deletions | additions       diff --git a/section_Proofs_The_bacterial_density__.tex b/section_Proofs_The_bacterial_density__.tex  index 6ffcfd4..fcc9f49 100644  --- a/section_Proofs_The_bacterial_density__.tex  +++ b/section_Proofs_The_bacterial_density__.tex 
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\begin{equation}  \begin{align}  \dot{h}_x = \frac{\dot{H_x}}{H_{\rm tot}} &= \frac{H_{\rm tot} - H_x}{H_{\rm tot}} \Gamma_x^* \tilde{b}_a \\  &= (1 - h_x) \Gamma_x \left(1 - \frac{\tilde{b_x}}{\tilde{b}_{x, \rm \frac{\tilde{b}}{\tilde{b}_{\rm  max}} \right) \tilde{b}_a \frac{m b_a}{h_a}  \end{align}  \end{equation}  \begin{equation}  \begin{align}  \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}  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 $\Gamma_x$.  \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 (1-b) + \tilde{r} h_r \Gamma_b (1-b) - b_a \Gamma_x \left(1 - \frac{\tilde{b_x}}{\tilde{b}_{x, \rm \frac{\tilde{b}}{\tilde{b}_{\rm  max}}\right) \end{align}  \end{equation}  \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}_x}{\tilde{b}_{x, \rm \frac{\tilde{b}}{\tilde{b}_{\rm  max}}\right) \end{equation}