Skip to main content
SpringerLink
Log in

Existence of Positive Periodic Solutions for a Neutral Delay Predator–Prey Model with Hassell–Varley Type Functional Response and Impulse

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

In this paper, we discuss a neutral delay predator–prey model with Hassel–Varley type functional response and impulse is investigated. By using Mawhin coincidence degree theory, we obtain some sufficient conditions for the existence of positive periodic solutions. We extend some known work.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Hassell, M., Varley, G.: New inductive population model for insect parasites and its bearing on biological control. Nature 223, 1133–1136 (1969)

    Article  Google Scholar 

  2. Wang, K.: Periodic solutions to a delayed predator-prey model with Hassell–Varley type functional response. Nonlinear Anal. Real World Appl. 12, 137–145 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Kuang, Y.: On neutral delay logistic Gauss-type predator-prey systems. Dyn. Stab. Syst. 6, 173–189 (1991)

    MATH  Google Scholar 

  4. Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, Boston (1993)

    MATH  Google Scholar 

  5. Zhang, F., Zheng, C.: Positive periodic solutions for the neutral ratio-dependent predator–prey model. Comput. Math. Appl. 61, 2221–2226 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Samoikleno, A.M., Perestyuk, N.A.: Impulsive Differential Equations. World Scientific, Singapore (1995)

    Book  Google Scholar 

  7. Zavalishchhin, S.T., Sesekin, A.N.: Dynamic Impulse Systems, Theory and Applications. Kluwer Academic Publishers Group, Dordrecht (1997)

    Google Scholar 

  8. Huo, H.: Existence of positive periodic solutions of a neutral delay Lotka–Volterra system with impulses. Comput. Math. Appl. 48, 1833–1846 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dai, B., Su, H., Hu, D.: Periodic Solution of a delayed ratio-dependent predator–prey model with monotonic functional response and impulse. Nonlinear Anal. 70, 126–134 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Wang, Q., Dai, B.: Existence of positive periodic solutions for a neutral population model with delays and impulse. Nonlinear Anal. 69, 3919–3930 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Du, Z., Feng, Z.: Periodic solutions of a neutral impulsive predator–prey model with Beddington–DeAngelis functional response with delays. J. Comput. Appl. Math. 258, 87–98 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lv, Y., Du, Z.: Existence and global attractivity of a positive periodic solution to a Lotka–Volterra model with mutual interference and Holling III type functional response. Nonlinear Anal. Real World Appl. 12, 3654–3664 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Du, Z., Lv, Y.: Permanence and almost periodic solution of a Lotka–Volterra model with mutual interference and time delays. Appl. Math. Model. 37(3), 1054–1068 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Terry, A.J.: Predator–prey models with component Allee effect for predator reproduction. J. Math. Biol. 71, 1325–1352 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ducrot, A., Langlais, M.: A singular reaction–diffusion system modelling prey–predator interactions: invasion and co-extinction waves. J. Differ. Equ. 253, 502–532 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fan, M., Kuang, Y.: Dynamics of a nonautonomous predator–prey system with the Beddington–DeAngelis functional response. J. Math. Anal. Appl. 295, 15–39 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gaines, R.E., Mawhin, J.L.: Coincidence Degree and Nonlinear Differential Equations. Springer, Berlin (1977)

    Book  MATH  Google Scholar 

  18. Lu, S.: On the existence of positive periodic solutions to a Lotka–Volterra cooperative population model with multiple delays. Nonlinear Anal. 68, 1746–1753 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lu, S.: On the existence of positive periodic solutions for neutral functional differential equation with multiple deviating arguments. J. Math. Anal. Appl. 280, 321–333 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu, People’s Republic of China

    Xiao Chen & Zengji Du

Authors
  1. Xiao Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zengji Du
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zengji Du.

Additional information

This work is supported by the Natural Science Foundation of China (Grant No. 11471146), and partially supported by PAPD of Jiangsu Province.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, X., Du, Z. Existence of Positive Periodic Solutions for a Neutral Delay Predator–Prey Model with Hassell–Varley Type Functional Response and Impulse. Qual. Theory Dyn. Syst. 17, 67–80 (2018). https://doi.org/10.1007/s12346-017-0223-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12346-017-0223-6

Keywords

Search

Navigation