive synchronization of Chua’s circuits with fully unknown parameters, Chaos Soliton Fractals 28 (2006), 219–227.Crossref

  • [27]

    A. E. Matouk, Dynamical analysis, feedback control and synchronization of Liu dynamical system, Nonlinear Anal. Theory Methods Appl. 69 (2008), 3213–3224.Crossref

  • [28]

    A. E. Matouk, H. N. Agiza, Bifurcations, chaos and synchronization in ADVP circuit with parallel resistor, J. Math. Anal. Appl. 341 (2008), 259–269.Crossref

  • [29]

    H. N. Agiza, E. M. Elabbasy, H. EL-Metwally, A. A. Elsadany, Chaotic dynamics of a discrete prey-predator model with Holling type II, Nonlinear Anal. Real World Appl. 10 (2009), 116–119.Crossref

  • [30]

    E. M. Elabbasy, H. N. Agiza, H. A. El-Metwally, A. A. Elsadany, Bifurcation analysis, chaos and control in the Burgers mapping, Int. J. Nonlinear Sci. 4 (2007), 171–185.

  • [31]

    A. A. Elsadany, H. A. El-Metwally, E. M. Elabbasy, H. N. Agzia, Chaos and bifurcation of a nonlinear discrete prey-predator system, Comput. Ecol. Softw. 2 (2012), 169–180.Google Scholar

  • [32]

    A. A. Elsadany, Competition analysis of a triopoly game with bounded rationality, Chaos Solitons Fractals. 45 (2012), 1343–1348.Crossref

  • [33]

    A. S. Hegazi, A. E. Matouk, Chaos synchronization of the modified autonomous Van der Pol-Duffing circuits via active control, Appl. Chaos Nonlinear Dynam. Sci. Eng. 3 (2013), 185–202.

  • [34]

    E. M. Elabbasy, A. A. Elsadany, Y. Zhang, Bifurcation analysis and chaos in a discrete reduced Lorenz system, Appl. Math. Comput. 228 (2014), 184–194.

  • [35]

    A. A. Elsadany, A. E. Matouk, Dynamical behaviors of fractional-order Lotka-Volterra predator-prey model and its discretization, Appl. Math. Comput. 49 (2015), 269–283.Google Scholar

  • [36]

    A. E. Matouk, A. A. Elsadany, E. Ahmed, H. N. Agiza, Dynamical behavior of fractional-order Hastings-Powell food chain model and its discretization, Commun. Nonlinear Sci. Numer. Simul. 27 (2015), 153–167.Crossref

  • [37]

    A. E. Matouk, A. A. Elsadany, Dynamical analysis, stabilization and discretization of a chaotic fractional-order GLV model, Nonlin. Dynam. 85 (2016), 1597–1612.Crossref

  • [38]

    A. S. Hegazi, A. E. Matouk, Dynamical behaviors and synchronization in the fractional order hyperchaotic Chen system, Appl. Math. Lett. 24 (2011), 1938–1944.Crossref

  • [39]

    A. S. Hegazi, E. Ahmed, A. E. Matouk, On chaos control and synchronization of the commensurate fractional order Liu system, Commun. Nonlinear Sci. Numer. Simul. 18 (2013), 1193–1202.Crossref

  • [40]

    A. E. Matouk, A. A. Elsadany, Achieving synchronization between the fractional-order hyperchaotic Novel and Chen systems via a new nonlinear control technique, Appl. Math. Lett. 29 (2014), 30–35.Crossref

  • [41]

    A. Elsaid, D. F. M. Torres, S. Bhalekar, A. Elsadany, A. Elsonbaty, Hyperchaotic fractional-Order Systems and Their Applications, Complexity. 2017 (2017), 1 page.Google Scholar

  • [42]

    A. M. A. El-Sayed, A. Elsonbaty, A. A. Elsadany, A. E. Matouk, Dynamical analysis and circuit simulation of a new fractional-order hyperchaotic system and its discretization, Int. J. Bifurcation Chaos. 26 (2016), 35pages.

  • [43]

    A. J. Lotka, Elements of Physical Biology. Williams and Wilkins, Baltimore, 1925.Google Scholar

  • [44]

    V. Volterra, Variazioni e fluttuazioni del numero di individui in specie animali conviventi. Mem. Acad. Lincei. 2 (1926), 31–113.

  • [45]

    C. S. Holling, The functional response of predator to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Can. 45 (1965), 1–60.

  • [46]

    H. I. Freedman, Deterministic mathematical models in population ecology. Marcel Dekker, New York, 1980.Google Scholar

  • [47]

    W. Sokol, J. A. Howell, kinetics of phenol oxidation by washed cells, Biotechnol. Bioeng. 23 (1980), 2039–2049.

  • [48]

    A. M. A. El-Sayed, S. M. Salman, On a discretization process of fractional order Riccatis differential equation, J. Fract. Calc. Appl. 4 (2013), 251–259.

  • [49]

    Z. F. El-Raheem, S. M. Salman, On a discretization process of fractional-order logistic differential equation. J. Egypt. Math. Soc. 22 (2014), 407–412.Crossref

  • [50]

    D. Matignon, Stability results for fractional differential equations with applications to control processing, Comput. Eng. Syst. Appl. 2 (1996), 963.

  • [51]

    X. Liu, D. Xiao, Complex dynamic behaviors of a discrete-time predator-prey system, Chaos Solitons Fractals. 32 (2007), 80–94.Crossref

  • [52]

    S. Elaydi, Discrete Chaos, second edition: with applications in science and engineering, Chapman and Hall/CRC, Boca Raton, 2008.

About the article

Received: 2017-07-09

Accepted: 2019-12-01

Published Online: 2019-01-26

Published in Print: 2019-04-26


Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 20, Issue 2, Pages 125–136, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2017-0152.

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