Export file:

Format

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

Content

  • Citation Only
  • Citation and Abstract

Bifurcation analysis in a singular Beddington-DeAngelis predator-prey model with two delays and nonlinear predator harvesting

School of Science, Lanzhou University of Technology, Lanzhou, Gansu 730050, P. R. China

Special Issues: Differential Equations in Mathematical Biology

In this paper, a differential algebraic predator-prey model including two delays, Beddington-DeAngelis functional response and nonlinear predator harvesting is proposed. Without considering time delay, the existence of singularity induced bifurcation is analyzed by regarding economic interest as bifurcation parameter. In order to remove singularity induced bifurcation and stabilize the proposed system, state feedback controllers are designed in the case of zero and positive economic interest respectively. By the corresponding characteristic transcendental equation, the local stability of interior equilibrium and existence of Hopf bifurcation are discussed in the different case of two delays. By using normal form theory and center manifold theorem, properties of Hopf bifurcation are investigated. Numerical simulations are given to demonstrate our theoretical results.
  Figure/Table
  Supplementary
  Article Metrics

Keywords bioeconomic system; predator-prey model; nonlinear predator harvesting; Beddington-DeAngelis functional response; singularity induced bifurcation; Hopf bifurcation

Citation: Xin-You Meng, Yu-Qian Wu. Bifurcation analysis in a singular Beddington-DeAngelis predator-prey model with two delays and nonlinear predator harvesting. Mathematical Biosciences and Engineering, 2019, 16(4): 2668-2696. doi: 10.3934/mbe.2019133

References

  • 1. M. Kot, Elements of Mathematical Biology, Cambridge University Press, Cambridge, 2001.
  • 2. S. Levin, T. Hallam and J. Cross, Applied Mathematical Ecology, Springer, New York, 1990.
  • 3. R. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, 1993.
  • 4. C. Ji, D. Jing and N. Shi, A note on a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbatio, J. Math. Anal. Appl., 377 (2011), 435–440.
  • 5. T. Kar and H. Matsuda, Global dynamics and controllability of a harvested prey-predator system with Holling type III functional response, Nonlinear Anal. Hybrid Syst., 1 (2007), 59–67.
  • 6. X. Meng, J. Wang, H. Huo, Dynamical behaviour of a nutrient-plankton model with Holling type IV, delay, and harvesting, Discrete Dyn. Nat. Soc., 2018 (2018), Article ID 9232590.
  • 7. X. Meng, H. Huo, H. Xiang, et al., Stability in a predator-prey model with Crowley-Martin function and stage structure for prey, Appl. Math. Comput., 232 (2014), 810–819.
  • 8. W. Yang, Diffusion has no influence on the global asymptotical stability of the Lotka-Volterra pre-predator model incorporating a constant number of prey refuges, Appl. Math. Comput., 223 (2013), 278–280.
  • 9. Y. Zhu and K.Wang, Existence and global attractivity of positive periodic solutions for a predatorprey model with modified Leslie-Gower Holling-type II schemes, J. Math. Anal. Appl., 384 (2011), 400–408.
  • 10. J. Liu, Dynamical analysis of a delayed predator-prey system with modified Leslie-Gower and Beddington-DeAngelis functional response, Adv. Difference Equ., 2014 (2014), 314–343.
  • 11. X. Liu and Y. Wei, Dynamics of a stochastic cooperative predator-prey system with Beddington- DeAngelis functional response, Adv. Difference Equ., 2016 (2016), 21–39.
  • 12. C. Li, X. Guo and D. He, An impulsive diffusion predator-prey system in three-species with Beddington-DeAngelis response, J. Appl. Math. Comput., 43 (2013), 235–248.
  • 13. T. Ivanov and N. Dimitrova, A predator-prey model with generic birth and death rates for the predator and Beddington-DeAngelis functional response, Math. Comput. Simulat., 133 (2017), 111–123.
  • 14. Q. Meng and L. Yang, Steady state in a cross-diffusion predator-prey model with the Beddington- DeAngelis functional response, Nonlinear Anal.: Real World Appl., 45 (2019), 401–413.
  • 15. W. Liu, C. Fu and B. Chen, Hopf bifurcation and center stability for a predator-prey biological economic model with prey harvesting, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 3989– 3998.
  • 16. F. Conforto, L. Desvillettes and C. Soresina, About reaction-diffusion systems involving the Holling-type II and the Beddington-DeAngelis functional responses for predator-prey models, Nonlinear Differ. Equ. Appl., 25 (2018), 24.
  • 17. X. Sun, R. Yuan and L. Wang, Bifurcations in a diffusive predator-prey model with Beddington- DeAngelis functional response and nonselective harvesting, J. Nonlinear Sci., 29 (2019), 287–318.
  • 18. J. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol., 44 (1975), 331–340.
  • 19. D. DeAngilis, R. Goldstein and R. Neill, A model for tropic interaction, Ecology, 56 (1975), 881–892.
  • 20. H. Freedman, Deterministic Mathematical Models in Population Ecology, Marcel Dekker, New York, 1980.
  • 21. G. Seifert, Asymptotical behavior in a three-component food chain model, Nonlinear Anal. Theory Methods Appl., 32 (1998), 749–753.
  • 22. C. Liu, Q. Zhang, X. Zhang, et al., Dynamical behavior in a stage-structured differential-algebraic prey-predator model with discrete time delay and harvesting, J. Comput. Appl. Math., 231 (2009), 612–625.
  • 23. X. Meng, H. Huo, X. Zhang, et al., Stability and hopf bifurcation in a three-species system with feedback delays, Nonlinear Dyn., 64 (2011), 349–364.
  • 24. X. Meng and Y. Wu, Bifurcation and control in a singular phytoplankton-zooplankton-fish model with nonlinear fish harvesting and taxation, Int. J. Bifurcat. Chaos, 28 (2018), 1850042(24 pages).
  • 25. H. Xiang, Y. Wang and H. Huo, Analysis of the binge drinking models with demographics and nonlinear infectivity on networks, J. Appl. Anal. Comput., 8 (2018), 1535–1554.
  • 26. K. Chakraborty, M. Chakraboty and T. Kar, Bifurcation and control of a bioeconomic model of a prey-predator system with a time delay, Nonlinear Anal. Hybrid Syst., 5 (2011), 613–625.
  • 27. W. Liu, C. Fu and B. Chen, Hopf birfucation for a predator-prey biological economic system with Holling type II functional response, J. Franklin Inst., 348 (2011), 1114–1127.
  • 28. G. Zhang, B. Chen, L. Zhu, et al., Hopf bifurcation for a differential-algebraic biological economic system with time delay, Appl. Math. Comput, 218 (2012), 7717–7726.
  • 29. T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433–463.
  • 30. G. Zhang, Y. She and B. Chen, Hopf bifurcation of a predator prey system with predator harvesting and two delays, Nonlinear Dyn., 73 (2013), 2119–2131.
  • 31. J. Zhang, Z. Jin, J. Yan, et al., Stability and Hopf bifurcation in a delayed competition system, Nonlinear Anal. Theory Methods Appl., 70 (2009), 658–670.
  • 32. Z. Lajmiri, R. K. Ghaziani and I. Orak, Bifurcation and stability analysis of a ratio-dependent predator-prey model with predator harvesting rate, Chaos Soliton Fract., 106 (2018), 193–200.
  • 33. M. Liu, X. He and J. Yu, Dynamics of a stochastic regime-switching predator-prey model with harvesting and distributed delays, Nonlinear Anal. Hybrid Syst., 28 (2018), 87–104.
  • 34. T. Das, R. Mukerjee and K. Chaudhuri, Harvesting of a prey-predator fishery in the presence of toxicity, Appl. Math. Model., 33 (2009), 2282–2292.
  • 35. R. Gupta and P. Chandra, Bifurcation analysis of modied Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, J. Math. Anal. Appl., 398 (2013), 278–295.
  • 36. P. Srinivasu, Bioeconomics of a renewable resource in presence of a predator, Nonlinear Anal. Real World Appl., 2 (2001), 497–506.
  • 37. G. Lan, Y. Fu, C. Wei, et al., Dynamical analysis of a ratio-dependent predator-prey model with Holling III type functional response and nonlinear harvesting in a random environment, Adv. Differ. Equ., 2018 (2018), 198.
  • 38. R. Gupta and P. Chandra, Dynamical complexity of a prey-predator model with nonlinear predator harvesting, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 423–443.
  • 39. J. Liu and L. Zhang, Bifurcation analysis in a prey-predator model with nonlinear predator harvesting, J. Franklin Inst., 353 (2016), 4701–4714.
  • 40. K. Chakraborty, S. Jana and T. Kar, Global dynamics and bifurcation in a stage structured preypredator fishery model with harvesting, Appl. Math. Comput., 218 (2012), 9271–9290.
  • 41. H. Gordon, The economic theory of a common property resource: the fishery, Bull. Math. Biol., 62 (1954), 124–142.
  • 42. C. Liu, Q. Zhang and X. Duan, Dynamical behavior in a harvested differential-algebraic preypredator model with discrete time delay and stage structure, J. Franklin Inst., 346 (2009), 1038– 1059.
  • 43. X. Zhang and Q. Zhang, Bifurcation analysis and control of a class of hybrid biological economic models, Nonlinear Anal. Hybrid Syst., 3 (2009), 578–587.
  • 44. C. Liu, N. Lu, Q. Zhang, et al., Modelling and analysis in a prey-predator system with commercial harvesting and double time delays, J. Appl. Math. Comput., 281 (2016), 77–101.
  • 45. M. Li, B. Chen and H. Ye, A bioeconomic differential algebraic predator-prey model with nonlinear prey harvesting, Appl. Math. Model., 42 (2017), 17–28.
  • 46. P. Leslie and J. Gower, The properties of a stochastic model for the predator prey type of interaction between two species, Biometrika, 47 (1960), 219–234.
  • 47. R. Cantrell and C. Cosner, On the dynamics of predator-prey models with the Beddington- DeAngelis functional response, J. Math. Anal. Appl., 257 (2001), 206–222.
  • 48. L. Dai, Singular Control System, Springer, New York, 1989.
  • 49. V. Venkatasubramanian, H. Schattler and J. Zaborszky, Local bifurcations and feasibility regions in differential-algebraic systems, IEEE Trans. Automat. Control, 40 (1995), 1992–2013.
  • 50. Q. Zhang, C. Liu and X. Zhang, A singular bioeconomic model with diffusion and time delay, J. Syst. Sci. Complex., 24 (2011), 277–190.
  • 51. J. Hale, Theory of Functional Differential Equations, Springer, New York, 1997.
  • 52. B. Hassard, N. Kazarinoff and Y. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.
  • 53. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983.
  • 54. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993.
  • 55. H. Freedman and V. S. H. Rao, The trade-off between mutual interference and time lags in predator-prey systems, Bull. Math. Biol., 45 (1983), 991–1004.

 

Reader Comments

your name: *   your email: *  

© 2019 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