$$\frac{dx}{dt} = \mu_{m} \left[ {1 - \frac{x}{{x_{m} }}} \right]x$$ Equation (1) $$x = \frac{{x_{m} x_{0} e^{{\mu_{m} t}} }}{{x_{m} - x_{0} + x_{0} e^{{\mu_{m} t}} }}$$ Equation (2) $$p = \alpha \frac{{x_{m} x_{0} e^{{\mu_{m} t}} }}{{x_{m} - x_{0} + x_{0} e^{{\mu_{m} t}} }} + \beta \frac{{x_{0} }}{{x_{m} }}\ln \left( {1 - \left( {\frac{{x_{0} }}{{x_{m} }}} \right)} \right)\left( {1 - e^{{\mu_{m} t}} } \right)$$ Equation (3) $$p = \alpha A(t) + \beta B(t)$$ Equation (4) $$s = \gamma \frac{{x_{m} x_{0} e^{{\mu_{m} t}} }}{{x_{m} - x_{0} + x_{m} x_{0} e^{{\mu_{m} t}} }} - \eta \frac{{x_{0} }}{{\mu_{0} }}\ln \left( {1 - \left( {\frac{{x_{0} }}{{x_{m} }}} \right)} \right)\left( {1 - e^{{\mu_{m} t}} } \right)$$ Equation (5) $$s = \gamma A(t) - \eta B(t)$$ Equation (6) $$A(t) = \frac{{x_{m} x_{0} e^{\mu }_{m} t}}{{x_{m} - x_{0} + x_{m} x_{0} e^{\mu }_{m} t}}$$ Equation (7) $$B(t) = \frac{{x_{0} }}{{\mu_{0} }}\frac{{x_{0} }}{{x_{m} }}\ln \left( {1 - \left( {\frac{{x_{0} }}{{x_{m} }}} \right)} \right)\left( {1 - e^{{\mu_{m} t}} } \right)$$ Equation (8) $$q_{{s_{1} }} = \frac{1}{{Y{x \mathord{\left/ {\vphantom {x s}} \right. \kern-0pt} s}}}U_{s} + M$$ Equation (9) $$q_{{s_{2} }} = \frac{1}{{Y{x \mathord{\left/ {\vphantom {x s}} \right. \kern-0pt} s}}}U_{s}$$ Equation (10) $$q_{{s_{3} }} = \frac{1}{{Y{x \mathord{\left/ {\vphantom {x s}} \right. \kern-0pt} s}}}U_{s}$$ Equation (11) $$q_{{p_{{}} }} = \frac{1}{{Y{p \mathord{\left/ {\vphantom {p s}} \right. \kern-0pt} s}}}U_{s}$$ Equation (12)