Model Mathematics

V_1 = 0.1 V_max Ca = \{\begin{matrix} Ca_m {(1 - (ⅇ^{(- a_c) {time}^{2}}))}^{2} if time ≦ t_d \\ Ca_m {((1 - (ⅇ^{(- a_c) {time}^{2}})) ⅇ^{(- b_c) {(time - t_d)}^{2}})}^{2} otherwise \end{matrix} C_2 = C_20 pi_n phi_A_1 pi_n = \{\begin{matrix} pi_min if 0.75 ≦ n \land n ≦ 1 \\ {pi_min}^{2 n - 0.5} if 0.25 ≦ n \land n < 0.75 \\ 1 otherwise \end{matrix} phi_A_1 = ⅇ^{(- q_k) A_1} S = 0.5 l_1 + S_0 A = \frac{A_1 S}{1} \frac{d}{d time} (A_1) = C_1 Ca (TnC - A_1) - (C_20 pi_n phi_A_1 A_1) n = n_1 n_2 n_1 = \{\begin{matrix} 0 if g_1 l_1 + g_2 < 0 \\ g_1 l_1 + g_2 if 0 ≦ g_1 l_1 + g_2 \land g_1 l_1 + g_2 ≦ 1 \\ 1 otherwise \end{matrix} \frac{d}{d time} (n_2) = q_V (m_0 G_V - n_2) q_V = \{\begin{matrix} q_1 - (\frac{q_2 V}{V_max}) if V ≦ 0 \\ q_3 otherwise \end{matrix} G_V = 1 + \frac{0.6 V}{V_max} V = - dl_1_dt F_V = \frac{a (1 + \frac{V}{V_max})}{a - (\frac{V}{V_max})} p_V = \frac{F_V}{G_V} P_CE = lambda S A_1 n p_V P_PE = beta_2 (ⅇ^{alpha_2 l_2} - 1) P_SE = beta_1 (ⅇ^{alpha_1 (l_2 - l_1)} - 1)

dl_1_dt = 0 l_1 = \{\begin{matrix} 0 if 200 ≦ time \land time ≦ 201 \\ 0 otherwise \end{matrix} l_2 = l_1 + 1.87