zbMATH — the first resource for mathematics

Pattern formation in generalized Turing systems. I: Steady-state patterns in systems with mixed boundary conditions. (English) Zbl 0829.92001
Summary: Turing’s model of pattern formation [A. M. Turing, Philos. Trans. R. Soc. Lond., Ser. B 237, 37-72 (1952)] has been extensively studied analytically and numerically, and there is recent experimental evidence that it may apply in certain chemical systems. The model is based on the assumption that all reacting species obey the same type of boundary condition pointwise on the boundary. We call these scalar boundary conditions. Here we study mixed or nonscalar boundary conditions, under which different species satisfy different boundary conditions at any point on the boundary, and show that qualitatively new phenomena arise in this case.
For example, we show that there may be multiple solutions at arbitrarily small lengths under mixed boundary conditions, whereas the solution is unique under homogeneous scalar boundary conditions. Moreover, even when the same solution exists under scalar and mixed boundary conditions, its stability may be different in the two cases. We also show that mixed boundary conditions can reduce the sensitivity of patterns to domain changes.

92C15 Developmental biology, pattern formation
35K57 Reaction-diffusion equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
Full Text: DOI
[1] Ashkenazi, M., Othmer, H. G.: Spatial patterns in coupled biochemical oscillators. J. Math. Biol. 5, 305-350 (1978) · Zbl 0381.92006
[2] Babloyantz, A., Bellemans, A.: Pattern regulation in reaction-diffusion systems ? the problem of size invariance. Bull. Math. Biol. 47, 475-487 (1985) · Zbl 0568.92001
[3] Benson, D. L., Sherratt, J. A., Maini, P. K.: Diffusion driven instability in an inhomogeneous domain. Bull. Math. Biol. 55, 365-384 (1993) · Zbl 0758.92003
[4] Benson, D. L., Maim, P. K., Sherratt, J. A.: Pattern formation in heterogeneous domains. In: Othmer, H. G., Maini, P. K., Murray, J. D. (eds.) Experimental and Theoretical Advances in Biological Pattern Formation. London: Plenum 1993
[5] Brümmer, F., Zempel, G., Buhle, P., Stein, J-C., Hulser, D. F.: Retinoic acid modulates gap junction permeability: a comparative study of dye spreading and ionic coupling in cultured cells. Exp. Cell Res. 196, 158-163 (1991) · doi:10.1016/0014-4827(91)90245-P
[6] Castets, V., Dulos, E., De Kepper, P.: Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern. Phys. Rev. Lett. 64(24), 2953-2956 (1990) · doi:10.1103/PhysRevLett.64.2953
[7] Child, C. M.: Patterns and Problems of Development. University of Chicago Press, 1941
[8] Conway, E., Hoff D., Smoller, J.: Large time behavior of solutions of nonlinear reaction diffusion equations. SIAM J. Appl. Math. 35(1), 1-16 (July 1978) · Zbl 0383.35035 · doi:10.1137/0135001
[9] Crick, F. H.: Diffusion in embryogenesis. Nature 225, 420-422 (1970) · doi:10.1038/225420a0
[10] 10.Dillon, R., Othmer, H. G.: Control of gap junction permeability can control pattern formation in limb development. In: Othmer, H. G., Maini, P. K., Murray, J. D. (eds.) Experimental and Theoretical Advances in Biological Pattern Formation. London: Plenum 1993
[11] Doedel, E.: AUTO: Software for continuation and bifurcation problems in ordinary differential equations. Technical report, California Institute of Technology, 1986
[12] Driesch, H.: Entwicklungsmechanische Studien. Z. Wiss. Zool. 53, 160-184 (1892)
[13] Driesch, H.: Entwicklungsmechanische Studien. Z. Wiss. Zool. 55, 3-61 (1893)
[14] Epstein, I. R., Lengyel, I., Kádár, S., Kagan, M., Yokoyama, M.: New systems for pattern formation studies. Physica A, 188, 26-33 (1992) · doi:10.1016/0378-4371(92)90249-P
[15] French, V., Bryant, P. J., Bryant, S. V.: Pattern regulation in epimorphic fields. Science 193, 969-981 (1977) · doi:10.1126/science.948762
[16] Goodwin, B. C., Kaufflnan, S. A.: Spatial harmonics and pattern specification in early Drosophila development. Part I. Bifurcation sequences and gene expression. J. Theor. Biol. 144, 303-319 (1990) · doi:10.1016/S0022-5193(05)80078-5
[17] Hunding, A., Sorensen, P. G.: Size adaptation of Turing prepatterns. J. Math. Biol. 26, 27-39 (1988) · Zbl 0631.92003
[18] Lacalli, T. C., Harrison, L. G.: The regulatory capacity of Turing’s model for morphogenesis with application to slime moulds. J. Theor. Biol. 70, 273-295 (1978) · doi:10.1016/0022-5193(78)90377-6
[19] Lengyel, I., Epstein, I.R.: A chemical approach to designing Turing patterns in reaction-diffusion systems. Proc. Natl. Acad. Sci. 89, 3977-3979 (1992). · Zbl 0745.92002 · doi:10.1073/pnas.89.9.3977
[20] Meinhardt, H.: Modes of Biological Pattern Formation. London: Academic Press 1982
[21] Moler, C. B., Stewart, G. W.: An algorithm for generalized matrix eigenproblems. SIAM J. Numer. Anal. 10, 241-256 (1973) · Zbl 0253.65019 · doi:10.1137/0710024
[22] Murray, J. D.: Mathematical Biology. Berlin Heidelberg New York: Springer 1989 · Zbl 0682.92001
[23] Othmer, H. G.: Interactions of Reaction and Diffusion in Open Systems. PhD thesis, Minneapolis: University of Minnesota 1969
[24] Othmer, H. G.: Current problems in pattern formation. In: Levin, S. A. (ed.) (Some mathematical questions in biology VIII. Lect. Math. Life Sci., vol. 9, pp. 57-85) Providence, RI: Am. Math. Soc. 1977
[25] Othmer, H. G.: Applications of bifurcation theory in the analysis of spatial and temporal pattern formation. In: Gurel, O., Rössler, O. K. (eds.) Bifurcation theory and applications in scientific disciplines, pp. 64-77. New York: New York Academy of Sciences 1979 · Zbl 0436.35011
[26] Othmer, H. G.: Synchronized and differentiated modes of cellular dynamics. In: Haken, H. (eds.) Dynamics of Synergetic Systems. Berlin Heidelberg New York: Springer 1980 · Zbl 0433.92004
[27] Othmer, H. G.: The interaction of structure and dynamics in chemical reaction networks. In: Ebert, K. H., Deuflhard, P., Jager, W. (eds.) Modelling of Chemical Reaction Systems, pp. 1-19 Berlin Heidelberg New York: Springer 1981
[28] Othmer, H. G., Aldridge, J.: The effects of cell density and metabolite flux on cellular dynamics. J. Math. Biol. 5, 169-200 (1978) · Zbl 0398.92007
[29] Othmer, H. G., Pate, E. F.: Scale invariance in reaction-diffusion models of spatial pattern formation. Proc. Nat. Acad. Sci. 77, 4180-4184 (1980) · doi:10.1073/pnas.77.7.4180
[30] Othmer, H. G., Scriven, L. E.: Interactions of reaction and diffusion in open systems. Ind. Eng. Chem. Fund 8, 302-315 (1969) · doi:10.1021/i160030a020
[31] Ouyang, Q., Swinney, H. L.: Transition from a uniform state to hexagonal and striped patterns. Nature 352, 610-612 (1991) · doi:10.1038/352610a0
[32] Pate, E., Othmer, H. G.: Applications of a model for scale-invariant pattern formation in developing systems. Differentiation 28, 1-8 (1984) · doi:10.1111/j.1432-0436.1984.tb00259.x
[33] Pearson, J. E., Horsthemke, W.: Turing instabilities with nearly equal diffusion coefficients. J. Chem. Phys. 90(3), 1588-1599 (1989) · doi:10.1063/1.456051
[34] Turing, A. M.: The chemical basis of morphogenesis. Philos., Trans. R. Soc. Lond Ser. B 237, 37-72 (1952) · Zbl 1403.92034 · doi:10.1098/rstb.1952.0012
[35] Ward, R. C.: The combination shift QZ algorithm. SIAM J. Numer. Anal. 12, 835-853 (1975) · Zbl 0342.65022 · doi:10.1137/0712062
[36] Wolpert, L.: Positional information and the spatial pattern of cellular differentiation. J. Theor. Biol. 25, 1-47 (1969) · doi:10.1016/S0022-5193(69)80016-0
[37] Wolpert, L.: Positional information and pattern formation. Curr. Top. Dev. Biol. 6, 183-224 (1971) · doi:10.1016/S0070-2153(08)60641-9
[38] Wright, D. A., Lawrence, P. A.: Regeneration of the segment boundary in Oncopeltus. Dev. Biol. 85, 317-327 (1981) · doi:10.1016/0012-1606(81)90263-3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.