Question

# The following question consider the Gompertz equation P(t)'=\alpha \ln (\frac{K}{P(t)})P(t)?, a modification for logistic growth, which is often used for modeling cancer growth, specifically the number of tumor cells.

Modeling
The following question consider the Gompertz equation, a modification for logistic growth, which is often used for modeling cancer growth, specifically the number of tumor cells. When does population increase the fastest for the Gompertz equation
$$\displaystyle{P}{\left({t}\right)}'=\alpha{\ln{{\left({\frac{{{K}}}{{{P}{\left({t}\right)}}}}\right)}}}{P}{\left({t}\right)}?$$

2021-08-01
The maximum population can be found by solving $$\displaystyle{P}'={0}$$ for P while the fastest growth can be reached by equating the differentiation of the population rate by zero, then solving
(i.e) by solving $$\displaystyle{P}\text{}{0}$$ for P as follows
$$\displaystyle{P}\text{}{\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left(\alpha{\ln{{\left({\frac{{{k}}}{{{P}}}}\right)}}}{P}\right)}={0}$$
$$\displaystyle\Rightarrow{\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left({\ln{{\left({\frac{{{k}}}{{{P}}}}\right)}}}{P}\right)}={0}$$
$$\displaystyle\Rightarrow{\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left({P}{\ln{{k}}}-{P}{\ln{{P}}}\right)}={0}$$
$$\displaystyle\Rightarrow{P}'{\ln{{k}}}-{P}'{\ln{{P}}}-\neg{\left\lbrace{P}\right\rbrace}{\frac{{{P}'}}{{\neg{\left\lbrace{P}\right\rbrace}}}}={0}$$ (product rule and chain rule)
$$\displaystyle\Rightarrow{P}'{\ln{{k}}}-{P}'{\ln{{P}}}-{P}'={0}$$
$$\displaystyle\Rightarrow{\ln{{k}}}-{\ln{{P}}}-{1}={0}$$ (dividing by P')
$$\displaystyle\Rightarrow{\ln{{P}}}={\ln{{k}}}-{1}$$
$$\displaystyle\Rightarrow{\ln{{P}}}={\ln{{k}}}-{\ln{{e}}}$$ (logatithmic properties)
$$\displaystyle\Rightarrow{\ln{{P}}}={\ln{{\left({\frac{{{k}}}{{{e}}}}\right)}}}$$ (logatithmic properties)
$$\displaystyle\Rightarrow{P}={\frac{{{k}}}{{{e}}}}$$