Phyllotaxis in a Keller-Segel model

Plants often exhibit regular arrangements of seeds or leaves, known as phyllotaxis, and can famously exhibit Fibonacci spirals, as in sunflower heads or on pine cones. While there are many models which can reproduce a spiral formation, the actual mechanism of arrangement remains unclear. Here, we test a more general class of model, based on the Keller-Segel model, in which a diffusing chemical signal produces its own transport signal and can form local instabilities. By modeling the plant as a disk with a signal source at the center and radial growth, we show that we can reproduce both Fibonacci spirals and alternating patterns, depending on a balance between growth and source. Finally, using linear stability analysis, we show how the pattern arises due to a balance between the growth of instabilities and the growth of the plant. Overall, this work demonstrates how the Keller-Segel model can reproduce a wide range of observed patterns, and may act as a phenomenological description of a number of processes in plants.