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Taylor Dunn edited section_Proofs_The_bacterial_density__.tex
over 8 years ago
Commit id: 004914fb5176b51a47fca64dce8e7cdcae7439db
deletions | additions
diff --git a/section_Proofs_The_bacterial_density__.tex b/section_Proofs_The_bacterial_density__.tex
index 10972fa..92d9fce 100644
--- a/section_Proofs_The_bacterial_density__.tex
+++ b/section_Proofs_The_bacterial_density__.tex
...
\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^*
\frac{B_a}{H_a} \tilde{b}_a \\
&= (1 - h_x) \Gamma_x \left(1 -
\frac{\tilde{b}}{\tilde{b}_{\rm \frac{\tilde{b_x}}{\tilde{b}_{x, \rm max}} \right) \frac{m b_a}{h_a}
\end{align}
\end{equation}
...
\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}}{\tilde{b}_{\rm \frac{\tilde{b_x}}{\tilde{b}_{x, \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}}{\tilde{b}_{\rm \frac{\tilde{b}_x}{\tilde{b}_{x, \rm max}}\right)
\end{equation}