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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.

MSC:
92C15 Developmental biology, pattern formation
35K57 Reaction-diffusion equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
Software:
AUTO; AUTO-86
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