Photo-gyrotactic bioconvection

Abstract

Many microorganisms exhibit taxes, biased swimming motion relative to a directional stimulus. Aggregations of cells with densities dissimilar to the medium in which they swim can induce hydrodynamic instabilities and bioconvection patterns. Here, three novel and mechanistically distinct models of the interaction of the two dominant taxes in suspensions of swimming phototrophic algae are presented: phototaxis, swimming towards or away from light, and gyrotaxis, a balance between viscous and gravitational torques. The descriptions are accordant with, and extend, recent rational models of bioconvection. In particular, the first model is for photokinesis–gyrotaxis, the second varies the cells' centre-of-mass offset, and the third introduces a reactive phototactic torque associated with the propulsive flagellar apparatus. Equilibria and linear-stability analysis in a layer of finite depth are analysed in detail using analytical and numerical methods. Results indicate that the first two models, despite their different roots, remarkably are in agreement. Penetrative and oscillatory modes are found and explained. Dramatically different behaviour is obtained for the model with phototactic torques: instabilities arise even in the absence of fluid motion due to induced gradients of light intensity. Typically, the response of microorganisms to light is multifaceted and thus some combination of the three models is appropriate. Encouragingly, qualitative agreement is found with recent experimental measurements on the effects of illumination on dominant pattern wavelength in bioconvection experiments. The theory may be of some interest in the emergent field of bioreactor design.