Stability and Period-Doubling Bifurcation in a Modified Commensal Symbiosis Model with Allee Effect

Year 2022, Volume: 15 Issue: 1, 310 - 324, 27.03.2022

Seval Işık

https://doi.org/10.18185/erzifbed.879963

Abstract

In this article, the qualitative behaviour of discrete-time commensal symbiosis model which is obtained by implementing the forward Euler’s scheme is discussed in detail. Firstly, the local stability conditions of fixed points of the model are studied. It is proved that the considered model undergoes Period-Doubling bifurcation around coexistence fixed point with the help of bifurcation theory. In order to support the accuracy of obtained analytical finding, some parameter values have been determined and numerical simulations are carried out for these parameter values. Numerical simulations display new and rich nonlinear dynamical behaviours. More specifically, when the parameter 𝛿 is choosen as a bifurcation parameter, it is seen that the considered discrete-time commensal symbiosis model shows very rich nonlinear dynamical.

Keywords

Allee effect, stability, Period-Doubling bifurcation

Supporting Institution

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Project Number

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Thanks

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References

• Allee, W. C., 1931, “Cooperation among animals”, Am. J. Sociol., 37(3): 386-398.
• Chen L. J., Chen L. J., Li Z, 2009, “Permanance of a Delayed Discrete Mutualsim Model with Feedback Controls”, Math. Comput. Model., 50(7-8):1083-1089.
• Cheng, H. C., 2016, “Bifurcation Analysis of a Discrete-time Ratio-dependent Prey-predator Model with the Allee Effect”, Commun. Nonlinear Sci. Numer. Simul., 38: 288-302.
• Din Q., 2018, “Stability Bifurcation Analysis and Chaos for a Predator-Prey System”, J. Vib. Control, 1-15, doi:10.117/1077546318790871.
• Eskandari Z., Alidousti J., 2020, “Generalized Flip and strong Resonances Bifurcations of a Predator-Prey Model”, Int. J. Dyn. Control, doi:10.1007/s40435020-00637-8.
• Elaydi S. N.,1996, “An Introduction to Difference Equation”, Springer-Verlag, New York, NY, USA.
• Feng P., Kang Y., 2015, “Dynamics of a Modified Leslie-Gower Model with Double Allee Effects”, Nonlinear Dyn., 80 :1051-1062.
• M. X., Chen F. D., 2009, “Dynamic behaviors of the impulsive periodic multi-species predator-prey system”, Comput. Math. Appl., 57:248-265.
• Kangalgil F., Kartal S., 2018, “Stability and Bifurcation Analysis in a Host–parasitoid Model with Hassell Growth Functione”, Adv. Difference Equ., 240.
• Kangalgil F., 2019, “Flip Bifurcation and Stabililty in a Discrete Time Prey-Predator Model with Allee Effect”, Cumhuriyet Scie. J., 40(1): 141-149.
• Kangalgil F., 2019, “Neimark–Sacker bifurcation and stability analysis of a discrete-time prey–predator model with Allee effect in prey”, Adv. Difference Equ., 1-19.
• Kartal S., 2016, “Dynamics of a Plant-Herbivore Model with Differential-Difference Equations”, Appl. Interdiscip. Math.:3.
• Kartal S., Kar M., Kartal N., Gurcan F., 2016, “Modelling and Analysis of a Phytoplankton-Zooplankton System with Continuous and Discrete-time”, Math. Comput. Model. Dyn. Sys., 22(6):539-554.
• Khan A. Q., 2016, “Neimark-Sacker Bifurcation of a Two-dimensional Discrete-Time Predator-Prey Model”, Springer Plus, 5(126): 1-10.
• Kılıç H., 2020, “Göçe Bağlı Ayrık Zamanlı Av-Avcı Popülasyon Modelinin Kararlılığı”, MSc. Thesis, Sivas Cumhuriyet University, Institute of Science and Technology, Sivas.
• Kuznetsov Y. A., 1998, “Elements of Applied Bifurcation Theory”, 2nd edition, Springer-Verlag, New York, USA.
• Lin Q., 2018, “Stability Analysis of a Single Species Logistic Model with Allee Effect and Feedback Control,” Adv. Difference Equ., doi:10.1186/s13662-018-1647-2.
• Miao Z., Xie X., Pu L., 2015, “Dynamic Behaviors of a Periodic Lotka-Volterra Commensal Symbiosis Model with Impulsive, Commun”, Math. Biol. Neurosci, 3.
• Rana S. M. S., 2015, “Bifurcation and Complex Dynamics of a Discrete-Time Predator-Prey System”, Comput. Ecol. Software, 5(2):187-200.
• Selvam, A. George Maria, S. Britto Jacob, and R. Dhineshbabu, 2020, "Bifurcation and Chaos Control for Prey Predator Model with Step Size in Discrete Time." J. of Phys.: Conf. Ser., 1543.
• Sen M., Banarjee M. And Morozou A., 2012, “Bifurcation Analysis of a Ratio-Dependent Prey Predator Model with the Allee Effect”, Ecol. Complex., 11:12-27.
• Sun G. C., Wei W. L., 2003, “The qualitative analysis of commensal symbiosis model of two populations”, Math.Theory Appl., 23: 64-68.
• Wiggins S., 2003, “Introduction to Applied Nonlinear Dynamical System and Chaos”, 2. Springer-Verlag, New York, USA.
• Wilson E.O., Amin K. A., 1975, “Sociobiology: The New Synthesis”, Harvard University Press, ISBN 978-0-674-00089-6.
• Williams E. H., Mignucci-Giannoni A. A., Bunkley- Williams L., Bonde R. K., Self-Sullivan C., 2003, “Preen Diet. J. of Fish Biol.”, 63(5):1176-1183.
• Wu R., Li L., Lin Q., 2018, “A Holling Type Commensal Symbiosis Model Involving Allee Effect”, Commun. Math. Biol. Neurosci., doi:10.28919/cmbn/3679.
• X. D. Xie, Z. S. Miao, Y. L. Xue, 2015, “Positive periodic solution of a discrete Lotka-Volterra commensal symbiosis model”, Commun. Math. Biol. Neurosci, Article ID 2.
• Zhou, S. R., Liu, Y. F., Wang, G., 2005, “The stability of predator-prey systems subject to the Allee effects”, Theor. Popul. Biol., 67(1): 23-31.