deletions | additions       diff --git a/section_Proofs_subsection_Bacterial_density__.tex b/section_Proofs_subsection_Bacterial_density__.tex  index f018788..18245f6 100644  --- a/section_Proofs_subsection_Bacterial_density__.tex  +++ b/section_Proofs_subsection_Bacterial_density__.tex 
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\begin{equation*}  \begin{align}  \dot{N}_B =& \dot{b} N_B (0) = \dot{b} m N \\  =&- =& -  N \Gamma_0 \rho_B - N_R \Gamma_1 \rho_B = -N \rho_B (\Gamma_0 + \tilde{r} \Gamma_1) \end{align}  \end{equation*}