deletions | additions       diff --git a/section_Proofs_The_bacterial_density__.tex b/section_Proofs_The_bacterial_density__.tex  index dad760c..ae59e7e 100644  --- a/section_Proofs_The_bacterial_density__.tex  +++ b/section_Proofs_The_bacterial_density__.tex 
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Change in ruffle formation is another quantity whose rate $\Gamma_r$ depends on attached bacteria per cell $\tilde{b}_a$.  \begin{equation}  \dot{\tilde{r}} = \frac{\dot{R}}{H_a} = \Gamma_r^* \tilde{b}_a = \Gamma_r \left(1 - \frac{\tilde{r}}{\tilde{r}_{\rm max}}\right) m \frac{b_a}{h_a} \frac{m b_a}{h_a}  \end{equation}