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.
References
[1] A. A. Berryman, The origins and evolution of predator–prey theory, Ecology. 73 (1992), 1530–1535.10.2307/1940005Search in Google Scholar
[2] A. M. A. El-Sayed, Fractional-order diffusion-wave equation, Int. J. Theor. Phys. 35 (1996), 311–322.10.1007/BF02083817Search 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.2005.10.011Search 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.2009.05.004Search 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.2012.04.005Search 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.2012.06.017Search 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.2014.08.032Search 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), 588–601.10.1177/0959651813492326Search 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.21533Search 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.21502Search 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.2007.01.010Search 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), 304–311.10.2514/3.20641Search 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.1984.1103551Search 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)80115-8Search 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-2016-0036Search 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.2017.06.003Search 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.2006.08.013Search 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.2006.01.087Search in Google Scholar
[19] O. Heaviside, Electromagnetic Theory, Chelsea, New York, 1971.Search 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.82.1136Search 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.xSearch in Google Scholar
[22] F. Ben Adda, Geometric interpretation of the fractional derivative. J. Fract. Calc. 11 (1997), 21–52.Search in Google Scholar
[23] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal. 5 (2002), 367–386.Search 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;2Search 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.2013.09.001Search 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.2005.05.055Search 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.2007.09.029Search 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.2007.09.067Search 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.2007.08.029Search 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.Search 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.Search 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.2012.07.003Search 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_7Search 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.2013.11.088Search 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/s12190-014-0838-6Search 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.2015.03.004Search 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/s11071-016-2781-6Search 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.2011.05.025Search 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.2012.09.026Search 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.2013.10.010Search 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.10.1155/2017/7476090Search 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/S0218127416502229Search in Google Scholar
[43] A. J. Lotka, Elements of Physical Biology. Williams and Wilkins, Baltimore, 1925.Search in Google Scholar
[44] V. Volterra, Variazioni e fluttuazioni del numero di individui in specie animali conviventi. Mem. Acad. Lincei. 2 (1926), 31–113.Search 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/entm9745fvSearch in Google Scholar
[46] H. I. Freedman, Deterministic mathematical models in population ecology. Marcel Dekker, New York, 1980.Search 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.260230909Search 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.Search 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.2013.09.001Search in Google Scholar
[50] D. Matignon, Stability results for fractional differential equations with applications to control processing, Comput. Eng. Syst. Appl. 2 (1996), 963.Search 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.2005.10.081Search in Google Scholar
[52] S. Elaydi, Discrete Chaos, second edition: with applications in science and engineering, Chapman and Hall/CRC, Boca Raton, 2008.Search in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston