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Abstract:
The arrangement of plant organs - called phyllotaxis - is often highly regular. Surprisingly, these arrangements are not only qualitatively, but also quantitatively similar for different plants. Several abstract mathematical models invoking general principles like circular symmetry and mutual repulsion of plant organs reproduce these characteristic features of phyllotaxis. Two such models, belonging to the class of field models, are analyzed in the present study, they were introduced by Thornley (1975), and by Douady and Couder (1992). It is numerically demonstrated that in the limit of high organ density their bifurcation diagrams ("divergence spectrum") are self-similar under rescaling of two model parameters. Field models are known to give the classically expected behaviour only if the positioning of primordia is controlled exclusively by next-neighbor interactions (Kunz, 1995). In the general case, the bifurcation diagram of such models is topologically different and phyllotactic sequences converging to non-noble irrational divergences are possible. It is discussed how in these systems self-similarity and scaling together with the Farey tree structure of the divergence spectrum lead to a universal selection of asymptotic divergences with periodic continued fraction expansion [a1,a2,...,aN,κ1,κ2,...,κP-], where the overlining denotes infinite repetition. Moreover, it is argued that for solutions easily accessible during growth, divergences are favored where all numbers N, P, a 1, a 2, ..., and κ 1, κ 2, ... are small, thereby selecting in particular the noble limit divergences of the classically observed unijugate phyllotactic sequences. © 2012.
