A reaction diffusion model for understanding phyllotactic formation

Phyllotactic patterns in plants are well known to be related to the golden ratio. Actually, many mathematical models using the theoretical inhibitory effect were proposed to reproduce these phyllotactic patterns. In 1996, Douady and Couder introduced a model using magnetic repulsion and succeeded in reproducing phyllotactic patterns numerically. On the other hand, it was recently revealed in biological experiments that a plant hormone, auxin, regulates the phyllotactic formation as an activator (Reinhardt et al., Nature 426:255-260, 2003). Then, there arises a natural question as to how the inhibitory effect can be related to the auxin. In this paper, a reaction diffusion model is proposed by taking account of auxin behavior in plant tips. The relationship between Douady and Couder's model and our model is shown by singular limit analysis. It also provides us with the potential function corresponding to the inhibitory effect, and the bifurcation diagram.