# Bioconvection in Microalgae: Review of Mathematical Models
## 1 Introduction  
In 1911, [PERSON] [1] described in fine detail the aggregation behavior of the unicellular alga _Euglena viridis_ and defined it as the mechanical result of the action of gravity combined with the cohesive force acting upon microorganisms heavier than water, which comes into play as soon as the microalgae are in a sufficiently crowded condition to allow gravity to act [1]. On the basis of his observation of the seasonal and diurnal behaviors of _Euglena_, the British botanist performed some experiments to follow the appearance and evolution of the aggregation network in different kinds of vessels. He examined the influences of light, vessel shape and depth, exposure to air, temperature, and specific gravity. He described the aggregation phenomenon as the result of a constant cyclic movement downwards and upwards. If cells are present in large numbers, as is the case when the conditions for aggregation are fulfilled, they soon become massed together; their freedom of movement gradually becomes more and more impeded, and very soon, they find themselves sinking downwards under gravitational influence due to the specific gravity of the living _Euglena_ being greater than that of water. This downward movement through the water generates a movement in the opposite direction. The friction between the cell stream and the water sets up a vortical motion by which many of the cells become separated from the main mass; once free, but still with a vertical orientation, they swim upwards again, where they will soon experience mass conditions leading again to sinking. It is in this way that the cyclic downward and upward movements persisting during aggregation are maintained. He concluded that aggregation, though not necessarily involving communication betweencells, is of benefit to _Euglena_ in that it results in a constant dissemination of the cells through the liquid environment and prevents accumulation in such dense masses that would have a detrimental effect on their existence by interfering with their assimilative and respiratory functions. [PERSON] observed that aggregation also occurs in the dark, relatively quickly, with cells forming more irregular and distantly spaced groups with respect to light conditions, and attributed it to mechanical forces, including gravity and cohesive forces acting on the single organisms that are heavier than water. About 50 years later, [PERSON] [2] coined the term \"bioconvection\" to describe these peculiar patterns formed in shallow suspension by the collective behavior of swimming organisms [2]. [PERSON] observed that under certain conditions, suspensions of swimming microorganisms that are denser than the fluid they swim in, such as the ciliate _Tetrahymena pyriformis_, would spontaneously form regular patterns, such as polygonal cells or parallel bands. He noted that many microorganisms, such as certain algae and protozoa, show a directional bias and tend to swim upward. As these organisms concentrate near the upper surface of the fluid, they create a top-heavy, unstable density distribution, which in turn leads to convection, where denser fluid (containing more organisms) sinks, while less dense fluid rises, creating circulation patterns that resemble the Benard cells typical of thermal convection (hence the term bioconvection pattern) [3; 4]. The requirement of a minimum critical depth and a minimum critical density for pattern formation support the resemblance to Benard cells, with the density of motile organisms being the analog of the energy input in the thermal case [5; 6; 7]. Confirming what was already stated by [PERSON], [PERSON] [2] concluded that the combination of upward swimming and fluid convection results in self-organized patterns visible at the macroscopic scale. To date, the detailed descriptions provided by [PERSON] [1] and [PERSON] [2] are still valid and representative of bioconvection in single-celled swimming microorganisms.