Skip to main content
SpringerLink
Log in

Spatial patterns of a predator-prey model with cross diffusion

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this paper, spatial patterns of a Holling–Tanner predator-prey model subject to cross diffusion, which means the prey species exercise a self-defense mechanism to protect themselves from the attack of the predator are investigated. By using the bifurcation theory, the conditions of Hopf and Turing bifurcation critical line in a spatial domain are obtained. A series of numerical simulations reveal that the typical dynamics of population density variation is the formation of isolated groups, such as spotted, stripe-like, or labyrinth patterns. Our results confirm that cross diffusion can create stationary patterns, which enrich the finding of pattern formation in an ecosystem.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Tanner, J.T.: The stability and the intrinsic growth rates of prey and predator populations. Ecology 56, 855–867 (1975)

    Article  Google Scholar 

  2. Wollkind, D.J., Collings, J.B., Logan, J.A.: Metastability in a temperature-dependent model system for predator-prey mite outbreak interactions on fruit flies. Bull. Math. Biol. 50, 379–409 (1988)

    MathSciNet  MATH  Google Scholar 

  3. Saez, E., Gonzalez-Olivares, E.: Dynamics of a predator-prey model. SIAM J. Appl. Math. 59, 1867–1878 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Hsu, S.-B., Huang, T.-W.: Global stability for a class of predator-prey systems. SIAM J. Appl. Math. 55, 763–783 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  5. Sun, G.-Q., Jin, Z., Liu, Q.-X., Li, L.: Spatial pattern in an epidemic system with cross-diffusion of the susceptible. J. Biol. Syst. 17, 141–152 (2009)

    Article  MathSciNet  Google Scholar 

  6. Hanski, I., Gilpin, M.E.: Metapopulation Biology. Academic Press, San Diego (1997)

    MATH  Google Scholar 

  7. 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)

    Article  MATH  Google Scholar 

  8. Okubo, A.: Diffusion and Ecological Problems: Mathematical Models. Springer, Berlin (1980)

    MATH  Google Scholar 

  9. Sun, G.-Q., Jin, Z., Li, L., Li, B.-L.: Self-organized wave pattern in a predator-prey model. Nonlinear Dyn. 60, 265–275 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Lou, Y., Ni, W.M.: Diffusion vs cross-diffusion: an elliptic approach. J. Differ. Equ. 154, 157–190 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dubey, B., Das, B., Hussain, J.: A predator-prey interaction model with self and cross-diffusion. Ecol. Model. 141, 67–76 (2001)

    Article  Google Scholar 

  12. Sun, X.-K., Huo, H.-F., Xiang, H.: Bifurcation and stability analysis in predator-prey model with a stage-structure for predator. Nonlinear Dyn. 58, 497–513 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Murray, J.D.: Mathematical Biology, 2nd edn. Springer, Berlin; New York (1993)

    Book  MATH  Google Scholar 

  14. Chung, J.M., Peacock-López, E.: Bifurcation diagrams and Turing patterns in a chemical self-replicating reaction-diffusion system with cross diffusion. J. Chem. Phys. 127, 174903 (2007)

    Article  Google Scholar 

  15. Li, L., Jin, Z., Sun, G.-Q.: Spatial pattern of an epidemic model with cross-diffusion. Chin. Phys. Lett. 25, 3500–3503 (2008)

    Article  Google Scholar 

  16. Sun, G.-Q., Jin, Z., Liu, Q.-X., Li, L.: Pattern formation induced by cross-diffusion in a predator-prey system. Chin. Phys. B 17, 3936–3941 (2008)

    Article  Google Scholar 

  17. Sun, G.-Q., Jin, Z., Zhao, Y.-G., Liu, Q.-X., Li, L.: Spatial pattern in a predator-prey system with both self- and cross-diffusion. Int. J. Mod. Phys. C 20, 71–84 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ipsen, M., Hynne, F., Soensen, P.: Amplitude equations for reaction-diffusion systems with a Hopf bifurcation and slow real modes. Physica D 136, 66–92 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pena, B., Perez-Garcia, C.: Stability of Turing patterns in the Brusselator model. Phys. Rev. E 64, 056213 (2001)

    Article  MathSciNet  Google Scholar 

  20. Sun, G.-Q., Li, L., Jin, Z., Li, B.-L.: Effect of noise on the pattern formation in an epidemic model. Numer. Methods Partial Differ. Equ. 1168–1179 (2010)

  21. Sun, G.-Q., Jin, Z., Liu, Q.-X., Li, B.-L.: Rich dynamics in a predator-prey model with both noise and periodic force. Biosystems 100, 14–22 (2010)

    Article  Google Scholar 

  22. Sun, G.-Q., Jin, Z., Li, L., Liu, Q.-X.: The role of noise in a predator-prey model with Allee effect, J. Biol. Phys. 35, 185–196 (2009)

    Article  Google Scholar 

  23. Li, L., Jin, Z.: Pattern dynamics of a spatial predator-prey model with noise. Nonlinear Dyn. 67, 1737–1744 (2012)

    Article  Google Scholar 

  24. Sun, G.-Q., Jin, Z., Liu, Q.-X., Li, L.: Dynamical complexity of a spatial predator-prey model with migration. Ecol. Model. 219, 248–255 (2008)

    Article  Google Scholar 

Download references

Acknowledgements

The research was partially supported by the National Natural Science Foundation of China under Grants (11171314, 10901145, and 11147015), Program for Basic Research (2010011007), and International and Technical Cooperation Project (2010081005).

Author information

Authors and Affiliations

  1. Department of Mathematics, North University of China, Taiyuan, Shan’xi, 030051, People’s Republic of China

    Gui-Quan Sun, Zhen Jin & Li Li

  2. Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK

    Mainul Haque

  3. Ecological Complexity and Modeling Laboratory, Department of Botany and Plant Sciences, University of California, Riverside, CA, 92521-0124, USA

    Bai-Lian Li

Authors
  1. Gui-Quan Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhen Jin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Li Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Mainul Haque
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Bai-Lian Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gui-Quan Sun.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sun, GQ., Jin, Z., Li, L. et al. Spatial patterns of a predator-prey model with cross diffusion. Nonlinear Dyn 69, 1631–1638 (2012). https://doi.org/10.1007/s11071-012-0374-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-012-0374-6

Keywords

Search

Navigation