The Mathematical Biology of Diatoms

8Modeling the Synchronization of the Movement of Bacillaria paxillifer by a Kuramoto Model with Time Delay

Thomas Harbich

Independent Researcher, Am Brüdenrain, Weissach im Tal, Germany

Abstract

Bacillaria paxillifer (also known as Bacillaria paradoxa) forms band-shaped colonies in which the diatoms can actively move relative to each other. Often a synchronization of sections or the whole colony is observed. According to the tracking data of adjacent diatoms, an easily calculable function is proposed for its relative positions as a function of time. Observations made on small colonies and individual diatoms indicate that the propulsion systems of neighboring diatoms interact in such a way that these systems can be regarded as one oscillator. The interaction includes the control of the own movement as well as the recognition of the position of the adjacent diatom. The correlated motion of the colony is assumed to be due to the interaction of neighboring oscillators, which presupposes communication between them. The hypothesis is made that the coupling of the oscillators according to the Kuramoto model is suitable for describing the movement of Bacillaria colonies. The phase shifts always observable in colonies require the assumption of a time delay in the coupling. This reproduces the typical observed dynamic movement patterns such as linear, bowed and S-shaped colonies. The Kuramoto model predicts that for oscillator frequencies which are near to each other, the movement from ...