Multiperiodicity to a Certain Delayed Predator–Prey Model

  • Yang-Yang Li
  • Xiang-Lai ZhuoEmail author
  • Feng-Xue Zhang


The delayed predator–prey system with generalized non-monotonic functional responses and stage structure was investigated in the present paper. By virtue of Mawhin’s coincidence degree and the application of inequalities technique, we are successful to generate some novel conditions to guarantee that the system has at least two positive periodic solutions. It is shown that all parameters of the system have effects on the existence of positive periodic solutions and the period of the coefficients can also affect the existence of positive periodic solutions. In the end, an illustrative example is presented to the feasibility of the main results.


Multiperiodicity Predator–prey model Stage structure Generalized non-monotone functional response 

Mathematics Subject Classification

34C25 92B05 



The authors would like to thank the editors and anonymous reviewers for their constructive suggestions towards upgrading the quality of the manuscript.


  1. 1.
    Andrews, J.F.: A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates. Biotech. Bioeng 10, 707–723 (1968)Google Scholar
  2. 2.
    Bian, F., Zhao, W., Song, Y., et al.: Dynamical analysis of a class of prey–predator model with Beddington–DeAngelis functional response, stochastic perturbation, and impulsive toxicant input. Complexity 2017, 1–18 (2017). zbMATHGoogle Scholar
  3. 3.
    Cao, J., Feng, G., Wang, Y.: Multistability and multiperiodicity of delayed Cohen–Grossberg neural networks with a general class of activation functions. Phys. D Nonlinear Phenom. 237, 1734–1749 (2008)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Chen, F., Wang, H., Lin, Y., Chen, W.: Global stability of a stage-structured predator–prey system. Appl. Math. Comput. 223, 45–53 (2013)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cheng, H., Zhang, T.: A new predator–prey model with a profitless delay of digestion and impulsive perturbation on the prey. Appl. Math. Comput. 217(22), 9198–9208 (2011)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Chi, M., Zhao, W.: Dynamical analysis of multi-nutrient and single microorganism chemostat model in a polluted environment. Adv. Differ. Equ. 2018(1), 120 (2018)MathSciNetGoogle Scholar
  7. 7.
    Devi, S.: Effects of prey refuge on a ratio-dependent predator–prey model with stage-structure of prey population. Appl. Math. Model. 37, 4337–4349 (2013)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Dong, X., Bai, Z., Zhang, S.: Positive solutions to boundary value problems of p-Laplacian with fractional derivative. Bound. Value Prob. 5(1), 1–15 (2017)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Feng, T., Meng, X., Liu, L., et al.: Application of inequalities technique to dynamics analysis of a stochastic eco-epidemiology model. J. Inequal. Appl. 2016(1), 327 (2016)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Gains, R.E., Mawhin, J.L.: Coincidence Degree and Nonlinear Differential Equations. Springer, Berlin (1977)Google Scholar
  11. 11.
    Georgescu, P., Hsieh, Y.H.: Global dynamics of a predator–prey model with stage structure for the predator. SIAM J. Appl. Math. 67, 1379–1395 (2007)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Han, M.: On the maximum number of periodic solutions of piecewise smooth periodic equations by average method. J. Appl. Anal. Comput. 7(2), 788–794 (2017)MathSciNetGoogle Scholar
  13. 13.
    Han, M., Hou, X., Sheng, L., Wang, C.: Theory of rotated equations and applications to a population model. Discret. Contin. Dyn. Syst. A 38(4), 2171–2185 (2018)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Huang, C., Cao, J.: Comparative study on bifurcation control methods in a fractional-order delayed predator–prey system. Sci. China Technol. Sci. 61(7), 1–10 (2018)Google Scholar
  15. 15.
    Huang, C., Cao, J., Xiao, M., et al.: Controlling bifurcation in a delayed fractional predator-prey system with incommensurate orders. Appl. Math. Comput. 293, 293–310 (2017)MathSciNetGoogle Scholar
  16. 16.
    Li, Z., Han, M., Chen, F.: Almost periodic solutions of a discrete almost periodic logistic equation with delay. Appl. Math. Comput. 232, 743–751 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Liu, G., Wang, X., Meng, X., Gao, S.: Extinction and persistence in mean of a novel delay impulsive stochastic infected predator-prey system with jumps. Complexity (2017).
  18. 18.
    Liu, X., Han, M.: Chaos and Hopf bifurcation analysis for a two species predator–prey system with prey refuge and diffusion. Nonlinear Anal. Real World Appl. 12(2), 1047–1061 (2011)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Liu, L., Meng, X.: Optimal harvesting control and dynamics of two-species stochastic model with delays. Adv. Differ. Equ. 2017(1), 18 (2017)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Lv, X., Wang, L., Meng, X.: Global analysis of a new nonlinear stochastic differential competition system with impulsive effect. Adv. Differ. Equ. 2017(1), 296 (2017)MathSciNetGoogle Scholar
  21. 21.
    Meng, X., Zhang, L.: Evolutionary dynamics in a Lotka–Volterra competition model with impulsive periodic disturbance. Math. Methods Appl. Sci. 39(2), 177–188 (2016)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Meng, X., Chen, L., Wu, B.: A delay SIR epidemic model with pulse vaccination and incubation times. Nonlinear Anal. Real World Appl. 11(1), 88–98 (2010)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Meng, X., Liu, R., Zhang, T.: Adaptive dynamics for a non-autonomous Lotka–Volterra model with size-selective disturbance. NoNonlinear Anal. Real World Appl. 16, 202–213 (2014)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Meng, X., Zhao, S., Zhang, W.: Adaptive dynamics analysis of a predator–prey model with selective disturbance. Appl. Math. Comput. 266, 946–958 (2015)MathSciNetGoogle Scholar
  25. 25.
    Wang, F., Kuang, Y., Ding, C., Zhang, S.: Stability and bifurcation of a stage-structured predator–prey model with both discrete and distributed delays. Chaos Solitons Fractals 46, 19–27 (2013)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Wang, J., Cheng, H., Meng, X., et al.: Geometrical analysis and control optimization of a predator–prey model with multi state-dependent impuls. Adv. Differ. Equ. 2017(1), 252 (2017)zbMATHGoogle Scholar
  27. 27.
    Wang, J., Cheng, H., Li, Y., Zhang, X.: The geometrical analysis of a predator–prey model with multi-stage dependent impulses. J. Appl. Anal. Comput. 8(2), 427–442 (2018)MathSciNetGoogle Scholar
  28. 28.
    Wolkowicz, G.S.K., Zhu, H., Campbell, S.: Bifurcation analysis of a predator–prey system with nonmonotonic functional response. SIAM J. Appl. Math. 63, 636–682 (2003)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Xia, Y., Han, M.: Multiple periodic solutions of a ratio-dependent predator–prey model. Chaos Solitons Fractals 39(3), 1100–1108 (2009)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Xia, Y., Cao, J., Cheng, S.: Multiple periodic solutions of a delayed stage-structured predator–prey model with non-monotone functional responses. Appl. Math. Model. 31, 1947–1959 (2007a)zbMATHGoogle Scholar
  31. 31.
    Xu, R., Chaplain, M.A.J., Davidson, F.A.: Permanence and periodicity of a delayed ratio- dependent predator–prey model with stage structure. J. Math. Anal. Appl. 303, 602–621 (2005)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Yu, P., Han, M., Xiao, D.: Four small limit cycles around a Hopf singular point in 3-dimensional competitive Lotka–Volterra systems. J. Math. Anal. Appl. 436(1), 521–555 (2016)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Zhang, T., Meng, X., Song, Y., et al.: A stage-structured predator–prey SI model with disease in the prey and impulsive effects. Math. Modell. Anal. 18(4), 505–528 (2013)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Zhang, T., Ma, W., Meng, X., et al.: Periodic solution of a prey–predator model with nonlinear state feedback control. Appl. Math. Comput. 266, 95–107 (2015)MathSciNetGoogle Scholar
  35. 35.
    Zhang, T., Meng, X., Zhang, T.: Global analysis for a delayed SIV model with direct and environmental transmissions. J. Appl. Anal. Comput. 6(2), 479 (2016)MathSciNetGoogle Scholar
  36. 36.
    Zhang, W., Bai, Z., Sun, S.: Extremal solutions for some periodic fractional differential equations. Adv. Differ. Equ. 2016(1), 1–8 (2016)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Zhang, T., Zhang, T., Meng, X.: Stability analysis of a Chemostat model with maintenance energy. Appl. Math. Lett. 68, 1–7 (2017)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Zhuo, X.: Global attractability and permanence for a new stage-structured delay impulsive ecosystem. J. Appl. Anal. Comput. 8(2), 457–457 (2018)MathSciNetGoogle Scholar
  39. 39.
    Zhuo, X., Zhang, F.: Stability for a new discrete ratio-dependent predator–prey system. Qual. Theory Dyn. Syst. 17(1), 189–202 (2018)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.College of Mathematics and Systems ScienceShandong University of Science and TechnologyQingdaoChina
  2. 2.College of Mining and Safety EngineeringShandong University of Science and TechnologyQingdaoChina

Personalised recommendations