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Homoclinic bifurcation of a state feedback impulsive controlled prey–predator system with Holling-II functional response

  • Meng ZhangEmail author
  • Lansun Chen
  • Zeyu Li
Original paper
  • 5 Downloads

Abstract

Keeping ecological balance between prey and predator population in ocean fishery is of great importance in fishery. A free developed two species prey–predator model with Holling-II functional response is carried out, and its fine case equilibrium is discussed. Then, the state feedback model which assumes the harvest of predator and supplement of prey as impulsive disturbance is presented to find the dynamical balance between prey and predator population. The existence of homoclinic cycle and order-1 periodic solution and the orbital asymptotical stability of the order-1 periodic solution are proved. Finally, some numerical simulations are displayed to confirm the results obtained in the paper.

Keywords

State feedback Impulsive model Homoclinic bifurcation Order-1 periodic solution 

Notes

Acknowledgements

We would like to sincerely thank the reviewers for their careful reading and constructive opinions of the original manuscript.

Funding

This work is supported by NSFC (No. 11701026) and the Fundamental Research Funds for Beijing University of Civil Engineering and Architecture, China (X18225, X19031) for M. Zhang, NSFC (No. 11671346, No. 61751317) for L. Chen. M. Zhang would like to thank the China Scholarship Council for financial support of her overseas study(No. 201808110071).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Christensen, V., Pauly, D.: ECOPATH II-a software for balancing steady-state ecosystem models and calculating network characteristics. Ecol. Model. 61(3–4), 169–185 (1992)CrossRefGoogle Scholar
  2. 2.
    Bogstad, B., Hauge, K.H., Ulltang, O.: MULTSPEC-a multi-species model for fish and marine mammals in the Barents sea. J. Northwest Atl. Fish. Sci. 22, 317–341 (1997)CrossRefGoogle Scholar
  3. 3.
    Clark, J.S., Gelfand, A.E.: Hierarchical Modelling for the Environmental Sciences: Statistical Methods and Applications. Oxford University Press, Oxford (2006)Google Scholar
  4. 4.
    Fulton, E.A., Link, J.S., Kaplan, I.C., et al.: Lessons in modelling and management of marine ecosystems: the Atlantis experience. Fish Fish. 12(2), 171–188 (2011)CrossRefGoogle Scholar
  5. 5.
    Brown, C.J., Parker, B., Ahmadia, G.N., et al.: The cost of enforcing a marine protected area to achieve ecological targets for the recovery of fish biomass. Biol. Conserv. 227, 259–265 (2018)CrossRefGoogle Scholar
  6. 6.
    Okamura, H., Ichinokawa, M., Komori, O.: Fish Population Dynamics, Monitoring, and Management. Springer, Tokyo (2018)Google Scholar
  7. 7.
    Wei, C., Chen, L.: Heteroclinic bifurcations of a prey–predator fishery model with impulsive harvesting. Int. J. Biomath. 6(05), 1350031 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Wei, C., Chen, L.: Dynamic analysis of mathematical model of ethanol fermentation with gas stripping. Nonlinear Dyn. 57(1–2), 13–23 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Wei, C., Chen, L.: Periodic solution of prey–predator model with Beddington–DeAngelis functional response and impulsive state feedback control. J. Appl. Math. 2012, 607105 (2012)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Wei, C., Chen, L.: Periodic solution and heteroclinic bifurcation in a predator–prey system with Allee effect and impulsive harvesting. Nonlinear Dyn. 76(2), 1109–1117 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Zhang, M., Song, G., Chen, L.: A state feedback impulse model for computer worm control. Nonlinear Dyn. 85(3), 1561–1569 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chen, L., Liang, X., Pei, Y.: The periodic solutions of the impulsive state feedback dynamical system. Commun. Math. Biol. Neurosci. 2018, 14 (2018)Google Scholar
  13. 13.
    Gritli, H., Belghith, S.: Walking dynamics of the passive compass-gait model under OGY-based state-feedback control: rise of the Neimark–Sacker bifurcation. Chaos, Solitons Fractals 110, 158–168 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gritli, H., Belghith, S.: Diversity in the nonlinear dynamic behavior of a one-degree-of-freedom impact mechanical oscillator under OGY-based state-feedback control law: order, chaos and exhibition of the border-collision bifurcation. Mech. Mach. Theory 124, 1–41 (2018)CrossRefGoogle Scholar
  15. 15.
    Xu, J., Tian, Y., Guo, H., et al.: Dynamical analysis of a pest management Leslie–Gower model with ratio-dependent functional response. Nonlinear Dyn. 93(2), 705–720 (2018)CrossRefzbMATHGoogle Scholar
  16. 16.
    Zhao, Z., Pang, L., Song, X.: Optimal control of phytoplankton-fish model with the impulsive feedback control. Nonlinear Dyn. 88(3), 2003–2011 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Guo, H., Chen, L., Song, X.: Qualitative analysis of impulsive state feedback control to an algae-fish system with bistable property. Appl. Math. Comput. 271, 905–92 (2015)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Elaiw, A.M.: Global properties of a class of virus infection models with multitarget cells. Nonlinear Dyn. 69, 423–435 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Elaiw, A.M., Abukwaik, R.M., Alzahrani, E.O.: Global properties of a cell mediated immunity in HIV infection model with two classes of target cells and distributed delays. Int. J. Biomath. 7(7), 119–143 (2014)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Muroya, Y., Enatsu, Y., Nakata, Y.: Monotone iterative techniques to SIRS epidemic models with nonlinear incidence rates and distributed delays. Nonlinear Anal.: Real World Appl. 12(4), 1897–1910 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gu, E., Tian, F.: Complex dynamics analysis for a duopoly model of common fishery resource. Nonlinear Dyn. 61(4), 579–590 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Zhang, X., Chen, L., Neumann, A.: The stage-structured predator–prey model and optimal harvesting policy. Math. Biosci. 168(2), 201–210 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rojas-Palma, A., Gonzalez-Olivares, E.: Optimal harvesting in a predator–prey model with Allee effect and sigmoid functional response. Appl. Math. Model. 36(5), 1864–1874 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Pal, D., Mahaptra, G., Samanta, G.: Optimal harvesting of prey–predator system with interval biological parameters: a bioeconomic model. Math. Biosci. 241(2), 181–187 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Tang, S., Chen, L.: Density-dependent birth rate, birth pulses and their population dynamic consequences. J. Math. Biol. 44(2), 185–199 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Tang, S., Xiao, Y., Chen, L., et al.: Integrated pest management models and their dynamical behaviour. Bull. Math. Biol. 67(1), 115–135 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Sun, G., Wu, Z., Wang, Z., et al.: Influence of isolation degree of spatial patterns on persistence of populations. Nonlinear Dyn. 83(1–2), 811–819 (2016)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Sun, G.: Mathematical modeling of population dynamics with Allee effect. Nonlinear Dyn. 85(1), 1–12 (2016)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Sun, G., Wang, C., Wu, Z.: Pattern dynamics of a Gierer–Meinhardt model with spatial effects. Nonlinear Dyn. 88(2), 1385–1396 (2017)CrossRefGoogle Scholar
  30. 30.
    Ye, Y.: Limit Cycle Theory. Shanghai Science and Technology Press, Shanghai (1984). (in Chinese)Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of ScienceBeijing University of Civil Engineering and ArchitectureBeijingChina
  2. 2.Department of MathematicsPurdue UniversityWest LafayetteUSA
  3. 3.Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  4. 4.Canvard CollegeBeijing Technology and Business UniversityBeijingChina

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