Chaotic solutions for irradiance-regulated buoyant motion of cyanobacteria

A very simple model for the irradiance-regulated buoyant motion of the cyanobacterium Oscillatoria agardhii is described. Removing the dimensions from this model enables us to find three independent parameters (a, b, c) that control the character of the motion. Small changes in the three parameters can cause large changes in the average irradiance incident upon a colony. The motion can be either periodic or chaotic depending on the values of these parameters. Motion with periods of 2 d is possible, in which case there can be subpopulations that move synchronously but 1 d out-of-phase with one another. The model is intrinsically nonlinear in any conceivable natural setting, but an analytic family of solutions exists for nighttime and an analytic family of approximate solutions for daytime can be found for velocity as a function of depth. The numerical solutions can be interpreted by switching between these analytic families of solutions. Vertical turbulent diffusivity causes a mean buoyant velocity that acts to oppose the downward diffusive flux. This turbulence results in the organism seeing less light, although its average depth is hardly changed because it is mixed deeper during the day and rises more at night.