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
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asked 2021-07-31
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
\(\displaystyle{P}{\left({t}\right)}'=\alpha{\ln{{\left({\frac{{{K}}}{{{P}{\left({t}\right)}}}}\right)}}}{P}{\left({t}\right)}?\)

Expert Answers (1)

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)
\(\displaystyle\Rightarrow{P}={\frac{{{k}}}{{{e}}}}\)
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