Acta Applicandae Mathematica

, Volume 48, Issue 1, pp 1–12 | Cite as

Effect of Cross-Diffusion on Pattern Formation – a Nonlinear Analysis

  • J. Chattopadhyay
  • P. K. Tapaswi


In this paper, we consider a model for the switching behaviour (determination) of a cell proposed by Meinhardt (1982) and observe that this two component system can create a pattern only in the presence of cross-diffusion. We also analyse the global behaviour of this model system by the Bendixson–Dulac criteria and Liapunov functional method.


Statistical Physic Component System Pattern Formation Nonlinear Analysis Functional Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Auchmuty, J. F. G. and Nicolis, G.: Bifurcation analysis of nonlinear reaction diffusion equations — I: Evolution equations and the steady-state solutions, Bull. Math. Biol. 37 (1975), 323–365.Google Scholar
  2. Auchmuty, J. F. G. and Nicolis, G.: Bifurcation analysis of nonlinear reaction diffusion equations — III: Chemical oscillations, Bull. Math. Biol. 38 (1976), 325–350.Google Scholar
  3. Alimirantis, Y. and Papageorgiou, S.: Cross-diffusional effects on chemical and biological pattern formation, J. Theoret. Biol. 151 (1991), 289–311.Google Scholar
  4. Babloyants, A. and Hiernaux, J.: Models for cell differentiation and generation of polarity in diffusion governed morphogenetic fields, Bull. Math. Biol. 37 (1975), 637–657.Google Scholar
  5. Berding, C. and Haken, H.: Pattern formation in morphogenesis, J. Math. Biol. 14 (1982), 133–151.Google Scholar
  6. Birkhoff, G. and Rota, G. C.: Ordinary Differential Equations, Ginn, Boston, 1982, p. 23.Google Scholar
  7. Chattopadhyay, J., Tapaswi, P. K. and Mukherjee, D.: Formation of a regular dissipative structure: a bifurcation and nonlinear analysis, Biosystems 26 (1992), 211–222.Google Scholar
  8. Chattopadhyay, J. and Tapaswi, P. K.: Order and disorder in biological systems through negative cross-diffusion of mitotic inhibitor — a mathematical model, Math. Comp. Modelling 17 (1993), 105–112.Google Scholar
  9. Chattopadhyay, J. and Tapaswi, P. K.: Morphogenetic prepattern during embryonic development — a nonlinear analysis, Appl. Math. Lett. 5 (1992), 19–22.Google Scholar
  10. Clark, C. W.: Mathematical Bioeconomics: The Optimal Managment of Renewable Resources, Wiley, New York, 1976.Google Scholar
  11. Gierer, A. and Meinhardt, H.: Applications of a theory of biological pattern formation based on lateral inhibition, J. Cell. Sci. 15 (1974), 321–376.Google Scholar
  12. Gierer, A. and Meinhardt, H.: A theory of biological pattern formation, Kybernetica 12 (1972), 30–39.Google Scholar
  13. Granero, M. I., Porati, A. and Zanacca, D.: Bifurcation analysis of pattern formation in diffusion governed morphogenetic field, J. Math. Biol. 4 (1977), 21–27.Google Scholar
  14. Haken, H. and Olbrich, H.: Analytical treatment of pattern formation in the Gierer-Meinhardt model of morphogenesis, J. Math. Biol. 6 (1978), 317–331.Google Scholar
  15. Hsu, Sze-Bi, Waltman, P. and Wolkowicz, G. S. K.: Global analysis of a model of plasmid-bearing, plasmid-free competition in a chemostat, J. Math. Biol. 32 (1994), 731–742.Google Scholar
  16. Hunding, A. and Sorensen, P. G.: Size adaptation of Turing prepatterns, J. Math. Biol. 26 (1988), 27–39.Google Scholar
  17. Huppert, H. E. and Hallworth, M. A.: J. Phys. Chem. 88 (1984), 2902–2905.Google Scholar
  18. Jorne, J.: Negative ionic cross-diffusion coefficient in electrolytic solution, J. Theoret. Biol. 55 (1975), 529–532.Google Scholar
  19. Lefever, R. and Prigogine, I.: Symmetry breaking instabilities in dissipative system — II, J. Chem. Phys. 48 (1968), 1695–1700.Google Scholar
  20. Levin, S. A. and Segel, L. A.: Pattern generation in space and aspect, SIAM Rev. 27 (1985), 45–67.Google Scholar
  21. Martinez, H. M.: Morphogenesis and chemical dissipative structures: a computer simulation case study, J. Theoret. Biol. 36 (1972), 479–501.Google Scholar
  22. McDougall, T. and Turner, J. S.: Influence of cross diffusion on finger double diffusive convection, Nature 299 (1982), 812–822.Google Scholar
  23. Meinhardt, H.: Models of Biological Pattern Formation, Academic Press, London, New York, 1982.Google Scholar
  24. Meinhardt, H.: Hierarchical inductions of cell states: a model for segmentation in Drosophila, J. Cell. Sci. Suppl. 4 (1986), 357–381.Google Scholar
  25. Murray, J. D.: Lectures on Nonlinear Differential Equation Models in Biology, Oxford University Press, 1977.Google Scholar
  26. Murray, J. D.: Mathematical Biology, Springer-Verlag, Heidelberg, 1989.Google Scholar
  27. Murray, J. D.: Discussion: Turing's theory of morphogenesis — its influence on modelling biological pattern and form, Bull. Math. Biol. 52 (1990), 119–152.Google Scholar
  28. Murray, J. D. and Oster, G. F.: Generation of biological pattern and form, IMA J. Math. Appl. Medic. Biol. 1 (1984), 51–75.Google Scholar
  29. Murray, J. D. and Maini, P. K.: A new approach to the generation of pattern and form in embryology, Sci. Prog. Oxford 70 (1986), 539–553.Google Scholar
  30. Nakamura, R.: The transport of histidine and methionine in rat brain slices, J. Biochem. 53 (1963), 314–332.Google Scholar
  31. Nicolis, G. and Auchmuty, J. F. G.: Dissipative structures, Catastrophes and pattern formation: a bifurcation analysis, Proc. Natl. Acad. Sci., U.S.A. 71 (1974), 2748–2751.Google Scholar
  32. Nicolis, G. and Prigogine, I.: Self-Organisation in Nonequilibrium Systems, Wiley, New York, 1977.Google Scholar
  33. Oster, G. F., Shubin, N., Murray, J. D. and Alberch, P.: Evolution and morphogenetic rules: the shape of the vertebrate limb in ontogeny and phylogeny, Evolution 45 (1988), 862–884.Google Scholar
  34. Othmer, H. C.: Current theories of pattern formation, In S. Levin (ed.) Lectures on Mathematics in Life Sciences 9, Amer. Math. Soc., Providence, 1977, pp. 55–86.Google Scholar
  35. Rosen, R.: Dynamical System Theory in Biology, Wiley-Intersciences, New York, 1979.Google Scholar
  36. Segel, L. A.: Modelling Dynamic Phenomena in Molecular and Cellular Biology, Cambridge University Press, 1984.Google Scholar
  37. Smoller, J.: Shock Waves and Reaction-Diffusion Systems, Springer-Verlag, New York, 1983.Google Scholar
  38. Tapaswi, P. K. and Saha, A. K.: Pattern formation and morphogenesis: a reaction diffusion model, Bull. Math. Biol. 48 (1986), 213–228.Google Scholar
  39. Turing, A. M.: The chemical basis of morphogenesis, Phil. Trans. R. Soc. London B 237 (1952), 37–72.Google Scholar
  40. Wolpert, L.: Positional informations and pattern formation, Phil. Trans. R. Soc. London B 295 (1981), 441–450.Google Scholar

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • J. Chattopadhyay
    • 1
  • P. K. Tapaswi
    • 1
  1. 1.Embryology Research UnitIndian Statistical Institute, 203CalcuttaIndia

Personalised recommendations