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Year 2023, Volume: 52 Issue: 4, 1029 - 1045, 15.08.2023

Özlem Ak Gümüş

https://doi.org/10.15672/hujms.1179682
Cited By: 1

Abstract

References

  • [1] A. Aldurayhim, A. A. Elsadany and A. Elsonbaty, On dynamic behavior of a discrete fractional-order nonlinear prey–predator model, Fractals 29 (08), p.2140037, 2021.
  • [2] W.C. Allee, Cooperation Among Animals, Henry Shuman, New York, 1951.
  • [3] W.C. Allee and E. Bowen, Studies in animal aggregations mass protection against colloidal silver among goldfishes, J. Exp. Zool. 61 (2), 185–207,1932.
  • [4] G. Chen and X. Dong, From Chaos to Order: Perspectives, Methodologies and Applications, World Scientific, Singapore, 1998.
  • [5] M.F. Danca, M. Fečkan, N. Kuznetsov and G. Chen, Rich dynamics and anticontrol of extinction in a prey–predator system, Nonlinear Dynam. 98 (2), 1421–1445, 2019.
  • [6] Q. Din, Controlling chaos in a discrete-time prey-predator model with Allee effects, Int. J. Dynam. Control 6, 858–872, 2018.
  • [7] Q. Din, Ö.A. Gümüş and H. Khalil, Neimark-sacker bifurcation and chaotic behaviour of a modified host–parasitoid model, Z. Naturforsch. A. 72 (1), 25–37, 2017.
  • [8] S. N. Elaydi, An Introduction to Difference Equations, 3rd edn., Springer-Verlag, New York, 2005.
  • [9] S. Elaydi, Discrete Chaos: With Applications in Science and Engineering, 2nd edn. Chapman and Hall/ CRC, Boca Raton, 2007.
  • [10] A.A. Elsadany, Q. Din and S.M. Salman, Qualitative properties and bifurcations of discrete-time Bazykin-Berezovskaya predator–prey model, Int. J. Biomath. 13 (06), p. 2050040, 2020.
  • [11] A.A. Elsadany, H.A. El-Metwally, E.M. Elabbasy and H.N. Agzia, Chaos and bifurcation of a nonlinear discrete prey–predator system, Comput. Ecol. Softw. 2 (3), 169–180, 2012.
  • [12] A.A. Elsadany and A. Matouk, Dynamical behaviors of fractional-order Lotka– Volterra predator-prey model and its discretization, J. Appl. Math. Comput. 49 (1), 269–283, 2015.
  • [13] Ö.A. Gümüş, Allee effect on the stability, PhD Thesis, Selçuk University, 2011.
  • [14] Ö.A. Gümüş, Global and local stability analysis in a nonlinear discrete time population model, Adv. Difference Equ. 299, 1687–1847, 2014.
  • [15] Ö.A. Gümüş, Neimark-Sacker bifurcation and stability a prey-predator system, Miskolc Math. Notes 21 (2), 873-875, 2020.
  • [16] Ö.A. Gümüş, Q. Cui, A.G.M. Selvam and A. Vianny, Global stability and bifurcation analysis of a discrete time SIR epidemic model, Miskolc Math. Notes 23 (1), 193-210, 2022.
  • [17] Ö.A. Gümüş and M. Fečkan, Stability, Neimark-Sacker bifurcation and chaos control for a prey-predator system with harvesting effect on predator, Miskolc Math. Notes 22 (2), 663-679, 2021.
  • [18] Ö.A. Gümüş, A.G.M. Selvam and R. Dhineshbabu, Bifurcation analysis and chaos control of the population model with harvest, Int. J. Nonlinear Anal. Appl. 13 (1), 115-125, 2022.
  • [19] Ö.A. Gümüş, A.G.M. Selvam and R. Janagaraj, Stability of Modified Host-Parasitoid Model with Allee Effect, Appl. Appl. Math. 15 (2), 1032-1045, 2020.
  • [20] Ö.A. Gümüş, A.G.M. Selvam and D. Vighnes, The effect of Allee factor on a nonlinear delayed population model with harvesting, J. Sci. Arts 22 (1), 159-176, 2022.
  • [21] F. Kangalgil and S. Işık, Controlling chaos and Neimark-Sacker bifurcation in a discrete-time predator-prey system, Hacet. J. Math. Stat. 49 (5), 1761-1776, 2020.
  • [22] S. Kapcak, Stability and bifurcation of predator-prey models with the Allee effect, Ph.D. Thesis, Izmir University of Economics, 2013.
  • [23] Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, NY, 1998.
  • [24] B. Li and Z. He, Bifurcations and chaos in a two-dimensional discrete Hindmarsh- Rose model, Nonlinear Dynam. 76 (1):697–715, 2014.
  • [25] X. Liu and D. Xiao, Complex dynamic behaviors of a discrete-time predator–prey system, Chaos Solitons Fractals 32 (1): 80–94, 2007.
  • [26] A.J. Lotka, Elements of Mathematical Biology, New York, Dover, 1956.
  • [27] S. Lynch, Dynamical Systems with Applications Using Mathematica, Birkhäuser, Boston, 2007.
  • [28] R.M. May, Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos, Scienc, 186, (4164) 645-647, 1974.
  • [29] R.M. May, Simple mathematical models with very complicated dynamics, Nature 261, 459-467, 1976.
  • [30] H. Merdan and Ö.A. Gümüş, Stability analysis of a general discrete-time population model involving delay and Allee effects, Appl. Math. Comput. 219, 1821–1832, 2012.
  • [31] C. Robinson, Dynamical Systems, Stability, Symbolic Dynamics and Chaos, 2nd edn. CRC Press, Boca Raton 1999.
  • [32] A.G.M. Selvam, S.M. Jacob and R. Dhineshbabu, Bifurcation and Chaos Control for Discrete Fractional-Order Prey–Predator Model with Square Root Interaction, Int. Conf. Math. Anal. Comput. 344 345-358, 2019.
  • [33] A.G.M. Selvam, R. Janagaraj and A. Hlafta, Bifurcation behaviour of a discrete differential algebraic prey–predator system with Holling type II functional response and prey refuge, AIP Conf. Proc. 2282 (1), p. 020011, 2020.
  • [34] A.G.M. Selvam, R. Janagaraj and D. Vignesh, Allee effect and Holling type - II response in a discrete fractional order prey - predator model, J. Phys.: Conf. Ser. 1139 (1), p.012003, 2018.
  • [35] A.G.M. Selvam, R. Janagaraj and D. Vignesh, Discretızation and chaos control in a fractional order predator-prey harvesting model, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 70 (2), 950–964, 2021.
  • [36] A.G.M. Selvam, D. Vignesh and R. Janagaraj, Bifurcation Analysis and Chaos Control for a Discrete Fractional-Order Prey–Predator System, Int. Conf. Math. Anal. Comput. 344, 205-219, 2019.
  • [37] A. Singh, A.A. Elsadany and A. Elsonbaty, Complex dynamics of a discrete fractionalorder Leslie-Gower predator-prey model, Math. Methods Appl. Sci. 42 (11), 3992- 4007, 2019.
  • [38] V. Voltera, Opere mathemtichem: Menorie e Note, Rome, 1962.
  • [39] W.X. Wang, Y.B. Zhang, and Liu, C.-Z., Analysis of a discrete time predator-prey with Allee effect, Ecol. Complex. 8 (1), 81-85, 2011.
  • [40] C. Xiang, M. Lu and J. Huang, Degenerate Bogdanov-Takens bifurcation of codimension 4 in Holling-Tanner model with harvesting, J. Differential Equations 314, 370-417, 2022.
  • [41] D. Xiao and S. Ruan, Codimension two bifurcations in a predator–prey system with group defense, Internat. J. Bifur. Chaos appl. Sci. Engrg. 11 (08), 2123-2131, 2001.
  • [42] A.M. Yousef, S.Z. Rida, H.M. Ali and A.S. Zaki, Stability, co-dimension two bifurcations and chaos control of a host-parasitoid model with mutual interference, Chaos Solitons Fractals 166, 112923, 2023.
  • [43] L. Zhang and C. Zhang, Codimension one and two bifurcations of a discrete stagestructured population model with self-limitation, J. Difference Equ. Appl. 24 (8), 1210- 1246, 2018.
  • [44] S. Zhou, Y. Liu and G. Wang, The stability of predator–prey systems subject to the Allee effects, Theoret. Popul. Biol. 67 (1), 23–31 2005.
  • [45] H. Zhu, S.A. Campbell and G.S. Wolkowicz, Bifurcation analysis of a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math. 63 (2), 636-682, 2003.

Bifurcation analysis and chaos control of a discrete-time prey-predator model with Allee effect

Year 2023, Volume: 52 Issue: 4, 1029 - 1045, 15.08.2023

Özlem Ak Gümüş

https://doi.org/10.15672/hujms.1179682
Cited By: 1

Abstract

In this study, a discrete-time prey-predator model based on the Allee effect is presented. We examine the parametric conditions for the local asymptotic stability of the fixed points of this model. Furthermore, with the use of the center manifold theorem and bifurcation theory, we analyze the existence and directions of period-doubling and Neimark-Sacker bifurcations. The plots of maximum Lyapunov exponents provide indications of complexity and chaotic behavior. The feedback control approach is presented to stabilize the unstable fixed point. Numerical simulations are performed to support the theoretical results.

Keywords

prey-predator model fixed point stability bifurcation theory Neimark-Sacker bifurcation chaos control flip bifurcation

References

  • [1] A. Aldurayhim, A. A. Elsadany and A. Elsonbaty, On dynamic behavior of a discrete fractional-order nonlinear prey–predator model, Fractals 29 (08), p.2140037, 2021.
  • [2] W.C. Allee, Cooperation Among Animals, Henry Shuman, New York, 1951.
  • [3] W.C. Allee and E. Bowen, Studies in animal aggregations mass protection against colloidal silver among goldfishes, J. Exp. Zool. 61 (2), 185–207,1932.
  • [4] G. Chen and X. Dong, From Chaos to Order: Perspectives, Methodologies and Applications, World Scientific, Singapore, 1998.
  • [5] M.F. Danca, M. Fečkan, N. Kuznetsov and G. Chen, Rich dynamics and anticontrol of extinction in a prey–predator system, Nonlinear Dynam. 98 (2), 1421–1445, 2019.
  • [6] Q. Din, Controlling chaos in a discrete-time prey-predator model with Allee effects, Int. J. Dynam. Control 6, 858–872, 2018.
  • [7] Q. Din, Ö.A. Gümüş and H. Khalil, Neimark-sacker bifurcation and chaotic behaviour of a modified host–parasitoid model, Z. Naturforsch. A. 72 (1), 25–37, 2017.
  • [8] S. N. Elaydi, An Introduction to Difference Equations, 3rd edn., Springer-Verlag, New York, 2005.
  • [9] S. Elaydi, Discrete Chaos: With Applications in Science and Engineering, 2nd edn. Chapman and Hall/ CRC, Boca Raton, 2007.
  • [10] A.A. Elsadany, Q. Din and S.M. Salman, Qualitative properties and bifurcations of discrete-time Bazykin-Berezovskaya predator–prey model, Int. J. Biomath. 13 (06), p. 2050040, 2020.
  • [11] A.A. Elsadany, H.A. El-Metwally, E.M. Elabbasy and H.N. Agzia, Chaos and bifurcation of a nonlinear discrete prey–predator system, Comput. Ecol. Softw. 2 (3), 169–180, 2012.
  • [12] A.A. Elsadany and A. Matouk, Dynamical behaviors of fractional-order Lotka– Volterra predator-prey model and its discretization, J. Appl. Math. Comput. 49 (1), 269–283, 2015.
  • [13] Ö.A. Gümüş, Allee effect on the stability, PhD Thesis, Selçuk University, 2011.
  • [14] Ö.A. Gümüş, Global and local stability analysis in a nonlinear discrete time population model, Adv. Difference Equ. 299, 1687–1847, 2014.
  • [15] Ö.A. Gümüş, Neimark-Sacker bifurcation and stability a prey-predator system, Miskolc Math. Notes 21 (2), 873-875, 2020.
  • [16] Ö.A. Gümüş, Q. Cui, A.G.M. Selvam and A. Vianny, Global stability and bifurcation analysis of a discrete time SIR epidemic model, Miskolc Math. Notes 23 (1), 193-210, 2022.
  • [17] Ö.A. Gümüş and M. Fečkan, Stability, Neimark-Sacker bifurcation and chaos control for a prey-predator system with harvesting effect on predator, Miskolc Math. Notes 22 (2), 663-679, 2021.
  • [18] Ö.A. Gümüş, A.G.M. Selvam and R. Dhineshbabu, Bifurcation analysis and chaos control of the population model with harvest, Int. J. Nonlinear Anal. Appl. 13 (1), 115-125, 2022.
  • [19] Ö.A. Gümüş, A.G.M. Selvam and R. Janagaraj, Stability of Modified Host-Parasitoid Model with Allee Effect, Appl. Appl. Math. 15 (2), 1032-1045, 2020.
  • [20] Ö.A. Gümüş, A.G.M. Selvam and D. Vighnes, The effect of Allee factor on a nonlinear delayed population model with harvesting, J. Sci. Arts 22 (1), 159-176, 2022.
  • [21] F. Kangalgil and S. Işık, Controlling chaos and Neimark-Sacker bifurcation in a discrete-time predator-prey system, Hacet. J. Math. Stat. 49 (5), 1761-1776, 2020.
  • [22] S. Kapcak, Stability and bifurcation of predator-prey models with the Allee effect, Ph.D. Thesis, Izmir University of Economics, 2013.
  • [23] Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, NY, 1998.
  • [24] B. Li and Z. He, Bifurcations and chaos in a two-dimensional discrete Hindmarsh- Rose model, Nonlinear Dynam. 76 (1):697–715, 2014.
  • [25] X. Liu and D. Xiao, Complex dynamic behaviors of a discrete-time predator–prey system, Chaos Solitons Fractals 32 (1): 80–94, 2007.
  • [26] A.J. Lotka, Elements of Mathematical Biology, New York, Dover, 1956.
  • [27] S. Lynch, Dynamical Systems with Applications Using Mathematica, Birkhäuser, Boston, 2007.
  • [28] R.M. May, Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos, Scienc, 186, (4164) 645-647, 1974.
  • [29] R.M. May, Simple mathematical models with very complicated dynamics, Nature 261, 459-467, 1976.
  • [30] H. Merdan and Ö.A. Gümüş, Stability analysis of a general discrete-time population model involving delay and Allee effects, Appl. Math. Comput. 219, 1821–1832, 2012.
  • [31] C. Robinson, Dynamical Systems, Stability, Symbolic Dynamics and Chaos, 2nd edn. CRC Press, Boca Raton 1999.
  • [32] A.G.M. Selvam, S.M. Jacob and R. Dhineshbabu, Bifurcation and Chaos Control for Discrete Fractional-Order Prey–Predator Model with Square Root Interaction, Int. Conf. Math. Anal. Comput. 344 345-358, 2019.
  • [33] A.G.M. Selvam, R. Janagaraj and A. Hlafta, Bifurcation behaviour of a discrete differential algebraic prey–predator system with Holling type II functional response and prey refuge, AIP Conf. Proc. 2282 (1), p. 020011, 2020.
  • [34] A.G.M. Selvam, R. Janagaraj and D. Vignesh, Allee effect and Holling type - II response in a discrete fractional order prey - predator model, J. Phys.: Conf. Ser. 1139 (1), p.012003, 2018.
  • [35] A.G.M. Selvam, R. Janagaraj and D. Vignesh, Discretızation and chaos control in a fractional order predator-prey harvesting model, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 70 (2), 950–964, 2021.
  • [36] A.G.M. Selvam, D. Vignesh and R. Janagaraj, Bifurcation Analysis and Chaos Control for a Discrete Fractional-Order Prey–Predator System, Int. Conf. Math. Anal. Comput. 344, 205-219, 2019.
  • [37] A. Singh, A.A. Elsadany and A. Elsonbaty, Complex dynamics of a discrete fractionalorder Leslie-Gower predator-prey model, Math. Methods Appl. Sci. 42 (11), 3992- 4007, 2019.
  • [38] V. Voltera, Opere mathemtichem: Menorie e Note, Rome, 1962.
  • [39] W.X. Wang, Y.B. Zhang, and Liu, C.-Z., Analysis of a discrete time predator-prey with Allee effect, Ecol. Complex. 8 (1), 81-85, 2011.
  • [40] C. Xiang, M. Lu and J. Huang, Degenerate Bogdanov-Takens bifurcation of codimension 4 in Holling-Tanner model with harvesting, J. Differential Equations 314, 370-417, 2022.
  • [41] D. Xiao and S. Ruan, Codimension two bifurcations in a predator–prey system with group defense, Internat. J. Bifur. Chaos appl. Sci. Engrg. 11 (08), 2123-2131, 2001.
  • [42] A.M. Yousef, S.Z. Rida, H.M. Ali and A.S. Zaki, Stability, co-dimension two bifurcations and chaos control of a host-parasitoid model with mutual interference, Chaos Solitons Fractals 166, 112923, 2023.
  • [43] L. Zhang and C. Zhang, Codimension one and two bifurcations of a discrete stagestructured population model with self-limitation, J. Difference Equ. Appl. 24 (8), 1210- 1246, 2018.
  • [44] S. Zhou, Y. Liu and G. Wang, The stability of predator–prey systems subject to the Allee effects, Theoret. Popul. Biol. 67 (1), 23–31 2005.
  • [45] H. Zhu, S.A. Campbell and G.S. Wolkowicz, Bifurcation analysis of a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math. 63 (2), 636-682, 2003.
There are 45 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Özlem Ak Gümüş 0000-0003-2610-8565

Publication Date August 15, 2023
Published in Issue Year 2023 Volume: 52 Issue: 4

Cite

APA Ak Gümüş, Ö. (2023). Bifurcation analysis and chaos control of a discrete-time prey-predator model with Allee effect. Hacettepe Journal of Mathematics and Statistics, 52(4), 1029-1045. https://doi.org/10.15672/hujms.1179682
AMA Ak Gümüş Ö. Bifurcation analysis and chaos control of a discrete-time prey-predator model with Allee effect. Hacettepe Journal of Mathematics and Statistics. August 2023;52(4):1029-1045. doi:10.15672/hujms.1179682
Chicago Ak Gümüş, Özlem. “Bifurcation Analysis and Chaos Control of a Discrete-Time Prey-Predator Model With Allee Effect”. Hacettepe Journal of Mathematics and Statistics 52, no. 4 (August 2023): 1029-45. https://doi.org/10.15672/hujms.1179682.
EndNote Ak Gümüş Ö (August 1, 2023) Bifurcation analysis and chaos control of a discrete-time prey-predator model with Allee effect. Hacettepe Journal of Mathematics and Statistics 52 4 1029–1045.
IEEE Ö. Ak Gümüş, “Bifurcation analysis and chaos control of a discrete-time prey-predator model with Allee effect”, Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 4, pp. 1029–1045, 2023, doi: 10.15672/hujms.1179682.
ISNAD Ak Gümüş, Özlem. “Bifurcation Analysis and Chaos Control of a Discrete-Time Prey-Predator Model With Allee Effect”. Hacettepe Journal of Mathematics and Statistics 52/4 (August 2023), 1029-1045. https://doi.org/10.15672/hujms.1179682.
JAMA Ak Gümüş Ö. Bifurcation analysis and chaos control of a discrete-time prey-predator model with Allee effect. Hacettepe Journal of Mathematics and Statistics. 2023;52:1029–1045.
MLA Ak Gümüş, Özlem. “Bifurcation Analysis and Chaos Control of a Discrete-Time Prey-Predator Model With Allee Effect”. Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 4, 2023, pp. 1029-45, doi:10.15672/hujms.1179682.
Vancouver Ak Gümüş Ö. Bifurcation analysis and chaos control of a discrete-time prey-predator model with Allee effect. Hacettepe Journal of Mathematics and Statistics. 2023;52(4):1029-45.

Cited By

A STUDY ON STABILITY, BIFURCATION ANALYSIS AND CHAOS CONTROL OF A DISCRETE-TIME PREY-PREDATOR SYSTEM INVOLVING ALLEE EFFECT
Journal of Applied Analysis & Computation
https://doi.org/10.11948/20220532

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