# Bioconvection in Microalgae: Review of Mathematical Models
## 2 Models
This phenomenon has particular relevance in marine environments, where it influences sediment ecology and nutrient transport patterns. In industrial applications, bioconvection principles have found significant use in microbial-enhanced oil recovery, where microorganisms are employed to improve oil extraction from reservoirs. The biotechnology sector has leveraged the understanding of these processes to optimize bioreactor design and improve biofilter efficiency, leading to advances in wastewater treatment, biofuel production, and microalgae cultivation. The medical field has also benefited from insights into bioconvection in porous media, particularly in the development of drug delivery systems and the study of bacterial infection patterns. Understanding how microorganisms move through porous biological tissues has improved our ability to design controlled release systems and combat biofilm formation. To continue with models of bioconvection in non-porous media, special attention has been paid to the bioconvective behavior of _Euglena_ by the group of [PERSON] [54], who investigated the effect that light exerts on pattern formation and proposed a model for it. These authors performed experimental observations of the cell behavior in a sealed container schematically divided into a thin upper layer and a thick lower layer and described a novel bioconvective localized pattern induced by positive phototaxis. They derived a two-component system of reaction-diffusion equations with a nonlocal term (this term represented the localization of the pattern since patterns were not formed in the whole container) to model the macroscopic pattern formed by _Euglena_ under stationary illumination, stressing the similarities between this pattern and Rayleigh-Benard convection [5; 6; 7]. The model considered not only upward swimming due to negative phototaxis and downward sinking due to gravity, but also the diffusion in the two layers of the container due to horizontal swimming and lateral movement in the upper layer based on the nonlocal interaction of cells (\"self-shading\" effect). It is based on a quasi-one-dimensional system with two layers (upper and lower) and incorporates both vertical and lateral phototaxis. It uses two coupled partial differential equations for population vertical dynamics in the two layers:For the upper layer,  
\[\frac{\partial P_{\rm u}}{\partial t}=-\frac{aP2_{\rm u}}{\delta}+bP_{1}+\frac{D_ {\rm u}\partial 2P_{\rm u}}{\partial x2}+(C_{+}-C_{-})P_{\rm u} \tag{10}\]  
For the lower layer,  
\[\frac{\partial n}{\partial t}=D\
abla 2n-\
abla\cdot(n(u+V_{g}))\]0  
where \(P_{u}\) and \(P_{1}\) are the cell populations in the upper and lower layers, \(P=P_{u}+P_{1}\) (total population), a and b are constants for downward/upward average velocities, \(\delta\) is the suspension depth, and \(D_{u}\) and \(D_{1}\) are the diffusion coefficients, while C\({}_{+}\) and C\({}_{-}\) represent the lateral phototaxis effects towards the right (+) and left (\(-\)) directions. The equations show that the downward movement (aP\({}^{2}\)u/\(\delta\) term) increases with a higher upper-layer population density, while the upward movement (bP\({}_{1}\) term) is proportional to the lower-layer population, with diffusive spreading being present in both layers (D terms). As for the lateral phototaxis term, it is expressed as follows:  
\[\frac{\partial n}{\partial t}=D\
abla 2n-\
abla\cdot(n(u+V_{g}))\]1  
where \(c\) is the coefficient relating cell density to light intensity, and \(\lambda\) is the decay length of light propagation. The lateral phototaxis term (C\({}_{+}(x)-C_{-}(x)\)) is particularly interesting because it uses a convolution integral to account for light effects from neighboring cells (light scattered from the cells proportional to their density), and an exponential decay (e\({}^{-1\,x-y^{1}/\lambda}\)) that models how light intensity diminishes with distance \(|x-y|\) (Lambert-Beer law). This creates a nonlocal interaction between cells through light. In their model, [PERSON] et al.