Skip to main content
SpringerLink
Log in

Traveling Wave Solution of a Reaction–Diffusion Predator–Prey System

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

Abstract

In this paper, we discuss a reaction–diffusion predator–prey model with nonlocal delays. By using the theory of dynamical systems, specifically based on geometric singular perturbation theory, center manifold theorem and Fredholm theory, we construct an invariant manifold for the associated predator–prey equation and use this invariant manifold to obtain a heteroclinic orbit between two non-negative equilibrium points. Furthermore, we establish the existence result of traveling wave solution for the predator–prey model.

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. 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 

  2. Marray, J.: Mathematical Biology, 2nd edn. Springer, New York (1998)

    Google Scholar 

  3. Lin, X., Du, Z., Lv, Y.: Global asymptotic stability of almost periodic solution for a multispecies competition-predator system with time delays. Appl. Math. Comput. 219, 4908–4923 (2013)

    MathSciNet  MATH  Google Scholar 

  4. 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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Huang, Y.L., Lin, G.: Traveling wave solutions in a diffusive system with two preys and one predator. J. Math. Anal. Appl. 418, 163–184 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Du, Z., Feng, Z., Zhang, X.: Traveling wave phenomena of n-dimensional diffusive predator–prey systems. Nonlinear Anal. Real World Appl. 41, 288–312 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bravo, J.L., Fernández, M., Gámez, M., Granados, B., Tineo, A.: Existence of a polycycle in non-Lipschitz Gause-type predator–prey model. J. Math. Anal. Appl. 373, 512–520 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Du, Z., Feng, Z., Zhang, X.: Traveling wave phenomena of n-dimensional diffusive predator-prey systems. Nonlinear Anal. Real World Appl. 41, 288–312 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jones, D.S., Sleeman, B.D.: Differential Equations and Mathematical Biology. Chapman & Hall/CRC, London (2003)

    MATH  Google Scholar 

  10. Ge, Z., He, Y.: Traveling wavefronts for a two-species predator–prey system with diffusion terms and stage structure. Appl. Math. Model. 33, 1356–1365 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gourley, S.A., Britton, N.F.: A predator–prey reaction–diffusion system with nonlocal effects. J. Math. Biol. 34, 297–333 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gourley, S.A., Kuang, Y.: Wavefronts and global stability in a time-delayed population model with stage structure. Proc. R. Soc. Lond. 459, 1563–1579 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gourley, S.A., Ruan, S.: Convergence and traveling fronts in functional equations with nonlocal terms: a competition model. SIAM J. Math. Anal. 35, 806–822 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Thieme, H.R., Zhao, X.Q.: A non-local delayed and diffusive predator–prey model. Nonlinear Anal. Real World Appl. 2, 145–160 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wang, Z., Li, W.: Monotone traveling fronts of a food-limited population model with nonlocal delay. Nonlinear Anal. Real World Appl. 8, 699–712 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Xu, R., Ma, Z.: Global stability of a reaction–diffusion predator–prey model with a nonlocal delay. Math. Comput. Model. 50, 194–206 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhang, J., Peng, Y.: Traveling waves of the diffusive Nicholson’s blowfies equation with strong generic delay kernel and non-local effect. Nonlinear Anal. 68, 1263–1270 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Schaaf, K.: Asymptotic behavior and traveling wave solutions for parabolic function differetial eqution. Trans. Am. Math. Soc. 302, 587–615 (1987)

    MATH  Google Scholar 

  19. Li, K., Li, X.: Traveling wave solutions in a delayed diffusion competiton system. Nonlinear Anal. 75, 3705–3722 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lin, G., Li, W.T., Ma, M.J.: Traveling wave solutions in delayed reaction diffusion systems with applications to muli-species models. Discrete Contin. Dyn. Syst. 13, 393–414 (2010)

    Article  MATH  Google Scholar 

  21. Ruan, S., Xiao, D.: Stability of steady states and existence of traveling waves in a vector-disease model. Proc. R. Soc. Edinb. 134A, 991–1011 (2004)

    Article  MATH  Google Scholar 

  22. Ai, S.: Traveling wave fronts for generalized Fisher equations with spatio-temporal delays. J. Differ. Equ. 232, 104–133 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ashwin, P., Bartuccelli, M.V., Bridges, T.J., Gourley, S.A.: Traveling fronts for the KPP equation with spatio-temporal delay. Z. Angew. Math. Phys. 53, 103–122 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Fenical, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31, 53–98 (1979)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors express their sincere thanks to the anonymous reviewers for their valuable comments and careful corrections for improving the quality of the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Hankou Road 22nd, Nanjing, 210093, Jiangsu, People’s Republic of China

    Jiang Liu

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

    Jiang Liu, Dongcheng Xu & Zengji Du

Authors
  1. Jiang Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dongcheng Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. 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 Nos. 11471146, 11771185 and 11671176).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, J., Xu, D. & Du, Z. Traveling Wave Solution of a Reaction–Diffusion Predator–Prey System. Qual. Theory Dyn. Syst. 18, 57–67 (2019). https://doi.org/10.1007/s12346-018-0276-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12346-018-0276-1

Keywords

Mathematics Subject Classification

Search

Navigation