Abstract
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.
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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.
Auchmuty, J. F. G. and Nicolis, G.: Bifurcation analysis of nonlinear reaction diffusion equations — III: Chemical oscillations, Bull. Math. Biol. 38 (1976), 325–350.
Alimirantis, Y. and Papageorgiou, S.: Cross-diffusional effects on chemical and biological pattern formation, J. Theoret. Biol. 151 (1991), 289–311.
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.
Berding, C. and Haken, H.: Pattern formation in morphogenesis, J. Math. Biol. 14 (1982), 133–151.
Birkhoff, G. and Rota, G. C.: Ordinary Differential Equations, Ginn, Boston, 1982, p. 23.
Chattopadhyay, J., Tapaswi, P. K. and Mukherjee, D.: Formation of a regular dissipative structure: a bifurcation and nonlinear analysis, Biosystems 26 (1992), 211–222.
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.
Chattopadhyay, J. and Tapaswi, P. K.: Morphogenetic prepattern during embryonic development — a nonlinear analysis, Appl. Math. Lett. 5 (1992), 19–22.
Clark, C. W.: Mathematical Bioeconomics: The Optimal Managment of Renewable Resources, Wiley, New York, 1976.
Gierer, A. and Meinhardt, H.: Applications of a theory of biological pattern formation based on lateral inhibition, J. Cell. Sci. 15 (1974), 321–376.
Gierer, A. and Meinhardt, H.: A theory of biological pattern formation, Kybernetica 12 (1972), 30–39.
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.
Haken, H. and Olbrich, H.: Analytical treatment of pattern formation in the Gierer-Meinhardt model of morphogenesis, J. Math. Biol. 6 (1978), 317–331.
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.
Hunding, A. and Sorensen, P. G.: Size adaptation of Turing prepatterns, J. Math. Biol. 26 (1988), 27–39.
Huppert, H. E. and Hallworth, M. A.: J. Phys. Chem. 88 (1984), 2902–2905.
Jorne, J.: Negative ionic cross-diffusion coefficient in electrolytic solution, J. Theoret. Biol. 55 (1975), 529–532.
Lefever, R. and Prigogine, I.: Symmetry breaking instabilities in dissipative system — II, J. Chem. Phys. 48 (1968), 1695–1700.
Levin, S. A. and Segel, L. A.: Pattern generation in space and aspect, SIAM Rev. 27 (1985), 45–67.
Martinez, H. M.: Morphogenesis and chemical dissipative structures: a computer simulation case study, J. Theoret. Biol. 36 (1972), 479–501.
McDougall, T. and Turner, J. S.: Influence of cross diffusion on finger double diffusive convection, Nature 299 (1982), 812–822.
Meinhardt, H.: Models of Biological Pattern Formation, Academic Press, London, New York, 1982.
Meinhardt, H.: Hierarchical inductions of cell states: a model for segmentation in Drosophila, J. Cell. Sci. Suppl. 4 (1986), 357–381.
Murray, J. D.: Lectures on Nonlinear Differential Equation Models in Biology, Oxford University Press, 1977.
Murray, J. D.: Mathematical Biology, Springer-Verlag, Heidelberg, 1989.
Murray, J. D.: Discussion: Turing's theory of morphogenesis — its influence on modelling biological pattern and form, Bull. Math. Biol. 52 (1990), 119–152.
Murray, J. D. and Oster, G. F.: Generation of biological pattern and form, IMA J. Math. Appl. Medic. Biol. 1 (1984), 51–75.
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.
Nakamura, R.: The transport of histidine and methionine in rat brain slices, J. Biochem. 53 (1963), 314–332.
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.
Nicolis, G. and Prigogine, I.: Self-Organisation in Nonequilibrium Systems, Wiley, New York, 1977.
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.
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.
Rosen, R.: Dynamical System Theory in Biology, Wiley-Intersciences, New York, 1979.
Segel, L. A.: Modelling Dynamic Phenomena in Molecular and Cellular Biology, Cambridge University Press, 1984.
Smoller, J.: Shock Waves and Reaction-Diffusion Systems, Springer-Verlag, New York, 1983.
Tapaswi, P. K. and Saha, A. K.: Pattern formation and morphogenesis: a reaction diffusion model, Bull. Math. Biol. 48 (1986), 213–228.
Turing, A. M.: The chemical basis of morphogenesis, Phil. Trans. R. Soc. London B 237 (1952), 37–72.
Wolpert, L.: Positional informations and pattern formation, Phil. Trans. R. Soc. London B 295 (1981), 441–450.
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Chattopadhyay, J., Tapaswi, P.K. Effect of Cross-Diffusion on Pattern Formation – a Nonlinear Analysis. Acta Applicandae Mathematicae 48, 1–12 (1997). https://doi.org/10.1023/A:1005764514684
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DOI: https://doi.org/10.1023/A:1005764514684