Model Mathematics

\frac{d}{d time} (T) = (s - (muT + kv T V)) + r T \frac{V}{C + V} + \int_{0}^{a1} gamma Ti d a s = 5.0 + \frac{5.0}{1.0 + V}

\frac{\partial}{\partial time} (Ti) = - (muTi Ti + r Ti \frac{V}{C + V} + gamma Ti + dTi_da) \frac{\partial}{\partial a} (Ti) = dTi_da

\frac{\partial}{\partial time} (V) = (N r \frac{V}{C + V} \int_{0}^{amax} Ti d a - (kT T V)) + \frac{gV V}{b + V}

gamma = \{\begin{matrix} \frac{c p}{1.0 - (ⅇ^{- (k p)})} ⅇ^{- (k time)} if a \geq 0.0 \land a ≦ a1 \land time \geq 0.0 \land time ≦ p \\ \frac{c p}{1.0 - (ⅇ^{- (k p)})} ⅇ^{- (k (time - p))} if a \geq 0.0 \land a ≦ a1 \land time \geq p \land time ≦ 2.0 p \\ 1.0 if a > a1 \end{matrix}