Nonlinear Dynamics

, Volume 81, Issue 4, pp 2155–2163 | Cite as

Pattern dynamics of a predator–prey reaction–diffusion model with spatiotemporal delay

  • Jian Xu
  • Gaoxiang Yang
  • Hongguang Xi
  • Jianzhong Su
Original Paper


Using the tool of Turing instability for partial differential equations, we investigate the spatiotemporal distributions for solutions of a predator–prey-type reaction–diffusion model with spatiotemporal delay. The linear stability conditions of Turing instability, which induce bifurcation patterns in this model, are obtained. Moreover, according to these conditions, we numerically calculate the bifurcation diagrams by using time delay and the predator rate as parameters. The effects of two parameters in the different bifurcation diagrams are also demonstrated through numerical computations and lead to some spatiotemporal patterns of this model, which enrich the pattern formation of predator–prey models.


Predator–prey model Spatiotemporal patterns Turing instability Spatiotemporal delay 



The authors thank the anonymous referees for their valuable comments on improvement of the presentation of this work. This work is supported by the State Key Program of National Natural Science Foundation of China under Grant No.11032009 and National Natural Science Foundation of China under Grant No.11272236.

Conflict of interest

The authors declare that the work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


  1. 1.
    Wu, J.: Theory and Applications of Partial Functional Differential Equations. Springer, New York (1996)CrossRefMATHGoogle Scholar
  2. 2.
    Murray, J.D.: Mathematical Biology: Spatial Models and Biomedical Applications. Springer, New York (2003)MATHGoogle Scholar
  3. 3.
    Ouyang, Q.: Patterns Formation in Reaction Diffusion Systems. Shanghai Sci-tech Education Publishing House, Shanghai (2010)Google Scholar
  4. 4.
    Huffaker, C.B.: Experimental studies on predation: dispersion factors and predator-prey oscillations. Hilgardia 27, 343–383 (1958)CrossRefGoogle Scholar
  5. 5.
    Zhang, J.F., Li, W.T., Yan, X.P.: Multiple bifurcations in delayed predator-prey diffusion system with a functional response. Nonlinear Anal. RWA 11, 2708–2725 (2010)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Su, Y., Wei, J., Shi, J.: Bifurcation analysis in a delayed diffusive Nicholson’s blowflies equation. Nonlinear Anal. RWA 11, 1692–1703 (2010)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Zuo, W., Wei, J.: Stability and Hopf bifurcation in a diffusive predator-prey system with delay effect. Nonlinear Anal. RWA 12, 1998–2011 (2011)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Zhang, J.F., Li, W.T., Yan, X.P.: Bifurcation and spatiotemporal patterns in a homogeneous diffusion competition system with delays. Int. J. Biomath. 5, 1250049–1250072 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Yan, X.P., Li, W.T.: Stability and Hopf bifurcation for a delayed cooperative system with diffusion effects. Int. J. Bifurc. Chaos 18, 441–453 (2008)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Chen, S., Shi, J., Wei, J.: Global stability and Hopf bifurcation in a delayed diffusive Leslie-Gower predator-prey system. Int. J. Bifurc. Chaos 22, 1250061 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ruan, S.: Turing instability and traveling waves in diffusive plankton models with delayed nutrient recycling. IMA J. Appl. Math. 61, 15–32 (1998)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Banerjee, M., Banerjee, S.: Turing instabilities and spatio-temporal chaos in ratio-depent Holling-Tanner model. Math. Biosci. 236, 64–76 (2012)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Zhang, X., Sun, G., Jin, Z.: Spatial dynamics in a predator-prey model with Beddington-DeAngelis functional response. Phys. Rev. E 85, 021924 (2012)CrossRefGoogle Scholar
  14. 14.
    Baurmann, M., Gross, T., Feudel, U.: Instabilities in spatially extended predator-prey systems: spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcation. J. Theor. Biol. 245, 220–229 (2007)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Metxnern, M., Wit, A.D., Bose, S., Scholl, E.: Generic spatio-temporal dynamics near codimension Turing-Hopf bifurcation. Phys. Rev. E 55, 6690–6697 (1997)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Wang, B., Wang, A.L., Liu, Y.J.: Analysis of a spatial predator-prey model with delay. Nonlinear Dyn. 62, 601–608 (2010)CrossRefMATHGoogle Scholar
  17. 17.
    Zhang, T., Xing, Y., Zang, H., Han, M.: Spatio-temporal patterns in a predator-prey model with hyperbolic mortality. Nonlinear Dyn. 78, 265–277 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Zhang, T., Zang, H.: Delay-induced Turing instability in reaction-diffusion equations. Phys. Rev. E 90, 052908 (2014)CrossRefGoogle Scholar
  19. 19.
    Sun, G.-Q., Zhang, G., Jin, Z., Li, L.: Predator cannibalism can give rise to regular spatial pattern in a predator-prey system. Nonlinear Dyn. 58, 75–84 (2009)CrossRefGoogle Scholar
  20. 20.
    Li, A.-W.: Impact of noise on pattern formation in a predator-prey model. Nonlinear Dyn. 66, 689–694 (2011)CrossRefGoogle Scholar
  21. 21.
    Britton, N.F.: Aggregation and the competitive exclusion principle. J. Theor. Biol. 136, 57–66 (1989)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Gourley, S.A., Chaplan, M.A.J., Davidson, F.A.: Spatio-temporal pattern formation in a nonlocal reaction-diffusion equation. Dyn. Syst. Int. J. 16, 173–192 (2001)CrossRefMATHGoogle Scholar
  23. 23.
    Britton, N.F.: Spatial structures and periodic traveling wave in an integro-differential reaction diffusion population model. SIAM J. Appl. Math. 50, 1663–1688 (1990)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Gourley, S.A., Britton, N.F.: A predator prey reaction diffusion system with nonlocal effect. J. Math. Biol. 34, 297–333 (1996)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Jian Xu
    • 1
  • Gaoxiang Yang
    • 1
  • Hongguang Xi
    • 2
  • Jianzhong Su
    • 2
  1. 1.School of Aerospace Engineering and Applied MechanicsTongji UniversityShanghaiPeople’s Republic of China
  2. 2.Department of MathematicsThe University of Texas at ArlingtonArlingtonUSA

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