Current interest in explanations of pattern formation in biological systems as an emergent dynamical property can be traced to the seminal paper by Turing (1952), who demonstrated that a system of reacting and diffusing chemical species, called morphogens, can interact so as to produce stable nonuniform concentration patterns in space. Recently it has been suggested that a Turing model can explain the development of pigmentation patterns on species of growing angelfish such as Pomacanthus semicirculatus, which exhibit readily-observed changes in the number, size and orientation of colored stripes during the development of the juvenile and adult stages (Kondo and Asai (1995), yet this model has been criticised for its failure to recreate a number of observations concerning stripe formation. We introduce a generalized Turing model that incorporates cell growth and cell movement and analyse the effects of these processes on pattern formation. We demonstrate that the model can explain important features of pattern formation in a growing system such as Pomacanthus., and further demonstrate how domain growth can be exploited in a manner to robustly generate Turing patterns on large-scale domain. The model relies on a novel use of the positional information provided by the morphogen distributions.

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1998-1999 Mathematics in Biology