weib.tex

\documentclass[12pt]{article}
\usepackage{a4full}
\begin{document}
\section*{Simple Weibull-Gompertz}

\begin{eqnarray}
  \dot{p}_C &=& [E_m] e L^3 (\dot{v}/ L - \dot{r})
\\
  \dot{r} &=&\frac{\dot{v} e/ L - (1 + L_T/ L) \dot{k}_M g} {e + g}
\\
  \dot{h} &=& \dot{h}_a m_D/ m_D^{\mbox{\tiny ref}}
\\
  \frac{d} {dt} m_Q &=& \eta_{QC} \dot{p}_C/ M_V - \dot{r} m_Q
\\
  \frac{d} {dt} m_D &=& 
   m_Q y_{DQ} \dot{k}_W + m_D \dot{k}_G - \dot{r} m_D
\\
  \frac{d} {dt} \dot{h} &=& \ddot{q} -
    \dot{h} (\dot{r} - \dot{k}_G); \quad 
  \ddot{q} = m_Q y_{DQ} \dot{k}_W \dot{h}_a/ m_D^{\mbox{\tiny ref}} 
\\
  \frac{d} {dt} \ddot{q} &=& \ddot{h}_a e (\dot{v}/ L - \dot{r})
    - \dot{r} \ddot{q}; \quad 
  \ddot{h}_a = \dot{h}_a \dot{k}_W y_{DQ} \frac{\eta_{QC}} {m_D^{\mbox{\tiny ref}}}
  \frac{[E_m]} {[M_V]}
\end{eqnarray}

\section*{Modified Weibull-Gompertz: $M_D$ acceleration $\propto \dot{p}_C$}

\begin{eqnarray}
  \frac{d} {dt} m_Q &=& \eta_{QC} \dot{p}_C/ M_V - \dot{r} m_Q
\\
  \frac{d} {dt} m_D &=& 
   m_Q y_{DQ} \dot{k}_W + 
   m_D \frac{s_G \dot{p}_C} {[E_m] L_m^3} - \dot{r} m_D
\\
  \frac{d} {dt} \dot{h} &=& \ddot{q} - \dot{h} (\dot{r} - 
  s_G \frac{L^3} {L_m^3} e (\dot{v}/ L - \dot{r}))
\\
  \frac{d} {dt} \ddot{q} &=& \ddot{h}_a e (\dot{v}/ L - \dot{r})
    - \dot{r} \ddot{q}
\end{eqnarray}


\section*{Modified Weibull-Gompertz: $M_Q$ acceleration $\propto \dot{p}_C$}

\begin{eqnarray}
  \frac{d} {dt} m_Q &=& \eta_{QC} \dot{p}_C/ M_V + 
  m_Q \frac{s_G \dot{p}_C} {[E_m] L_m^3} - \dot{r} m_Q
\\
  \frac{d} {dt} m_D &=& m_Q y_{DQ} \dot{k}_W - \dot{r} m_D
\\
  \frac{d} {dt} \dot{h} &=& \ddot{q} - \dot{r} \dot{h}; \quad 
  \ddot{q} = m_Q y_{DQ} \dot{k}_W \dot{h}_a/ m_D^{\mbox{\tiny ref}} 
\\
  \frac{d} {dt} \ddot{q} &=& (\ddot{q} \frac{L^3} {L_m^3} s_G + \ddot{h}_a) 
    e (\dot{v}/ L - \dot{r}) - \dot{r} \ddot{q}
\end{eqnarray}

For short growth periods, $\dot{r} = 0$, and constant food, $e = f$ 
\begin{eqnarray}
 \ddot{q}(t) &=& \ddot{a} (\exp(\dot{b} t) - 1); \quad 
 \ddot{a} = \frac{\ddot{h}_a L_m^3} {s_G (e L_m - L_T)^3}; \quad
 \dot{b} = s_G e \dot{v} \frac{(e L_m - L_T)^2} {L_m^3}
\\
 \dot{h}(t) &=& \ddot{a} ( \frac{\exp(\dot{b} t) - 1} {\dot{b}} - t)
\\
 S(t) &=& \exp \left( - \frac{\ddot{a}} {\dot{b}} \left( 
 \frac{\exp(\dot{b} t) - 1} {\dot{b}} - t - \dot{b} t^2/ 2
 \right) \right)
\end{eqnarray}

for $\dot{h}_G = s_G \dot{v} e (e L_m - L_T)^2 L_m^{-3}$
\begin{equation}\label{eqn:qhS_gomp}
 \ddot{q}(t) = 
 \ddot{q}(0) \exp(t \dot{h}_G); \quad
 \dot{h}(t) = \ddot{q}(0) \frac{\exp(t \dot{h}_G) - 1} {\dot{h}_G}; \quad
 \Pr \{\underline{a}_{\dagger} > t \} = 
 \exp(\frac{\ddot{q}(0)} {\dot{h}_G^2} (\exp(t \dot{h}_G) - 1 - t \dot{h}_G))
\end{equation}

\end{document}