Dynamical system approach to phyllotaxis

Abstract

This paper presents a bifurcation study of a model widely used to discuss phyllotactic patterns, i.e., leaf arrangements. Although stable patterns can be easily obtained by numerical simulations, a stability or bifurcation analysis is hindered by the fact that the model is defined by an algorithm and not a dynamical system, mainly because new active elements are added at each step, and thus the dimension of the "natural" phase space is not conserved. Here a construction is presented by which a well defined dynamical system can be obtained, and a bifurcation analysis can be carried out. Stable and unstable patterns are found by an analytical relation, in which the roles of different growth mechanisms determining the shape is clarified. Then bifurcations are studied, especially anomalous scenarios due to discontinuities embedded in the original model. Finally, an explicit formula for evaluation of the Jacobian, and thus the eigenvalues, is given. It is likely that problems of the above type often arise in biology, and especially in morphogenesis, where growing systems are modeled.