Team:ETH Zurich/Modeling/Analytical Approximations
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\begin{align} | \begin{align} | ||
- | \frac{\partial AHL | + | \frac{\partial AHL}{\partial t} = C_{agar} \cdot D_{AHL} \nabla^{2} AHL + DF \cdot (\alpha_{AHL} \cdot [LuxI]-d_{AHL,i} \cdot [AHL])_{MineCell} - DF \cdot (d_{AHL,i} \cdot [AHL])_{ReceiverCell} - d_{AHL,e} \cdot [AHL] |
\end{align} | \end{align} | ||
<br clear="all"/> | <br clear="all"/> | ||
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Revision as of 19:55, 24 October 2013
Steady State AHL Gradient Approximation
In the reaction-diffusion model we have formalized the gradient establishment and we solved the resulting partial differential equation numerically. Now, the aim is to derive a suitable analytical approximation.
Kolmogorov-Petrovsky-Piskounov Equation
A reaction–diffusion equation concerning the concentration of a single molecule in one spatial dimension is known as the Kolmogorov-Petrovsky-Piskounov Equation. In our case, the equation for AHL has the following general structure:
\begin{align} \frac{\partial AHL(\textbf{r},t)}{\partial t} = Diff(AHL(\textbf{r},t),\textbf{r}) + R(AHL(\textbf{r},t)) \end{align}
And a less general form, as derived for the reaction-diffusion model for AHL:
\begin{align}
\frac{\partial AHL}{\partial t} = C_{agar} \cdot D_{AHL} \nabla^{2} AHL + DF \cdot (\alpha_{AHL} \cdot [LuxI]-d_{AHL,i} \cdot [AHL])_{MineCell} - DF \cdot (d_{AHL,i} \cdot [AHL])_{ReceiverCell} - d_{AHL,e} \cdot [AHL]
\end{align}