Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Flip bifurcation of a discrete predator-prey model with modified Leslie-Gower and Holling-type III schemes

1 College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
2 College of Mining and Safety Engineering, Shandong University of Science and Technology, Qingdao 266590, China

The continuous predator-prey model is one of the main models studied in recent years. The dynamical properties of these models are so complex that it is an urgent topic to be studied. In this paper, we transformed a continuous predator-prey model with modified Leslie-Gower and Hollingtype III schemes into a discrete mode by using Euler approximation method. The existence and stability of fixed points for this discrete model were investigated. Flip bifurcation analyses of this discrete model was carried out and corresponding bifurcation conditions were obtained. Provided with these bifurcation conditions, an example was given to carry out numerical simulations, which shows that the discrete model undergoes flip bifurcation around the stable fixed point. In addition, compared with previous studies on the continuous predator-prey model, our discrete model shows more irregular and complex dynamic characteristics. The present research can be regarded as the continuation and development of the former studies.
  Figure/Table
  Supplementary
  Article Metrics

Keywords flip bifurcation; Euler approximation method; discrete predator-prey model; stability; center manifold theorem

Citation: Yangyang Li, Fengxue Zhang, Xianglai Zhuo. Flip bifurcation of a discrete predator-prey model with modified Leslie-Gower and Holling-type III schemes. Mathematical Biosciences and Engineering, 2020, 17(3): 2003-2015. doi: 10.3934/mbe.2020106

References

  • 1. N. Wang, M. Han, Relaxation oscillations in predator-prey model with distributed delay, Comput. Appl. Math., 37 (2018), 475-484.
  • 2. T. Zhang, X. Meng, T. Zhang, Global analysis for a delayed SIV model with direct and environmental transmissions, J. Appl. Anal. Comput., 6 (2016), 479-491.
  • 3. B. S. Chen, X. Lin, W. Zhang, T. Zhou, On the system entropy and energy dissipativity of stochastic systems and their application in biological systems, Complexity, 2018 (2018), 1628472.
  • 4. T. Zhang, N. Gao, T. Wang, H. Liu, Z. Jiang, Global dynamics of a model for treating microorganisms in sewage by periodically adding microbial flocculants, Math. Bio. Eng., 17 (2020), 179-201.
  • 5. M. Han, X. Hou, L. Sheng, C. Wang, Theory of rotated equations and applications to a population model, Discrete Contin. Dyn. Syst. A, 38 (2018), 2171-2185.
  • 6. M. Chi, W. Zhao, Dynamical analysis of multi-nutrient and single microorganism chemostat model in a polluted environment, Adv. Differ. Eq., 2018 (2018), 8719067.
  • 7. N. Gao, Y. Song, X. Wang, J. Liu, Dynamics of a stochastic SIS epidemic model with nonlinear incidence rates, Adv. Differ. Eq., 2019 (2019).
  • 8. X. Dong, Z. Bai, S. Zhang, Positive solutions to boundary value problems of p-Laplacian with fractional derivative, Boundary Value Probl., 2017 (2017).
  • 9. X. Meng, S. Zhao, W. Zhang, Adaptive dynamics analysis of a predator-prey model with selective disturbance, Appl. Math. Comput., 266 (2015), 946-958.
  • 10. T. Ma, X. Meng, Z. Chang, Dynamics and optimal harvesting control for a stochastic one-predatortwo-prey time delay system with jumps, Complexity, 2019 (2019), 5342031.
  • 11. X. Meng, F. Li, S. Gao, Global analysis and numerical simulations of a novel stochastic ecoepidemiological model with time delay, Appl. Math. Comput., 339 (2018), 701-726.
  • 12. X. Liang, R. Wang, Global well-posedness and dynamical behavior of delayed reaction-diffusion BAM neural networks driven by Wiener processes, IEEE Access, 6 (2018), 69265-69278.
  • 13. T. Zhang, T. Xu, J. Wang, Y. Song, Z. Jiang, Geometrical analysis of a pest management model in food-limited environments with nonlinear impulsive state feedback control, J. Appl. Anal. Comput., 9 (2019), 2261-2277.
  • 14. S. Gao, L. Luo, S. Yan, X. Meng, Dynamical behavior of a novel impulsive switching model for HLB with seasonal fluctuations, Complexity, 2018 (2018), 2953623.
  • 15. J. Wang, H. Cheng, Y. Li, X. Zhang, The geometrical analysis of a predator-prey model with multi-state dependent impulses, J. Appl. Anal. Comput., 8 (2018), 427-442.
  • 16. X.Lv, X. Meng, X. Wang, Extinction and stationary distribution of an impulsive stochastic chemostat model with nonlinear perturbation, Chaos, Solitons Fractals, 110 (2018), 273-279.
  • 17. F. Bian, W. Zhao, Y. Song, R. Yue, Dynamical analysis of a class of prey-predator model with beddington-deangelis functional response, stochastic perturbation, and impulsive toxicant input, Complexity, 2017 (2017), 3742197.
  • 18. J. Wang, H. Cheng, X. Meng, B. G. S. A. Pradeep, Geometrical analysis and control optimization of a predator-prey model with multi state-dependent impulse, Adv. Differ. Eq., 2017 (2017), 252.
  • 19. Z. Jiang, X. Bi, T. Zhang, B.G. S. A. Pradeep, Global Hopf bifurcation of a delayed phytoplanktonzooplankton system considering toxin producing effect and delay dependent coefficient, Math. Biosci. Eng., 16 (2019), 3807-3829.
  • 20. Y. Li, H. Cheng, Y. Wang, A Lycaon pictus impulsive state feedback control model with Allee effect and continuous time delay, Adv. Diff. Eq., 2018 (2018), 367.
  • 21. G. Liu, Z. Chang, X. Meng, Asymptotic analysis of impulsive dispersal predator-prey systems with Markov switching on finite-state space, J. Funct. Spaces, 2019 (2019), 1-18.
  • 22. X. Zhuo, Global attractability and permanence for a new stage-structured delay impulsive ecosystem, J. Appl. Anal. Comput., 8 (2018), 457-470.
  • 23. X. Feng, Y. Song, J. Liu, G. Wang, Permanence, stability, and coexistence of a diffusive predatorprey model with modified Leslie-Gower and B-D functional response, Adv. Differ. Eq., 2018 (2018), 314.
  • 24. X. Zhuo, F. Zhang, Stability for a new discrete ratio-dependent predator-prey system, Qual. Theory Dyn. Syst., 17 (2018), 189-202.
  • 25. Z. Bai, Z. Shuo, S. Sun, C. Yin, Monotone iterative method for fractional differential equations, Elect. J. Differ. Eq., 2016 (2016), 06.
  • 26. T. Zhang, W. Ma, X. Meng, T. Zhang, Periodic solution of a prey-predator model with nonlinear state feedback control, Appl. Math. Comput., 266 (2015), 95-107.
  • 27. Y. Li, X. Zhuo, F. Zhang, Multiperiodicity to a certain delayed predator-prey model, Qual. Theory Dyn. Syst., 18 (2019), 793-811.
  • 28. K. Liu, T. Zhang, L. Chen, State-dependent pulse vaccination and therapeutic strategy in an SI epidemic model with nonlinear incidence rate, Comput. Math. Methods Med., 2019 (2019), 3859815.
  • 29. X. Zhang, Q. Zhang, V. Sreeram, Bifurcation analysis and control of a discrete harvested preypredator system with Beddington-DeAngelis functional response, J. Franklin Inst., 347 (2010), 1076-1096.
  • 30. M. Han, V. G. Romanovski, X. Zhang, Equivalence of the Melnikov function method and the averaging method, Qual. Theory Dyn. Sys., 15 (2016), 471-479.
  • 31. M. Han, L. Sheng, Bifurcation of limit cycles in piecewise smooth systems via Melnikov function, J. Appl. Anal. Comput., 5 (2015), 809-815.
  • 32. M. Han, L. Sheng, X. Zhang, Bifurcation theory for finitely smooth planar autonomous differential systems, J. Differ. Eq., 264 (2018), 3596-3618.
  • 33. Q. Song, R. Yang, C. Zhang, L. Tang, Bifurcation analysis in a diffusive predator-prey system with michaelis-menten-type predator harvesting, Adv. Differ. Eq., 2018 (2018), 329.
  • 34. X. Liang, X. Zhuo, R. Wang, Global attractor of reaction-diffusion gene regulatory networks with S-Type delay, Neural Process. Lett., 2019 (2019), 1-21.
  • 35. F. Liu, R. Yang, L. Tang, Hopf bifurcation in a diffusive predator-prey model with competitive interference, Chaos, Solitons Fractals, 120 (2019), 250-258.
  • 36. Z. Li, M. Han, F. Chen, Almost periodic solutions of a discrete almost periodic logistic equation with delay, Appl. Math. Comput., 232 (2014), 743-751.
  • 37. J. Zhao, Y. Yan, Stability and bifurcation analysis of a discrete predator-prey system with modified Holling-Tanner functional response, Adv. Differ. Eq., 2018 (2018), 402.
  • 38. Y. A. Kuznetsov, Elements of Applied Bifurcation Throry, 2nd edition, Springer-Verlag, New York, 1998.
  • 39. J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical System and Bifurcation of Vector Fields, Springer-Verlag, New York, 2002.
  • 40. J. Carr, Application of Center Manifold Theory, Springer-Verlag, New York, 1981.
  • 41. S. Huang, S. Ruan, D. Xiao, Bifurcations in a discrete predator-prey model with nonmonotonic functional response, J. Math. Anal. Appl., 464 (2018), 201-230.
  • 42. L. Meng, Y. Han, Z. Lu, G. Zhang, Bifurcation, chaos, and pattern formation for the discrete predator-prey reaction-diffusion model, Discrete Dyn. Nat. Soc., 2019 (2019), 9592878.
  • 43. E. M. Elabbasy, A. A. Elsadany, Y. Zhang, Bifurcation analysis and chaos in a discrete reduced Lorenz system, Appl. Math. Comput., 228 (2014), 184-194.
  • 44. S. Dange, K. Manoj, S. Banerjee1, S. A. Pawar, S. Mondal, R. I. Sujith, Oscillation quenching and phase-flip bifurcation in coupled thermoacoustic systems, Chaos: Interdiscip. J. Nonlinear Sci., 29 (2019), 093135.
  • 45. H. Kitajima, T. Yazawa, Flip-flip bifurcation in a mathematical cardiac system, Int. J. Bifurcation Chaos, 29 (2019), 1950045.
  • 46. M. Chi, W. Zhao, Dynamical analysis of two-microorganism and single nutrient stochastic chemostat model with monod-haldane response function, Complexity, 2019 (2019), 8719067.

 

Reader Comments

your name: *   your email: *  

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved