International Journal of Nonlinear Analysis and Applications
Dynamical analysis, stability and discretization of fractional-order predator-prey model with negative feedback on two species
Document Type : Research Paper
Authors
1 Dep. of Applied Mathematics, Yadegar-e-Imam Khomeini (RAH), Shahr-rey Branch, Islamic Azad University, Tehran, Iran.
2 Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran
Abstract
The Lotka-Volterra model is an important model being employed in biological phenomena to investigate the nonlinear interaction among existing species. In this work, we first consider an integer order predator-prey model with negative feedback on both prey and predator. Then by introducing a fractional model into the existing one, we give them a specified memory. We also obtain its discretized counterpart. Finally, along with giving the biological interpretation of the system, the stability and dynamical analysis of the proposed model are investigated and the results are illustrated as well.
Keywords
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[26] G. Kotalczyk and F. Kruis, Fractional Monte Carlo time steps for the simulation of coagulation for parallelized flowsheet simulations, Chem. Eng. Res. Des. 136 (2018) 71–82.
[27] A. Matouk, Dynamical behaviors, linear feedback control and synchronization of the fractional-order Liu system, J. Nonlinear Syst. Appl. 1 (2010) 135–140.
[28] A. Matouk, Dynamical analysis, feedback control and synchronization of Liu dynamical system, Nonlinear AbalTheor. 69 (2008) 3213–3224.
[29] R.L. Magin, Fractional calculus in bioengineering, Begell House, 2006.
[30] F. Pitolli, A Fractional B-spline Collocation Method for the Numerical Solution of Fractional predator-prey Models, Fract. Fractional, 2 (2018) 1–16.
[31] L. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic Press, 1998.
[32] I. Petras, Fractional order nonlinear systems: modeling, analysis and simulation, Springer Science and Business Media, 2011.
[33] I. Petr´as, A note on the fractional-order Volta’s system, Commun. Nonlinear Sci. 15 (2010) 384-393.
[34] I. Petr´as, Chaos in the fractional-order Volta’s system: modeling and simulation. Nonlinear Dyn. 57 (2009) 157–170.
[35] M. Rahmani Doust, S. Gholizadeh, The Lotka-Volterra predator-prey Equations, CJMS, 3 (2014) 221–225.
[36] L.J. Sheua, H.K. Chen, J.H. Chen, L.M.Tam, W.C. Chen, K.T. Line and Y. Kang, Chaos in the Newton-Leipnik system with fractional-order, Chaos, Solitons and Fractals, 36 (2008) 98–103.
[37] N. Samardzija and L.D. Greller, Explosive route to chaos through a fractal torus in a generalized Lotka-Volterra model. Bull. Math. Biol. 50 (1988) 465–491.
[38] R. Satriyantara, A. Suryanto and N. Hidayat, Numerical Solution of a Fractional-Order predator-prey Model with Prey Refuge and Additional Food for Predator, J. Exp. Biol. 8 (2018) 66–70.
[39] N.T. Shawagfeh, Analytical approximate solutions for nonlinear fractional differential equations, J. Comput. Appl. Math. 131 (2002) 517-529.
[40] S. Vaidyanathan, Lotka-Volterra population biology models with negative feedback and their ecological monitoring, Int. J. PharmTech Res. 8 (2015) 974–981.
[41] S. Vaidyanathan, Adaptive biological control of generalized Lotka-Volterra three-species biological system, Int. J. Pharmtech Res. 8 (2015) 622–631.
[42] T. Zhou, Y. Tang and G. Chen, Chen’s attractor exists. Int. J. Bif. Chaos, 14 (2004) 3167–3177.
[2] E. Ahmed, A. El-Sayed and H.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.
[3] R. Barbosa, J.A. Tenreiro Machado, B.M. Vinagre and A.J. Calderon, Analysis of the Van der Pol oscillator containing derivatives of fractional-order, J. Vib. Control, 13 (2007) 1291–1301.
[4] D.A. Benson, S.W. Wheatcraft and M.M. Meerschaert, The fractional-order governing equation of L´evy motion, Water. res. 36 (2000) 1413–1423.
[5] N. Biranvand and A. Salari, Energy estimate for impulsive fractional advection dispersion equations in anomalous diffusions, J. Nonlinear Funct. Anal. 2018 (2018) Article ID 30.[6] B. Carmichael, H. Babahosseini, SN Mahmoodi and M. Agah, The fractional viscoelastic response of human breast tissue cells, Phys. Biol. 12 (2015) 1–14 .
[7] W.C. Chen, Nonlinear dynamics and chaos in a fractional-order financial system, Chaos, Solitons and Fractals, 36 (2008) 1305–1314.
[8] J.H. Chen and W.C. Chen, Chaotic dynamics of the fractionally damped van der Pol equation, Chaos, Solitons and Fractals, 35 (2008) 188-198.
[9] V. Daftardar-Gejji and S. Bhalekar, Chaos in fractional-ordered Liu system, Comput. Math. Appl. 59 (2010) 1117–1127.
[10] M. Dalir and M. Bashour, Applications of fractional calculus, App. Math. Sci. 21 (2010) 1021–1032.
[11] M. Das, A. Maiti and G. Samanta, Stability analysis of a predator-prey fractional-order model incorporating prey refuge, Ecol. Genet. Genom. 7 (2018) 33–46.
[12] W. Deng and C. Li, Chaos synchronization of the fractional L¨u system, Physica A, 353 (2005) 61–72.
[13] K. Diethelm, N.J. Ford and A.D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn. 29 (2002) 3–22.
[14] L. Edelstein-Keshet, Mathematical models in biology, SIAM, 1988.
[15] S. Elaydi and A.-A. Yakubu, Global Stability of Cycles: Lotka-Volterra Competition Model With Stocking, J. Differ. Equ. Appl. 8 (2002) 537–549.
[16] A.A. Elsadany, A.E. Matouk, A.G. Abdelwahab and H.S. Abdallah, Dynamical analysis, linear feedback control and synchronization of a generalized Lotka-Volterra system, Int. J. Dyn. Control, 6 (2018) 328–338.
[17] A. Elsadany and A. Matouk, Dynamical behaviors of fractional-order Lotka-Volterra predator-prey model and itsdiscretization, J. Appl. Math. Comput. 49 (2015) 269–283.
[18] A. El-Sayed, A. El-Mesiry and H. El-Saka, On the fractional-order logistic equation, Appl. Math. Lett. 20 (2007) 817–823.
[19] X. Fu, P. Zhang and J. Zhang, Forecasting and Analyzing Internet Users of China with Lotka-Volterra Model, Asia Pac. J. Oper. Res. 34 (2017) 1–18.
[20] Z.M. Ge and C.Y. Ou, Chaos in a fractional-order modified Duffing system, Chaos, Solitons and Fractals, 34 (2007) 262–291.
[21] I. Grigorenko and E. Grigorenko, Chaotic dynamics of the fractional Lorenz system, Phys. Rev. Lett. 91 (2003) 34–101.
[22] D. Jana, Chaotic dynamics of a discrete predator-prey system with prey refugem, J. Comput. Appl. Math. 224 (2013) 848–865.
[23] L. Jun-Guo, Chaotic dynamics and synchronization of fractional-order Genesio-Tesi systems, Chin. Phys. Lett. 14 (2005) 1517–1522.
[24] M. Jun-hai and C. Yu-shu, Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system (II), J. Appl. Math. Mech. 22 (2001) 1375–1382.
[25] R.Khoshsiar Ghaziani, J.Alidousti and A. Bayati, Eshkaftakib Stability and dynamics of a fractional-order LeslieGower prey-predator model Author links open overlay panel, Appl. Math. Modelling, 40(3) (2016) 2075-2086.
[26] G. Kotalczyk and F. Kruis, Fractional Monte Carlo time steps for the simulation of coagulation for parallelized flowsheet simulations, Chem. Eng. Res. Des. 136 (2018) 71–82.
[27] A. Matouk, Dynamical behaviors, linear feedback control and synchronization of the fractional-order Liu system, J. Nonlinear Syst. Appl. 1 (2010) 135–140.
[28] A. Matouk, Dynamical analysis, feedback control and synchronization of Liu dynamical system, Nonlinear AbalTheor. 69 (2008) 3213–3224.
[29] R.L. Magin, Fractional calculus in bioengineering, Begell House, 2006.
[30] F. Pitolli, A Fractional B-spline Collocation Method for the Numerical Solution of Fractional predator-prey Models, Fract. Fractional, 2 (2018) 1–16.
[31] L. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic Press, 1998.
[32] I. Petras, Fractional order nonlinear systems: modeling, analysis and simulation, Springer Science and Business Media, 2011.
[33] I. Petr´as, A note on the fractional-order Volta’s system, Commun. Nonlinear Sci. 15 (2010) 384-393.
[34] I. Petr´as, Chaos in the fractional-order Volta’s system: modeling and simulation. Nonlinear Dyn. 57 (2009) 157–170.
[35] M. Rahmani Doust, S. Gholizadeh, The Lotka-Volterra predator-prey Equations, CJMS, 3 (2014) 221–225.
[36] L.J. Sheua, H.K. Chen, J.H. Chen, L.M.Tam, W.C. Chen, K.T. Line and Y. Kang, Chaos in the Newton-Leipnik system with fractional-order, Chaos, Solitons and Fractals, 36 (2008) 98–103.
[37] N. Samardzija and L.D. Greller, Explosive route to chaos through a fractal torus in a generalized Lotka-Volterra model. Bull. Math. Biol. 50 (1988) 465–491.
[38] R. Satriyantara, A. Suryanto and N. Hidayat, Numerical Solution of a Fractional-Order predator-prey Model with Prey Refuge and Additional Food for Predator, J. Exp. Biol. 8 (2018) 66–70.
[39] N.T. Shawagfeh, Analytical approximate solutions for nonlinear fractional differential equations, J. Comput. Appl. Math. 131 (2002) 517-529.
[40] S. Vaidyanathan, Lotka-Volterra population biology models with negative feedback and their ecological monitoring, Int. J. PharmTech Res. 8 (2015) 974–981.
[41] S. Vaidyanathan, Adaptive biological control of generalized Lotka-Volterra three-species biological system, Int. J. Pharmtech Res. 8 (2015) 622–631.
[42] T. Zhou, Y. Tang and G. Chen, Chen’s attractor exists. Int. J. Bif. Chaos, 14 (2004) 3167–3177.
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Volume 12, Issue 2
November 2021Pages 729-741
- Receive Date: 09 February 2020
- Revise Date: 22 May 2020
- Accept Date: 24 May 2020