Abstract
Pattern formation in a unicomponent reaction-diffusion system with trigger type dynamics and combined boundary conditions is considered. The boundary permeabilities and reservoir concentrations as well as the dimension of the system are the control parameters. The whole assemblage of steady states, their bifurcations and changes under the variation of these parameters is described. Among all steady distributions possible for given values of the parameters, only the simplest ones prove to be asymptotically stable. The relation to catastrophe theory is discussed.
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References
Arnold, V. I.: Theory of ordinary differential equations (Advanced course). M. Nauka, 1978 (Russian)
Bass, F. G., Bochkov, V. S., Gurevich, Yu. G.: Influence of sample size on the volt-ampere characteristic in media with an ambiguous dependence of electron temperature on field strength. JETP 58, 1814–1824 (1970)
Belintsev, B. N., Livshits, M. A., Volkenstein, M. V.: The study of stability of inhomogeneous states in spatially distributed systems. Biofizika 23, 1056–1062 (1978) (Russian)
Belintsev, B. N., Livshits, M. A., Volkenstein, M. V.: Spatially inhomogeneous stationary forms in kinetic systems with reactions and diffusion. Positional differentiation. Biofizika 24, 117–123 (1979) (Russian)
Belintsev, B. N., Livshits, M. A., Volkenstein, M. V.: On the multi-stationary state transitions in the spatial kinetic systems. Z. Physik B 30, 211–218 (1978)
Belintsev, B. N., Livshits, M. A., Gurija, G. T., Volkenstein, M. V.: Transient processes in the development of multicellular organisms. Dokl. Akad. Nauk., USSR 244, 1239–1243 (1979) (Russian)
Crick, F.: Diffusion in embryogenesis. Nature 225, 420–422 (1970)
Chafee, N.: Asymptotic behaviour for solutions of a one-dimensional parabolic equation with homogeneous Neumann boundary conditions. J. Differential Equations 18, 111–134 (1975)
Edelstein, B. B.: The dynamics of cellular differentiation and associated pattern formation. J. Theoret. Biol. 37, 221–243 (1972)
Gierer, A., Meinhardt, H.: A theory of biological pattern formation. Kibernetika 12, 30–39 (1972)
Grigorov, L. N., Poljakova, Chernavskii, D. S.: Model investigation of the trigger schemes and of the differentiation process. Mol. Biol. (Russ.) 1, 410–418 (1967)
Hanson, M. P.: Dissipative structures in one dimensional pseudo-one component systems. J. Chem. Phys. 66, 5551–5556 (1977)
Jacob, F., Monod, J.: Genetic regulatory mechanisms in the synthesis of proteins. J. Mol. Biol. 3, 318–356 (1961)
Lavenda, B. H.: The theory of multi-stationary state transitions and biosynthetic control processes. Quart. Rev. Biophys. 5, 429–479 (1972)
Lemarchand, H., Nicolis, G.: Long range correlations and the onset of chemical instabilities. Physica A 82, 521–542 (1976)
Maginu, K.: Stability of stationary solutions of a semilinear parabolic partial differential equation, J. Math. Anal. Appl. 63, 224–243 (1978)
Nitzan, A., Ortoleva, P., Deutch, J., Ross, J.: Fluctuations and transitions at chemical instabilities. The analogy phase transitions. J. Chem. Phys. 61, 1056–1074 (1974)
Robertson, A., Cohen, M. H.: Control of developing fields. Ann. Rev. Biophys. Bioeng. 1, 409–464 (1972)
Sattinger, D. H.: Stability of bifurcating solutions by Leray-Schauder degree. Arch. Rational Mech. Anal. 43, 154–166(1971)
Thom, R.: Stabilité structurelle et morphogenese. New York: Benjamin 1972
Wolpert, L.: Positional information and the spatial pattern of cellular differentiation. J. Theor. Biol. 25, 1–47 (1969)
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Livshits, M.A., Gurija, G.T., Belintsev, B.N. et al. Positional differentiation as pattern formation in reaction-diffusion systems with permeable boundaries. Bifurcation analysis. J. Math. Biology 11, 295–310 (1981). https://doi.org/10.1007/BF00276898
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DOI: https://doi.org/10.1007/BF00276898