Summary
A certain interaction-diffusion equation occurring in morphogenesis is considered. This equation is proposed by Gierer and Meinhardt, which is introduced by Child's gradient theory and Turing's idea about diffusion driven instability. It is shown that slightly asymmetric gradients in the tissue produce stable striking patterns depending on its asymmetry, starting from uniform distribution of morphogens. The tool is the perturbed bifurcation theory. Moreover, from a mathematical point of view, the global existence of steady state solutions with respect to some parameters is discussed.
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References
Auchmuty, J. F. G., Nicolis, G.: Bifurcation Analysis of Nonlinear Reaction-Diffusion Equation-I. Evolution Equations and the Steady State Solutions, Bulletin Math. Biol. 37, 323–365 (1975)
Chueh, K. H., Conley, C. C., Smoller, J. A.: Positively Invariant Regions for Systems of Nonlinear Diffusion Equations, Indiana Univ. Math. J. 26, 373–392 (1977)
Conway, E., Hoff, D., Smoller, J. A.: Large Time Behavior of Solutions of Systems of Nonlinear Reaction-Diffusion Equations, SIAM J. Appl. Math. 35, 1–16 (1978)
Crandall, M. G. Rabinowitz, P. H.: Bifurcation from Simple Eigenvalues, J. Func. Anal. 8, 321–340 (1971)
Dieudonné, J.: Foundation of Modern Analysis, Academic Press, New York, 1960
Friedman, A.: Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N.J., 1964
Gierer, A., Meinhardt, H.: A Theory of Biological Pattern Formation, Kybernetik 12, 30–39 (1972)
Granero, M. I., Porati, A., Zanacca, D.: A Bifurcation Analysis of Pattern Formation in a Diffusion Governed Morphogenetic Field, J. Math. Biol. 4, 21–27 (1977)
Kielhöfer, H.: Stability and Semilinear Evolution Equations in Hilbert Space, Arch. Rat. Mech. Anal. 57, 150–165 (1974)
Mimura, M., Nishiura, Y., Yamaguti, M.: Some Diffusive Prey and Predator Systems and Their Bifurcation Problems, to appear in the Annals of the New York Academy of Sciences (1977)
Nirenberg, L.: Topics in Nonlinear Functional Analysis, Lecture Notes, Courant Institute of Mathematical Sciences, New York University, 1973
Rabinowitz, P. H.: Some Global Results for Nonlinear Eigenvalue Problems, J. Func. Anal. 7, 487–513 (1971)
Sather, D.: Branching of Solutions of Nonlinear Equations, Rocky Mount. J. Math. 3, 203–250 (1973)
Sattinger, D. H.: Stability of Bifurcating Solutions by Leray-Schauder Degree. Arch. Rat. Mech. Anal. 43, 154–166 (1971)
Turing, A. M.: The Chemical Basis of Morphogenesis, Phil. Trans. Roy. Soc. B237, 32, 37–72 (1952)
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Mimura, M., Nishiura, Y. Spatial patterns for an interaction-diffusion equation in morphogenesis. J. Math. Biology 7, 243–263 (1979). https://doi.org/10.1007/BF00275727
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DOI: https://doi.org/10.1007/BF00275727