Skip to main content
SpringerLink
Log in

Diffusion, Cross-diffusion and Competitive Interaction

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

The cross-diffusion competition systems were introduced by Shigesada et al. [J. Theor. Biol. 79, 83–99 (1979)] to describe the population pressure by other species. In this paper, introducing the densities of the active individuals and the less active ones, we show that the cross-diffusion competition system can be approximated by the reaction-diffusion system which only includes the linear diffusion. The linearized stability around the constant equilibrium solution is also studied, which implies that the cross-diffusion induced instability can be regarded as Turing’s instability of the corresponding reaction-diffusion system.

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. Amann, H. Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In: Schmeisser, H., Triebel, H. (eds.) Function Spaces, Differential Operators and Nonlinear Analysis. Teubner-Texte Math. vol. 133, pp. 9–126 (1993)

  2. Choi Y.S., Lui R., Yamada Y. (2003) Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with weak cross-diffusion. Discret. Contin. Dyn. Syst. 9, 1193–1200

    MATH  MathSciNet  Google Scholar 

  3. Hirsch M.W. (1982) Differential equations and convergence almost everywhere of strongly monotone semiflows. PAM Technical Report, University of California, Berkeley

    Google Scholar 

  4. Huang Y. (2005) How do cross-migration models arise? Math. Biosci. 195, 127–140

    Article  MATH  MathSciNet  Google Scholar 

  5. Iida, M., Ninomiya, H. A reaction-diffusion approximation to a cross-diffusion system. In: Chipot, M., Ninomiya, H. (eds.) Recent Advances on Elliptic and Parabolic Issues. World Scientific, pp. 145–164 (2006)

  6. Kan-on Y. (1993) Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics. Hiroshima Math. J. 23, 509–536

    MATH  MathSciNet  Google Scholar 

  7. Kareiva P., Odell G. (1987) Swarms of predators exhibit “preytaxis” if individual predator use area-restricted search. The Am. Nat. 130, 233–270

    Article  Google Scholar 

  8. Kishimoto K., Weinberger H.F. (1985) The spatial homogeneity of stable equilibria of some reaction-diffusion system on convex domains. J. Differ. Equ. 58, 15–21

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Lou Y., Ni W.-M., Yotsutani S. (2004) On a limiting system in the Lotka-Volterra competition with cross-diffusion. Discret. Contin. Dyn. Syst. 10, 435–458

    Article  MATH  MathSciNet  Google Scholar 

  11. Lou Y., Ni W.-M., Wu Y. (1998) On the global existence of a cross-diffusion system. Discrete Contin. Dyn. Syst. 4, 193–203

    Article  MATH  MathSciNet  Google Scholar 

  12. Matano H., Mimura M. (1983) Pattern formation in competition-diffusion systems in nonconvex domains. Publ. Res. Inst. Math. Sci. Kyoto Univ. 19, 1049–1079

    Article  MATH  MathSciNet  Google Scholar 

  13. Mimura M., Kawasaki K. (1980) Spatial segregation in competitive interaction-diffusion equations. J. Math. Biol. 9, 49–64

    Article  MATH  MathSciNet  Google Scholar 

  14. Mimura M., Nishiura Y., Tesei A., Tsujikawa T. (1984) Coexistence problem for two competing species models with density-dependent diffusion. Hiroshima Math. J. 14, 425–449

    MATH  MathSciNet  Google Scholar 

  15. Okubo A. (1980) Diffusion and Ecological Problems: Mathematical Models. Biomathematics. vol. 10, Springer, Berlin Heidelberg New York

    Google Scholar 

  16. Shigesada N., Kawasaki K., Teramoto E. (1979) Spatial segregation of interacting species. J. Theor. Biol. 79, 83–99

    Article  MathSciNet  Google Scholar 

  17. Turchin, P. Quantitative Analysis of Movement: Measuring and Modeling Population Redistribution in Animals and Plants. Sinauer (1998)

  18. Turing A.M. (1952) The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B 237, 37–72

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Humanities and Social Sciences, Iwate University, Morioka, 020-8550, Japan

    Masato Iida

  2. Department of Mathematics, School of Science and Technology, Meiji University, 1-1 Higashi Mita, Tama-ku, Kawasaki, 214-8571, Japan

    Masayasu Mimura

  3. Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu, 520-2194, Japan

    Hirokazu Ninomiya

Authors
  1. Masato Iida
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Masayasu Mimura
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hirokazu Ninomiya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hirokazu Ninomiya.

Additional information

M. Iida was partially supported by Grant-in-Aid for Scientific Research (No. (C)15540194), Japan Society for the Promotion of Science.

M. Mimura was partially supported by Grant-in-Aid for Scientific Research (No. (A)15204006), Japan Society for the Promotion of Science.

H. Ninomiya was Partially supported by Grant-in-Aid for Young Scientists (No. (B)15740076), Japan Society for the Promotion of Science.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Iida, M., Mimura, M. & Ninomiya, H. Diffusion, Cross-diffusion and Competitive Interaction. J. Math. Biol. 53, 617–641 (2006). https://doi.org/10.1007/s00285-006-0013-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-006-0013-2

Keywords

Mathematics Subject Classification (2000)

Search

Navigation