# Bioconvection in Microalgae: Review of Mathematical Models
## 2 Models
With the assumption that the cell volume fraction is low enough for hydrodynamic or cell-cell interactions to be neglected, the authors described the models for dilute suspensions of the unicellular, bottom-heavy alga _Chlamydomonas nivalis_ as this organism exhibits the key behaviors the model was designed to capture--specifically phototaxis, negative gravitaxis (swimming upward against gravity), and gyrotaxis (orientation due to the balance between viscous and gravitational torques on the cell). The model was also tested on the bacteria _Bacillus subtilis_, oxygen-consuming organisms that become active swimmers when the oxygen concentration in the environment exceeds a critical value. Since they swim up oxygen gradients (oxytaxis) and are denser than the surrounding medium, their swimming behavior leads to bioconvective instability. The mathematical framework these authors developed for _C. nivalis_ can be synthesized into the following governing equations:  
* Continuity equation (conservation of cells): \[\frac{\partial n}{\partial t}=D\
abla 2n-\
abla\cdot(n(u+V_{g}))\] (2)
* Fluid conservation (incompressibility): \[\
abla\cdot u=0\] (3)
* Momentum equation (Navier-Stokes with cell density effects): \[\rho(\frac{\partial u}{\partial t}+u\cdot\
abla u)=-\
abla P-g\rho(1+\alpha n )k+\mu\
abla 2u\] (4)  
where \(n\) is the cell concentration, \(u\) is the fluid velocity, \(V_{g}\) is the gyrotactic swimming velocity, \(D\) is the cell diffusivity tensor, \(\rho\) is the fluid density, \(P\) is the pressure, \(g\) is the gravitational acceleration, \(\alpha\) is the relative density difference parameter, \(\mu\) is the fluid viscosity, and \(k\) is the unit vector in the vertical direction. The mathematical constraints of this model limit its applicability to Newtonian fluids, taking into account only small perturbations from equilibrium; therefore, the model may not capture strongly nonlinear behaviors. The model created by [PERSON] and [PERSON] [42] captures cell transport through both advection and diffusion, the flow of the fluid driven by density differences, the directed swimming response to stimuli, and the coupling between cell distribution and fluid motion. It is only valid for dilute suspensions where cell-cell interactions are negligible, and where cell morphologies or swimming patterns are simple and constant. The main drawback of this model is that it does not account for cell division or death and for complex behavioral responses to multiple stimuli. Moreover, by assuming uniform cell properties across the population, it ignores potential chemical signaling between cells. Though representing a significant advance in understanding bioconvection, the model by [PERSON] and [PERSON] may need modification for systems with strong cell-cell interactions or complex behavioral responses. Within the framework of [PERSON] and [PERSON] [17], [PERSON] and [PERSON] [43] presented three novel approaches aiming to integrate the effects of two important behavioral traits, phototaxis and gyrotaxis, in the descriptive model of bioconvection in cell suspensions and their impact on pattern formation. The three proposed models take into consideration different aspects of cell swimming behavior: the swimming speed as a function of light intensity; the adjustment of the center-of-mass offset in response to light by modulating the symmetric flagellar beat; and a new reactive torque due to phototaxis, induced by an asymmetric flagellar beat pattern. The first model is named Photokinesis-Gyrotaxis. It focuses on how light intensity affects swimming velocity (\(V_{s}\)) while maintaining normal gravitaxis and gyrotaxis behaviors. Its photokinetic description is based on the assumption that the swimming velocity, vs, varies linearly with light intensity \(I\) according to  
\[V_{s}(I)=-\xi(I-I_{c}) \tag{5}\]  
where \(I_{c}\) is the critical intensity, and \(\xi=V_{n}/I_{c}\) (\(V_{n}\) is the swimming speed in the dark). Cells swim forward when \(I<I_{c}\), stop at \(I=I_{c}\), and swim backward when \(I>I_{c}\).