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

References

[1] A. A. Berryman, The origins and evolution of predator–prey theory, Ecology. 73 (1992), 1530–1535.10.2307/1940005Suche in Google Scholar

[2] A. M. A. El-Sayed, Fractional-order diffusion-wave equation, Int. J. Theor. Phys. 35 (1996), 311–322.10.1007/BF02083817Suche in Google Scholar

[3] A. E. M. El-Misiery, E. Ahmed, On a fractional model for earthquakes, Appl. Math. Comput. 178 (2006), 207–211.10.1016/j.amc.2005.10.011Suche in Google Scholar

[4] F. C. Meral, T. J. Royston, R. Magin, Fractional calculus in viscoelasticity: An experimental study, Commun. Nonlinear. Sci. Numer. Simul. 15 (2010), 939–945.10.1016/j.cnsns.2009.05.004Suche in Google Scholar

[5] D. A. Benson, M. M. Meerschaert, J. Revielle, Fractional calculus in hydrologic modeling: A numerical perspective, Adv. Water. Res. 51 (2013), 479–497.10.1016/j.advwatres.2012.04.005Suche in Google Scholar PubMed PubMed Central

[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), 63–74.10.1016/j.cnsns.2012.06.017Suche in Google Scholar

[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), 396–406.10.1016/j.cnsns.2014.08.032Suche in Google Scholar

[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), 588–601.10.1177/0959651813492326Suche in Google Scholar

[9] M. P. Aghababa, Fractional modeling and control of a complex nonlinear energy supply-demand system, Complexity 20 (2015), 74–86.10.1002/cplx.21533Suche in Google Scholar

[10] M. P. Aghababa, M. Borjkhani, Chaotic fractional-order model for muscular blood vessel and its control via fractional control scheme, Complexity 20 (2014), 37–46.10.1002/cplx.21502Suche in Google Scholar

[11] E. Ahmed, A. S. Elgazzar, On fractional order differential equations model for nonlocal epidemics, Phys A. 379 (2007), 607–614.10.1016/j.physa.2007.01.010Suche in Google Scholar

[12] R. L. Bagley, R. A. Calico, Fractional order state equations for the control of viscoelastically damped structures, J. Guid. Control Dyn. 14 (1991), 304–311.10.2514/3.20641Suche in Google Scholar

[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), 441–44410.1109/TAC.1984.1103551Suche in Google Scholar

[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), 253–265.10.1016/S0022-0728(71)80115-8Suche in Google Scholar

[15] A. M. Yousef, S. M. Salman, Backward Bifurcation in a Fractional-Order SIRS Epidemic Model with a Nonlinear Incidence Rate, IJNSNS 17 (2016), 343–420.10.1515/ijnsns-2016-0036Suche in Google Scholar

[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), 445–451.10.1016/j.joems.2017.06.003Suche in Google Scholar

[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), 817–823.10.1016/j.aml.2006.08.013Suche in Google Scholar

[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), 542–553.10.1016/j.jmaa.2006.01.087Suche in Google Scholar

[19] O. Heaviside, Electromagnetic Theory, Chelsea, New York, 1971.Suche in Google Scholar

[20] D. Kusnezov, A. Bulgac, G. D. Dang, Quantum levy processes and fractional kinetics, Phys. Rev. Lett. 82 (1999), 1136–1139.10.1103/PhysRevLett.82.1136Suche in Google Scholar

[21] M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geophys. J. R. Astron. Soc. 13 (1967), 529–539.10.1111/j.1365-246X.1967.tb02303.xSuche in Google Scholar

[22] F. Ben Adda, Geometric interpretation of the fractional derivative. J. Fract. Calc. 11 (1997), 21–52.Suche in Google Scholar

[23] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal. 5 (2002), 367–386.Suche in Google Scholar

[24] E. N. Lorenz, Deterministic non-periodic flows, J. Atmos. Sci. 20 (1963), 130–141.10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2Suche in Google Scholar

[25] D. Jana, Chaotic dynamics of a discrete predator-prey system with prey refuge, Appl. Math. Comput. 224 (2013), 848–865.10.1016/j.amc.2013.09.001Suche in Google Scholar

[26] H. N. Agiza, A. E. Matouk, Adaptive synchronization of Chua’s circuits with fully unknown parameters, Chaos Soliton Fractals 28 (2006), 219–227.10.1016/j.chaos.2005.05.055Suche in Google Scholar

[27] A. E. Matouk, Dynamical analysis, feedback control and synchronization of Liu dynamical system, Nonlinear Anal. Theory Methods Appl. 69 (2008), 3213–3224.10.1016/j.na.2007.09.029Suche in Google Scholar

[28] A. E. Matouk, H. N. Agiza, Bifurcations, chaos and synchronization in ADVP circuit with parallel resistor, J. Math. Anal. Appl. 341 (2008), 259–269.10.1016/j.jmaa.2007.09.067Suche in Google Scholar

[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), 116–119.10.1016/j.nonrwa.2007.08.029Suche in Google Scholar

[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.Suche in Google Scholar

[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.Suche in Google Scholar

[32] A. A. Elsadany, Competition analysis of a triopoly game with bounded rationality, Chaos Solitons Fractals. 45 (2012), 1343–1348.10.1016/j.chaos.2012.07.003Suche in Google Scholar

[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), 185–202.10.1007/978-3-642-34017-8_7Suche in Google Scholar

[34] E. M. Elabbasy, A. A. Elsadany, Y. Zhang, Bifurcation analysis and chaos in a discrete reduced Lorenz system, Appl. Math. Comput. 228 (2014), 184–194.10.1016/j.amc.2013.11.088Suche in Google Scholar

[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), 269–283.10.1007/s12190-014-0838-6Suche in 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), 153–167.10.1016/j.cnsns.2015.03.004Suche in Google Scholar

[37] A. E. Matouk, A. A. Elsadany, Dynamical analysis, stabilization and discretization of a chaotic fractional-order GLV model, Nonlin. Dynam. 85 (2016), 1597–1612.10.1007/s11071-016-2781-6Suche in Google Scholar

[38] A. S. Hegazi, A. E. Matouk, Dynamical behaviors and synchronization in the fractional order hyperchaotic Chen system, Appl. Math. Lett. 24 (2011), 1938–1944.10.1016/j.aml.2011.05.025Suche in Google Scholar

[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), 1193–1202.10.1016/j.cnsns.2012.09.026Suche in Google Scholar

[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), 30–35.10.1016/j.aml.2013.10.010Suche in Google Scholar

[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.10.1155/2017/7476090Suche in 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.10.1142/S0218127416502229Suche in Google Scholar

[43] A. J. Lotka, Elements of Physical Biology. Williams and Wilkins, Baltimore, 1925.Suche in Google Scholar

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

[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), 1–60.10.4039/entm9745fvSuche in Google Scholar

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

[47] W. Sokol, J. A. Howell, kinetics of phenol oxidation by washed cells, Biotechnol. Bioeng. 23 (1980), 2039–2049.10.1002/bit.260230909Suche in Google Scholar

[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.Suche in Google Scholar

[49] Z. F. El-Raheem, S. M. Salman, On a discretization process of fractional-order logistic differential equation. J. Egypt. Math. Soc. 22 (2014), 407–412.10.1016/j.joems.2013.09.001Suche in Google Scholar

[50] D. Matignon, Stability results for fractional differential equations with applications to control processing, Comput. Eng. Syst. Appl. 2 (1996), 963.Suche in Google Scholar

[51] X. Liu, D. Xiao, Complex dynamic behaviors of a discrete-time predator-prey system, Chaos Solitons Fractals. 32 (2007), 80–94.10.1016/j.chaos.2005.10.081Suche in Google Scholar

[52] S. Elaydi, Discrete Chaos, second edition: with applications in science and engineering, Chapman and Hall/CRC, Boca Raton, 2008.Suche in Google Scholar