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(x,t)}{\partial t} = Diff(AHL(x,t),x) + R(AHL(x,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} \frac{\partial^2 AHL}{\partial x^2} + DF \cdot (\alpha_{AHL} \cdot [LuxI]-d_{AHL,i} \cdot [AHL]) - d_{AHL,e} \cdot [AHL] \end{align}

For the analytical approximation of the AHL gradient in one dimension, we consider the boundary condition that the concentration at the reservoir (located at z = 0) stays constant:
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