deletions | additions       diff --git a/section_Proofs_The_bacterial_density__.tex b/section_Proofs_The_bacterial_density__.tex  index a01d408..f174bd0 100644  --- a/section_Proofs_The_bacterial_density__.tex  +++ b/section_Proofs_The_bacterial_density__.tex 
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\begin{equation}  \begin{align}  \dot{b}_a = \frac{\dot{B_a}}{B_{\mathrm{tot}}} &= \frac{H_{\rm tot} \Gamma_a \rho_B}{B_{\mathrm{tot}}} + \frac{R \Gamma_b \rho_B}{B_{\mathrm{tot}}} - \frac{B_a \Gamma_x^*}{B_{\mathrm{tot}}} \\  &= \Gamma_a b c + \tilde{r} h_rh_a  \Gamma_b b c - b_a \Gamma_x \left(1 - \frac{\tilde{b}_x}{\tilde{b}_{x, \rm max}}\right) \end{align}  \end{equation} 
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Change in ruffle formation is another quantity whose rate $\Gamma_r^*$ depends on attached bacteria per cell $\tilde{b}_a$ and is limited by a maximum $\tilde{r}_{\rm max}$.  \begin{equation}  \dot{\tilde{r}} = \frac{\dot{R}}{H_a} \frac{\dot{R}}{H_r}  = \frac{H_a}{H_a} \frac{B_a}{H_r} \Gamma_r^* = \Gamma_r \left(1 - \frac{\tilde{r}}{\tilde{r}_{\rm max}}\right) \frac{b_a m}{h_r}  %\dot{\tilde{r}} = \frac{\dot{R}}{H_r} = \frac{B_a}{H_r}  \Gamma_r^* \tilde{b}_a = \Gamma_r \left(1 - \frac{\tilde{r}}{\tilde{r}_{\rm max}}\right) \frac{b_a m}{h_a} \end{equation}