Dynamical system approach to phyllotaxis

Area of Science:

• Mathematical Biology
• Developmental Biology
• Dynamical Systems

Background:

• Phyllotactic patterns (leaf arrangements) are commonly modeled using algorithms.
• Standard dynamical system analysis is challenging due to the model's algorithmic nature and evolving phase space dimension.
• Existing numerical simulations provide stable patterns but lack detailed stability analysis.

Purpose of the Study:

• To develop a well-defined dynamical system from an algorithmic model of phyllotaxis.
• To perform a stability and bifurcation analysis on phyllotactic pattern formation.
• To clarify the roles of different growth mechanisms in determining plant structures.

Main Methods:

• Constructing a conserved-dimension dynamical system from the algorithmic model.
• Employing analytical relations to identify stable and unstable patterns.
• Investigating bifurcations, including anomalous scenarios arising from model discontinuities.
• Deriving an explicit formula for Jacobian evaluation and eigenvalue analysis.

Main Results:

• A method to transform an algorithmic model into a tractable dynamical system is presented.
• Stable and unstable phyllotactic patterns are identified through an analytical relation.
• The influence of various growth mechanisms on pattern formation is clarified.
• An explicit formula for Jacobian and eigenvalues facilitates stability analysis.

Conclusions:

• The developed dynamical system approach enables rigorous bifurcation analysis of algorithmic models.
• This framework is applicable to understanding pattern formation in biological morphogenesis.
• The study provides a foundation for analyzing complex growing systems in biology.