Journal of Mathematical Biology

, Volume 32, Issue 4, pp 345–393

Pattern formation in generalized Turing systems

I. Steady-state patterns in systems with mixed boundary conditions


  • R. Dillon
    • Department of MathematicsUniversity of Utah
  • P. K. Maini
    • Department of MathematicsUniversity of Utah
    • Centre for Mathematical BiologyMathematical Institute
  • H. G. Othmer
    • Department of MathematicsUniversity of Utah

DOI: 10.1007/BF00160165

Cite this article as:
Dillon, R., Maini, P.K. & Othmer, H.G. J. Math. Biol. (1994) 32: 345. doi:10.1007/BF00160165


Turing's model of pattern formation 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.

Key words

Spatial pattern formationsBifurcationTuring systems

Copyright information

© Springer-Verlag 1994