Skip to main content
SpringerLink
Log in

Predator cannibalism can give rise to regular spatial pattern in a predator–prey system

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

Abstract

One of the central issues in ecology is the study of spatial pattern in the distribution of organisms. Thus, in this paper, spatial pattern of a predator–prey system with predator cannibalism is considered. By mathematical analysis, we obtain the condition for emerging Turing pattern formation. Furthermore, numerical simulations reveal that large variety of different spatiotemporal dynamics emerge as the consequence of the interaction of Holling type II with predator cannibalism. The obtained results show predator cannibalism has great influence on the spatial pattern formation. In other words, the regular pattern is induced by predator cannibalism. Moreover, we find that although the environment is heterogeneous, the system still exhibits Turing pattern, which means the pattern is self-organized. It may help us better understand the dynamics of predator–prey interaction in a real environment.

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. Baurmann, M., Gross, T., Feudel, U.: Instabilities in spatially extended predator–prey systems: spatiotemporal patterns in the neighborhood of Turing–Hopf bifurcations. J. Theor. Biol. 245, 220–229 (2007)

    Article  MathSciNet  Google Scholar 

  2. Kuang, Y., Beretta, E.: Global qualitative analysis of a ratio-dependent predator–prey system. J. Math. Biol. 36, 389–406 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. Jones, L.E., Ellner, S.P.: Evolutionary tradeoff and equilibrium in an aquatic predator–prey system. Bull. Math. Biol. 66, 1547–1573 (2004)

    Article  MathSciNet  Google Scholar 

  4. Auger, P., de la Parra, R.B., Morand, S., Sanchez, E.: A predator–prey model with predators using hawk and dove tactics. Math. Biol. 177&178, 185–200 (2002)

    Google Scholar 

  5. Jost, C.: Comparing predator-prey models qualitatively and quanti tatively with ecological time-series data. Ph.D. thesis. Institute National Agronomique, Paris Grignon (1998)

  6. Real, L.A.: The kinetics of functional response. Am. Natl. 111, 289–300 (1977)

    Article  Google Scholar 

  7. Boukal, D.S., Sabelisc, M.W., Berec, L.: How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapses. Theor. Popul. Biol. 72, 136–147 (2007)

    Article  MATH  Google Scholar 

  8. van Baalen, M., Sabelis, M.W.: The milker–killer dilemma in spatially structured predator–prey interactions. Oikos 74, 391–400 (1995)

    Article  Google Scholar 

  9. Stephens, P.A., Sutherland, W.J., Freckleton, R.P.: What is the Allee effect? Oikos 87, 185–190 (1999)

    Article  Google Scholar 

  10. Fox, L.R.: Cannibalism in natural populations. Ann. Rev. Ecol. Syst. 6, 87–106 (1975)

    Article  Google Scholar 

  11. Meffe, G.K.: Density-dependent cannibalism in the endangered Sonoran topminnow (Poeciliopsis occidentalis). Ann. Rev. Ecol. Syst. 12, 500–503 (1984)

    Google Scholar 

  12. Elgar, M.A., Crespi, B.J. (eds.): Cannibalism: Ecology and Evolution among Diverse Taxa. Oxford University Press, Oxford (1992)

    Google Scholar 

  13. Medvinsky, A.B., Petrovskii, S.V., Tikhonova, I.A., Malchow, H., Li, B.L.: Spatiotemporal complexity of plankton and fish dynamics. SIAM Rev. 44, 311–370 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Alonso, D., Bartumeus, F., Catalan, J.: Mutual interference between predators can give rise to Turing spatial patterns. Ecology 83, 28–34 (2002)

    Article  Google Scholar 

  15. Segel, L.A., Jackson, J.L.: Dissipative structure: An explanation and an ecological example. J. Theor. Biol. 37, 545–559 (1972)

    Article  Google Scholar 

  16. Levin, S.A.: Dispersion and population interactions. Am. Nat. 108, 207–228 (1974)

    Article  Google Scholar 

  17. Ricklefs, R.E.: Environmental heterogeneity and plant species diversity: A hypothesis. Am. Nat. 111, 376–381 (1977)

    Article  Google Scholar 

  18. Okubo, A.: Horizontal dispersion and critical scales for phytoplankton patches. In: Steel, J.H. (ed.) Spatial Pattern in Plankton Communities, p. 21. Plenum, New York (1978)

    Google Scholar 

  19. Mimura, M., Murray, J.D.: On a diffusive prey–predator model which exhibits patchiness. J. Theor. Biol. 75, 249–262 (1979)

    Article  MathSciNet  Google Scholar 

  20. Okubo, A.: Diffusion and Ecological Problems: Modern Perspective. Interdisciplinary Applied Mathematics, vol. 14. Springer, Berlin (2001)

    Google Scholar 

  21. Murray, J.D.: A Pre-pattern formation mechanism for animal coat markings. J. Theor. Biol. 88, 161–199 (1981)

    Article  Google Scholar 

  22. Bascompte, J., Solé, R.V., Martínez, N.: Population cycles and spatial patterns in snowshoe hares: An individual-oriented simulation. J. Theor. Biol. 187, 213–222 (1997)

    Article  Google Scholar 

  23. Bascompte, J., Solé, R.V.: Spatiotemporal patterns in nature. Trends Ecol. Evolution 13, 173–174 (1998)

    Article  Google Scholar 

  24. Neuhauser, C.: Mathematical challenges in spatial ecology. Not. Am. Math. Soc. 47, 1304–1314 (2001)

    MathSciNet  Google Scholar 

  25. Baurmann, M.: Turing instabilities and pattern formation in a benthic nutrient-microorganism system. Math. Biosci. Eng. 1, 111–130 (2004)

    MATH  MathSciNet  Google Scholar 

  26. Lepänen, T.: Computational studies of sattern formation in Turing systems. Ph.D. thesis. Helsinki University of Technology, Finland (2004)

  27. Wang, W., Liu, Q.X., Jin, Z.: Spatiotemporal complexity of a ratio-dependent predator–prey system. Phys. Rev. E 75, 051903 (2007)

    Article  MathSciNet  Google Scholar 

  28. Morozov, A., Petrovskii, S., Li, B.L.: Spatiotemporal complexity of patchy invasion in a predator–prey system with the Allee effect. J. Theor. Biol. 238, 18–35 (2006)

    Article  MathSciNet  Google Scholar 

  29. Sherratt, J.A., Lambin, X., Sherratt, T.N.: The effects of the size and shape of landscape features on the formation of traveling waves in cyclic populations. Am. Nat. 162, 503–513 (2003)

    Article  Google Scholar 

  30. Ovaskainen, O., Sato, K., Bascompte, J., Hanski, I.: Metapopulation models for extinction threshold in spatially correlated landscapes. J. Theor. Biol. 215, 95–108 (2002)

    Article  Google Scholar 

  31. Bascompte, J., Rodríguez, M.A.: Self-disturbance as a source of spatiotemporal heterogeneity: The case of the tallgrass prairie. J. Theor. Biol. 204, 153–164 (2000)

    Article  Google Scholar 

  32. Bascompte, J., Solé, R.V.: Effects of habitat destruction in a prey–predator metapopulation model. J. Theor. Biol. 195, 383–393 (1998)

    Article  Google Scholar 

  33. Polis, G.A.: The evolution and dynamics of intraspecific predation. Ann. Rev. Ecol. Syst. 12, 225–251 (1981)

    Article  Google Scholar 

  34. Elgar, M.A., Crespi, B.J.: Ecology and evolution of cannibalism. In: Elgar, M.A., Crespi, B.J. (eds.) Cannibalism, p. 1. Oxford University Press, Oxford (1992)

    Google Scholar 

  35. Garvie, M.R.: Finite-difference schemes for reaction–diffusion equations modeling predator–prey interactions in MATLAB. Bull. Math. Biol. 69, 931–956 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  36. Morozov, A., Petrovskii, S., Li, B.L.: Bifurcations and chaos in a predator–prey system with the Allee effect. Proc. R. Soc. Lond. B 271, 1407–1414 (2004)

    Article  Google Scholar 

  37. Cushing, J.M., Costantino, R.F., Dennis, B., Desharnais, R.A., Henson, S.M.: Chaos in Ecology: Experimental Nonlinear Dynamics. Academic Press, San Diego (2003)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mechatronic Engineering, North University of China, Taiyuan, 030051, People’s Republic of China

    Gui-Quan Sun

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

    Gui-Quan Sun, Guang Zhang, Zhen Jin & Li Li

  3. Department of Mathematics, Tianjin University of Commerce, Tianjin, 300134, People’s Republic of China

    Guang Zhang

Authors
  1. Gui-Quan Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhen Jin
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Li Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhen Jin.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sun, GQ., Zhang, G., Jin, Z. et al. Predator cannibalism can give rise to regular spatial pattern in a predator–prey system. Nonlinear Dyn 58, 75–84 (2009). https://doi.org/10.1007/s11071-008-9462-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-008-9462-z

Keywords

Search

Navigation