Research Article
BibTex RIS Cite

Year 2022, Volume: 51 Issue: 2, 404 - 420, 01.04.2022

Seval Işık Figen Kangalgil

https://doi.org/10.15672/hujms.728889
Cited By: 5

Abstract

Supporting Institution

Sivas Cumhuriyet Üniversitesi

Project Number

EĞT-067

References

  • [1] H.N. Agiza, E.M. Elabbasy, H. El-Metwally and A.A. Elsandany, Chaotic dynamics of a discrete prey-predator model with Holling type II, Nonlinear Anal. Real World Appl. 10, 116-129, 2009.
  • [2] W.C. Allee, Animal Aggregations, a Study in General Sociology, University of Chicago Press, Chicago, IL, 1931.
  • [3] S. Biswas, D. Pal, G.S. Mahapatra and G.P. Samanta, Dynamics of a prey–predator system with herd behaviour in both and strong Allee effect in prey, Biophysics 65 (5), 826-835, 2020.
  • [4] L. Cheng and H. Cao, Bifurcation analysis of a discrete-time ratio-dependent predator- prey model with the Allee effect, Commun. Nonlinear Sci. Numer. Simul. 38, 288-302, 2016.
  • [5] Q. Din, Neimark-Sacker bifurcation and chaos control in Hassel-Varley model, J. Difference Equ. Appl. 23 (4), 741-762, 2016.
  • [6] Q. Din, Complexity and choas control in a discrete-time prey-predator model, Commun Nonlinear Sci. Numer. Simul. 49, 113-134, 2017.
  • [7] Q. Din, Bifurcation analysis and chaos control in a Host-parasitoid model, Mat. Meth- ods Appl. Sci. 40, 5391-5406, 2017.
  • [8] S.N. Elaydi, An Introduction to Difference Equations, Springer-Verlag, New York, USA, 1996.
  • [9] C.B. Fu, A.H. Tian, K.N. Yu, Y.H. Lin and H.T. Yau, Analyses and control of chaotic behavior in DC–DC converters, Math. Probl. Eng. 2018, Article ID 7439137, 2018.
  • [10] Z. He and X. Lai, Bifurcation and chaotic behavior of a discrete-time predator-prey system, Nonlinear Anal. Real World Appl. 12, 403-417, 2011.
  • [11] Z. Hu, Z. Teng and L. Zhang, Stability and bifurcation analysis of a discrete predator- prey model with nonmonotonic functional response, Nonlinear Anal. RealWorld Appl. 12, 2356-2377, 2011.
  • [12] P.K. Jha and S. Ghorai, Stability of prey-predator model with Holling type response function and selective harvesting, J. Appl. Comput. Math. 6 (3), 2017.
  • [13] F. Kangalgil and S. Isık, Controlling chaos and Neimark-Sacker bifurcation discrete- time predator-prey system, Hacet. J. Math. Stat. 49 (5), 1761-1776, 2020.
  • [14] Ş. Kartal, Dynamics of a plant-herbivore model with differential-difference equations, Cogents Math. 3, 1136198, 2016.
  • [15] Ş. Kartal, Flip and Neimark–Sacker bifurcation in a differential equation with piece- wise constant arguments model, J. Difference Equ. Appl. 23 (4), 763-778, 2017.
  • [16] Ş. Kartal and F. Gurcan, Global behaviour of a predator–prey like model with piecewise constant arguments, J. Biol. Dyn. 9 (1), 159-171, 2015.
  • [17] A.Q. Khan, Neimark-Sacker bifurcation of a two-dimensional discrete-time predator- prey model, Springer Plus 5, Article No: 5, 2016.
  • [18] A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett. 14 (6), 697-699, 2001.
  • [19] Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd edition, Springer- Verlag, New York, USA, 1998.
  • [20] P.H. Leslie, Some further notes on the use of matrices in population mathematics, Biometrika 35, 213-245, 1948.
  • [21] P.H. Leslie, A stochastic model for studying the properties of certain biological systems by numerical methods, Biometrika 45, 16-31, 1958.
  • [22] S. Li and T. Chen, Nonlinear dynamics in the switched reluctance motor drive with time-delay feedback control, IEEE Texas Power and Energy Conference (TPEC), 1–6, 2019.
  • [23] Y. Ling and Z. Liu, An improvement and proof of OGY method, Appl. Math. Mech. 19 (1), 1-8, 1998.
  • [24] X. Liu and D. Xiao, Complex dynamics behaviors of a discrete-time predator-prey system, Chaos Solitons Fractals, 32, 80-94, 2007.
  • [25] S. Lynch, Dynamical Systems with Applications using Mathematica, Birkhauser, Boston, 2007.
  • [26] P.S. Mandal, U. Kumar, K. Garain and R. Sharma, Allee effect can simplify the dynamics of a prey-predator model, J. Appl. Math. Comput. 63, 739-770, 2020.
  • [27] E. Ott, C. Grebogi and J.A. Yorke, Controlling chaos, Phys. Rev. Lett. 64 (11), 1196–1199, 1990.
  • [28] S.M. Rana and U. Kulsum, Bifurcation analysis and chaos control in a discrete-time predator-prey system of Leslie type with simplified Holling type IV functional response, Discrete Dyn. Nat. Soc. 2017, Article ID 9705985, 2017.
  • [29] S.M. Salman, A.M. Yousef and A.A. Elsadany, Stability, bifurcation analysis and chaos control of a discrete predator-prey system with square root functional response, Chaos Solitons Fractals, 93, 20-31, 2016.
  • [30] M. Sen, M. Banarjee and A. Morozou, Bifurcation analysis of a ratio-dependent prey- predator model with the Allee effect, Ecol. Complex. 11, 12-27, 2012.
  • [31] B. Tiwari and S.N. Raw, Dynamics of Leslie-Gower model with double Allee effect on prey and mutual interference among predators, Nonlinear Dyn. 103, 1229–1257, 2021.
  • [32] S. Wang and H. Yu, Complexity analysis of a modified pedator-prey system with Beddington–DeAngelis functional response and Allee-like effect on predator, Discrete Dyn. Nat. Soc. 2021, Article ID 5618190, 2021.
  • [33] S. Wiggins, Introduction to Applied Nonlinear Dynamical System and Chaos, 2, Springer-Verlag, New York, USA, 2003.
  • [34] S. Zhou, Y. Liu and G. Wang, The stability of predator-prey systems subject to the Allee Effects, Theor. Popul. Biol. 67, 23-31, 2005.

On the analysis of stability, bifurcation, and chaos control of discrete-time predator-prey model with Allee effect on predator

Year 2022, Volume: 51 Issue: 2, 404 - 420, 01.04.2022

Seval Işık Figen Kangalgil

https://doi.org/10.15672/hujms.728889
Cited By: 5

Abstract

In this paper, a discrete predator-prey model with Allee effect which is obtained by the forward Euler method has been investigated. The local stability conditions of the model at the fixed point have been discussed. In addition, it is shown that the model undergoes Neimark-Sacker bifurcation by using bifurcation theory. Then, the direction of Neimark-Sacker bifurcation has been given. The OGY method is applied in order to control chaos in considered model due to emergence of Neimark-Sacker bifurcation. Some numerical simulations such as phase portraits and bifurcation figures have been presented to support the theoretical results. Also, the chaotic features are justified numerically by computing Lyapunov exponents. Because of consistency with the biological facts, the parameter values have been taken from literature [Controlling chaos and Neimark-Sacker bifurcation discrete-time predator-prey system, Hacet. J. Math. Stat. 49 (5), 1761-1776, 2020].

Keywords

predator-prey model fixed point stability Neimark-Sacker bifurcation Allee effect OGY control method

Project Number

EĞT-067

References

  • [1] H.N. Agiza, E.M. Elabbasy, H. El-Metwally and A.A. Elsandany, Chaotic dynamics of a discrete prey-predator model with Holling type II, Nonlinear Anal. Real World Appl. 10, 116-129, 2009.
  • [2] W.C. Allee, Animal Aggregations, a Study in General Sociology, University of Chicago Press, Chicago, IL, 1931.
  • [3] S. Biswas, D. Pal, G.S. Mahapatra and G.P. Samanta, Dynamics of a prey–predator system with herd behaviour in both and strong Allee effect in prey, Biophysics 65 (5), 826-835, 2020.
  • [4] L. Cheng and H. Cao, Bifurcation analysis of a discrete-time ratio-dependent predator- prey model with the Allee effect, Commun. Nonlinear Sci. Numer. Simul. 38, 288-302, 2016.
  • [5] Q. Din, Neimark-Sacker bifurcation and chaos control in Hassel-Varley model, J. Difference Equ. Appl. 23 (4), 741-762, 2016.
  • [6] Q. Din, Complexity and choas control in a discrete-time prey-predator model, Commun Nonlinear Sci. Numer. Simul. 49, 113-134, 2017.
  • [7] Q. Din, Bifurcation analysis and chaos control in a Host-parasitoid model, Mat. Meth- ods Appl. Sci. 40, 5391-5406, 2017.
  • [8] S.N. Elaydi, An Introduction to Difference Equations, Springer-Verlag, New York, USA, 1996.
  • [9] C.B. Fu, A.H. Tian, K.N. Yu, Y.H. Lin and H.T. Yau, Analyses and control of chaotic behavior in DC–DC converters, Math. Probl. Eng. 2018, Article ID 7439137, 2018.
  • [10] Z. He and X. Lai, Bifurcation and chaotic behavior of a discrete-time predator-prey system, Nonlinear Anal. Real World Appl. 12, 403-417, 2011.
  • [11] Z. Hu, Z. Teng and L. Zhang, Stability and bifurcation analysis of a discrete predator- prey model with nonmonotonic functional response, Nonlinear Anal. RealWorld Appl. 12, 2356-2377, 2011.
  • [12] P.K. Jha and S. Ghorai, Stability of prey-predator model with Holling type response function and selective harvesting, J. Appl. Comput. Math. 6 (3), 2017.
  • [13] F. Kangalgil and S. Isık, Controlling chaos and Neimark-Sacker bifurcation discrete- time predator-prey system, Hacet. J. Math. Stat. 49 (5), 1761-1776, 2020.
  • [14] Ş. Kartal, Dynamics of a plant-herbivore model with differential-difference equations, Cogents Math. 3, 1136198, 2016.
  • [15] Ş. Kartal, Flip and Neimark–Sacker bifurcation in a differential equation with piece- wise constant arguments model, J. Difference Equ. Appl. 23 (4), 763-778, 2017.
  • [16] Ş. Kartal and F. Gurcan, Global behaviour of a predator–prey like model with piecewise constant arguments, J. Biol. Dyn. 9 (1), 159-171, 2015.
  • [17] A.Q. Khan, Neimark-Sacker bifurcation of a two-dimensional discrete-time predator- prey model, Springer Plus 5, Article No: 5, 2016.
  • [18] A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett. 14 (6), 697-699, 2001.
  • [19] Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd edition, Springer- Verlag, New York, USA, 1998.
  • [20] P.H. Leslie, Some further notes on the use of matrices in population mathematics, Biometrika 35, 213-245, 1948.
  • [21] P.H. Leslie, A stochastic model for studying the properties of certain biological systems by numerical methods, Biometrika 45, 16-31, 1958.
  • [22] S. Li and T. Chen, Nonlinear dynamics in the switched reluctance motor drive with time-delay feedback control, IEEE Texas Power and Energy Conference (TPEC), 1–6, 2019.
  • [23] Y. Ling and Z. Liu, An improvement and proof of OGY method, Appl. Math. Mech. 19 (1), 1-8, 1998.
  • [24] X. Liu and D. Xiao, Complex dynamics behaviors of a discrete-time predator-prey system, Chaos Solitons Fractals, 32, 80-94, 2007.
  • [25] S. Lynch, Dynamical Systems with Applications using Mathematica, Birkhauser, Boston, 2007.
  • [26] P.S. Mandal, U. Kumar, K. Garain and R. Sharma, Allee effect can simplify the dynamics of a prey-predator model, J. Appl. Math. Comput. 63, 739-770, 2020.
  • [27] E. Ott, C. Grebogi and J.A. Yorke, Controlling chaos, Phys. Rev. Lett. 64 (11), 1196–1199, 1990.
  • [28] S.M. Rana and U. Kulsum, Bifurcation analysis and chaos control in a discrete-time predator-prey system of Leslie type with simplified Holling type IV functional response, Discrete Dyn. Nat. Soc. 2017, Article ID 9705985, 2017.
  • [29] S.M. Salman, A.M. Yousef and A.A. Elsadany, Stability, bifurcation analysis and chaos control of a discrete predator-prey system with square root functional response, Chaos Solitons Fractals, 93, 20-31, 2016.
  • [30] M. Sen, M. Banarjee and A. Morozou, Bifurcation analysis of a ratio-dependent prey- predator model with the Allee effect, Ecol. Complex. 11, 12-27, 2012.
  • [31] B. Tiwari and S.N. Raw, Dynamics of Leslie-Gower model with double Allee effect on prey and mutual interference among predators, Nonlinear Dyn. 103, 1229–1257, 2021.
  • [32] S. Wang and H. Yu, Complexity analysis of a modified pedator-prey system with Beddington–DeAngelis functional response and Allee-like effect on predator, Discrete Dyn. Nat. Soc. 2021, Article ID 5618190, 2021.
  • [33] S. Wiggins, Introduction to Applied Nonlinear Dynamical System and Chaos, 2, Springer-Verlag, New York, USA, 2003.
  • [34] S. Zhou, Y. Liu and G. Wang, The stability of predator-prey systems subject to the Allee Effects, Theor. Popul. Biol. 67, 23-31, 2005.
There are 34 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Seval Işık 0000-0002-6523-7805

Figen Kangalgil 0000-0003-0116-8553

Project Number EĞT-067
Publication Date April 1, 2022
Published in Issue Year 2022 Volume: 51 Issue: 2

Cite

APA Işık, S., & Kangalgil, F. (2022). On the analysis of stability, bifurcation, and chaos control of discrete-time predator-prey model with Allee effect on predator. Hacettepe Journal of Mathematics and Statistics, 51(2), 404-420. https://doi.org/10.15672/hujms.728889
AMA Işık S, Kangalgil F. On the analysis of stability, bifurcation, and chaos control of discrete-time predator-prey model with Allee effect on predator. Hacettepe Journal of Mathematics and Statistics. April 2022;51(2):404-420. doi:10.15672/hujms.728889
Chicago Işık, Seval, and Figen Kangalgil. “On the Analysis of Stability, Bifurcation, and Chaos Control of Discrete-Time Predator-Prey Model With Allee Effect on Predator”. Hacettepe Journal of Mathematics and Statistics 51, no. 2 (April 2022): 404-20. https://doi.org/10.15672/hujms.728889.
EndNote Işık S, Kangalgil F (April 1, 2022) On the analysis of stability, bifurcation, and chaos control of discrete-time predator-prey model with Allee effect on predator. Hacettepe Journal of Mathematics and Statistics 51 2 404–420.
IEEE S. Işık and F. Kangalgil, “On the analysis of stability, bifurcation, and chaos control of discrete-time predator-prey model with Allee effect on predator”, Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 2, pp. 404–420, 2022, doi: 10.15672/hujms.728889.
ISNAD Işık, Seval - Kangalgil, Figen. “On the Analysis of Stability, Bifurcation, and Chaos Control of Discrete-Time Predator-Prey Model With Allee Effect on Predator”. Hacettepe Journal of Mathematics and Statistics 51/2 (April 2022), 404-420. https://doi.org/10.15672/hujms.728889.
JAMA Işık S, Kangalgil F. On the analysis of stability, bifurcation, and chaos control of discrete-time predator-prey model with Allee effect on predator. Hacettepe Journal of Mathematics and Statistics. 2022;51:404–420.
MLA Işık, Seval and Figen Kangalgil. “On the Analysis of Stability, Bifurcation, and Chaos Control of Discrete-Time Predator-Prey Model With Allee Effect on Predator”. Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 2, 2022, pp. 404-20, doi:10.15672/hujms.728889.
Vancouver Işık S, Kangalgil F. On the analysis of stability, bifurcation, and chaos control of discrete-time predator-prey model with Allee effect on predator. Hacettepe Journal of Mathematics and Statistics. 2022;51(2):404-20.

Cited By

Qualitative Analysis of the Discretization of a Continuous Fractional Order Prey-Predator Model with the Effects of Harvesting and Immigration in the Population
Complexity
https://doi.org/10.1155/2024/8855142
Fractional order prey–predator model incorporating immigration on prey: Complexity analysis and its control
International Journal of Biomathematics
https://doi.org/10.1142/S1793524523500511
Bifurcation and Chaos of a Nonlinear Discrete-Time Predator-Prey Model Involving the Nonlinear Allee Effect
Discrete Dynamics in Nature and Society
https://doi.org/10.1155/2023/5475999
Effect of immigration in a predator-prey system: Stability, bifurcation and chaos
AIMS Mathematics
https://doi.org/10.3934/math.2022791
Period-doubling Bifurcation and Stability in a Two Dimensional Discrete Prey-predator Model with Allee Effect and Immigration Parameter on Prey
Cumhuriyet Science Journal
https://doi.org/10.17776/csj.1026330

https://dergipark.org.tr/en/pub/hujms