Disk-stacking models are consistent with Fibonacci and non-Fibonacci structure in sunflowers

Abstract

This paper investigates a model of plant organ placement motivated by the appearance of large Fibonacci numbers in phyllotaxis, and provides the first large-scale empirical validation of this model. Specifically it evaluates the ability of Schwendener disk-stacking models to generate parastichy patterns seen in a large dataset of sunflower seedheads. We find that features of this data that the models can account for include a predominance of Fibonacci counts, usually in a pair of left and right counts on a single seedhead, a smaller but detectable frequency of Lucas and double Fibonacci numbers, a comparable frequency of Fibonacci numbers plus or minus one, and occurrences of pairs of roughly equal but non-Fibonacci counts in a `columnar' structure. A further observation in the dataset was an occasional lack of rotational symmetry in the parastichy spirals, and this paper demonstrates those in the model for the first time. Schwendener disk-stacking models allow Fibonacci structure by ensuring that a parameter of the model corresponding to the speed of plant growth is kept small enough. While many other models can exhibit Fibonacci structure, usually by specifying a rotation parameter to an extremely high precision, no other model has accounted for further, non-Fibonacci, features in the observed data. The Schwendener model produces these naturally in the region of parameter space just beyond where the Fibonacci structure breaks down, without any further parameter fitting. We also introduce stochasticity into the model and show that it while it can be responsible for the appearance of columnar structure, the disordered dynamics of the deterministic system near the critical region can also generate this structure.