deletions | additions       diff --git a/section_Proofs_The_bacterial_density__.tex b/section_Proofs_The_bacterial_density__.tex  index fcc9f49..ce39323 100644  --- a/section_Proofs_The_bacterial_density__.tex  +++ b/section_Proofs_The_bacterial_density__.tex 
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\dot{h}_r = \frac{\dot{H}_r}{H_{\rm tot}} &= \frac{H_{\rm tot} - H_r}{H_{\rm tot}} \Gamma_r^* \tilde{b}_a \\  &= (1 - h_r) \Gamma_r \left(1 - \frac{\tilde{r}}{\tilde{r}_{\rm max}} \right) \tilde{b}_a  \end{align}  \end{equaiton} \end{equation}  A more complicated quantity is the fraction of bacteria which are attached to host cells, which has three means of change. The first is by regular primary attachment with rate $\Gamma_a$, the second is secondary ruffle recruitment with rate $\Gamma_b$, and the third is a loss of attached bacteria as they invade with rate $\Gamma_x$.