# Bioconvection in Microalgae: Review of Mathematical Models
## 2 Models  
While [PERSON] did not provide a formal mathematical model in his paper [2], his description and explanation of bioconvection were crucial in establishing the field and inspiring future research and modeling efforts. In this section, we will describe some of the models developed from experimental observations to analyze the bioconvection phenomenon and simulate the different patterns generated by the collective motion of two microalgae, i.e., _Chlamydomonas_ and _Euglena_, the microorganisms of choice due to their complex behavior and multifaceted potentialities as cell models in photobiology [26; 27; 28; 29; 30; 31] and as sources of active biomolecules [32; 33; 34; 35]. We will not go into full detail of the complex mathematical formulations at the basis of the various models since this is beyond the scope of this review; still, we will provide simplified forms of the equations used by the authors to describe their approach. One of the most influential modern models for bioconvection was developed by [PERSON] and [PERSON] in the 1990s [17; 36], often referred to as the Pedley-Kessler continuum model formulated for dilute suspensions of swimming microorganisms with asymmetric mass distributions [17; 36]. It treats the suspension as a continuous medium rather than focusing on individual microorganisms, combining fluid dynamics with the behavior of swimming microorganisms, and accounting for randomness in swimming direction by incorporating gyrotaxis (orientation due to balance between viscous and gravitational torques) [37; 38; 39]. The model used a Fokker-Planck equation analogous to that obtained for colloid suspensions in the presence of rotational Brownian motion [40; 41] and was tested on swimming cells of _Chlamydomonas nivalis_. The Fokker-Planck equation [42] for bioconvection describes the probability distribution of microorganism positions and velocities in a fluid, where the organisms exhibit both random motion and directed swimming behavior (often in response to stimuli). The general form is  
\[\frac{\partial p}{\partial t}=-\
abla\;(vp)+\
abla\;(D\cdot\
abla\;p)-\
abla\; (V_{s}p) \tag{1}\]  
where \(p\) is the probability density function, \(v\) is the fluid velocity, \(D\) is the diffusion coefficient representing the random motion of the organisms, \(V_{s}\) is their swimming velocity, t is time, and \(\
abla\) is the gradient operator. This equation combines advection (second term), diffusion (third term), and active swimming (fourth term) to model how populations of swimming microorganisms distribute themselves in fluid environments. A key consideration is that while this model is powerful for understanding bulk behaviors and patterns in microorganism suspensions, it may not be suitable for studyingphenomena that depend heavily on individual organism characteristics or interactions. The model's success with _Chlamydomonas nivalis_ suggests that it works well for similar swimming microorganisms, but careful validation would be needed for application to other species. The Pedley-Kessler continuum model may not be fully applicable when suspensions become too dense or when strong chemical gradients are present. It may also become limited in the case of interactions of multiple species since it does not account for cell-cell signaling or chemical communication, so it may not capture complex behavioral responses to environmental cues. Still, its computational efficiency enables the simulation of large-scale systems and makes it suitable for studying collective behavior. Extensions and refinements of this model have been proposed afterwards. [PERSON] and [PERSON] [42] further developed and renewed the new continuum model of 1992 [17], stressing the rational aspect of the formulation, i.e., its systematical derivation from scientific data. They provided nonlinear extensions of previous models, introducing the effects of transport and dispersion by the flow, known as the \"taxis\" function (a biased motile response of the microorganism to directional stimuli).