Dynamical behaviors of Bazykin-Berezovskaya model with fractional-order and its discretization

Akrami, Mohammad Hossein

doi:10.22034/cmde.2020.30802.1460

Dynamical behaviors of Bazykin-Berezovskaya model with fractional-order and its discretization

Document Type : Research Paper

Author

Mohammad Hossein Akrami

Department of Mathematics, Yazd University, 89195-741 Yazd, Iran.

10.22034/cmde.2020.30802.1460

Abstract

‎This paper is devoted to study dynamical behaviors of the fractional-order BazykinBerezovskaya model and its discretization. The fractional derivative has been described in the Caputo sense. We show that the discretized system, exhibits more complicated dynamical behaviors than its corresponding fractional-order model. Specially, in the discretized model Neimark-Sacker and flip bifurcations and also chaos phenomena will happen. In the final part, some numerical simulations verify the analytical results.

Keywords

20.1001.1.23453982.2021.9.4.6.8

References

[1] 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(1) (2007), 542-553.
[2] W. C. Allee and E. Bowen, Studies in animal aggregations: mass protection against colloidal silver among goldfishes, Journal of Experimental Zoology, 61 (1932), 185–207.
[3] K. T. Alligood, D. S. Tim, and J. Yorke, Chaos, Springer Berlin Heidelberg, 1997.
[4] O. A. Arqub and A. El-Ajou, Solution of the fractional epidemic model by homotopy analysis method, Journal of King Saud University-Science, 25 (2013), 73-81.
[5] A. Atabaigi, Bifurcation and chaos in a discrete time predator-prey system of Leslie type with generalized Holling type III functional response, Journal of Applied Analysis and Computation, 7(2) (2017), 411-426.
[6] A. D. Bazykin, Nonlinear Dynamics of Interacting Populations, World Scientific, Singapore, 1998.
[7] A. Ben Saad and O. Boubaker, On Bifurcation Analysis of the Predator-Prey BB-model with Weak Allee Effect, 16th international conference on Sciences and Techniques of Automatic control & computer engineering - STA’2015, Monastir, Tunisia, December 21-23, 2015.
[8] R. L. Devaney, An introduction to chaotic dynamical systems, Addison-Wesley, 1989.
[9] S. Elaydi, An introduction to difference equations, Springer Science Business Media, New York, 2005.
[10] A. A. Elsadany and A. E. Matouk, Dynamical behaviors of fractional-order Lotka-Volterra predator-prey model and its discretization, Journal of Applied Mathematics and Computing, 49(2015), 269-283.
[11] E. Gonz´alez-Olivares, J. Mena-Lorca, A. Rojas-Palma, and J. D. Flores, Dynamical complexities in the Leslie-Gower predator-prey model as consequences of the Allee effect on prey Citation data, Applied Mathematical Modelling, 35(1) (2011), 366-381.
[12] J. Guckenheimer and P. Holmes, Nonlinear Oscillation, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 1983.
[13] J. He, S. Yu, and J. Cai, Numerical Analysis and Improved Algorithms for Lyapunov-Exponent Calculation of Discrete-Time Chaotic Systems, International Journal of Bifurcation and Chaos, 26(13) (2016), 1650219.
[14] C. Ionescu, A. Lopes, D. Copot, J. A. T. Machado, and J. H. T. Bates, The Role of Fractional Calculus in Modelling Biological Phenomena: A review, Communications in Nonlinear Science and Numerical Simulation, 51(2017), 141-159.
[15] S. Kartala and F. Gurcan, Stability and bifurcations analysis of a competition model with piece- wise constant arguments, Mathematical Methods in the Applied Sciences, 38(9) (2014), 1855– 1866.
[16] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer Science and Business Media, 2013.
[17] X. Liu and X. Dongmei, Complex dynamic behaviors of a discrete-time predator-prey system, Chaos Solitons and Fractals, 32(1) (2007), 80-94.
[18] G. Livadiotis and D. J. McComas, Measure of the departure of the q-metastable stationary states from equilibrium, Physica Scripta, 8(2010), 035003.
[19] A. J. Lotka, Elements of Physical Biology, Williams and Wilkins, Baltimore, 1925.
[20] D. Matignon, Stability results for fractional differential equations with applications to control processing, Computational Engineering in System Application, 2(1) (1996), 963-968.
[21] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, NJ, 2001.
[22] J. D. Murray, Mathematical Biology I: An Introduction, Springer Science & Business Media, 2007.
[23] I. Petr´aˇs, Fractional-order nonlinear systems: modeling, analysis and simulation, Springer Sci- ence & Business Media, 2011.
[24] E. C. Pielou, An Introduction to Mathematical Ecology, Wiley, New York, 1969.
[25] I. Podlubny, Fractional Differential Equations, New York, Academic Press, 1999.
[26] M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, American Naturalist, 97(1963), 209-223.
[27] V. K. Srivastava, S. Kumar, M. K. Awasthi, and B. K. Singh, Two-dimensional time fractional- order biological population model and its analytical solution, Egyptian Journal of Basic and Applied Sciences, 1(2014), 71-76.
[28] J. P. Tripathi, S. S. Meghwani, M. Thakur, and S. Abbas, A modified Leslie-Gower predator- prey interaction model and parameter identifiability, Communications in Nonlinear Science and Numerical Simulation, 54(2018), 331-346.
[29] G. A. K. Van Voorn, L. Hemerik, M. P. Boer, and B. W. Kooi, Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee effect, Mathematical Biosciences, 209(2007), 451-469.
[30] V. Volterra, Variazioni e fluttuazioni del numero di individui in specie animali conviventi, C. Ferrari, 1927.