deletions | additions       diff --git a/section_Proofs_The_bacterial_density__.tex b/section_Proofs_The_bacterial_density__.tex  index dd4d1c6..381ecf0 100644  --- a/section_Proofs_The_bacterial_density__.tex  +++ b/section_Proofs_The_bacterial_density__.tex 
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\begin{equation}  \dot{b}_x = \frac{\dot{B_x}}{B_{\rm tot}} = \frac{B_a}{B_{\rm tot}} \Gamma_x^* = b_a \Gamma_x \left(1 - \frac{\tilde{b}}{\tilde{b}_{\rm max}}\right)  \end{equation}  \subsection{Ruffle dynamic equation}  \begin{equation}  \dot{\tilde{r}} = \frac{\dot{R}}{H_r} = \frac{B_a}{H_r} \Gamma_r^* = m \frac{b_a}{h_r} \Gamma_r \left(1 - \frac{tilde{r}}{\tilde{r}_{\rm max}}\right)  \end{equation}