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.

Daniaal Sanchez

Daniaal Sanchez

Answered question

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
P(t)=αln(KP(t))P(t)?

Answer & Explanation

StrycharzT

StrycharzT

Skilled2021-08-01Added 102 answers

You may determine the largest population by solving P=0 for P While the population rate differentiation can be equaled to zero to achieve the fastest growth, then solving
(i.e) by solving P0 for P as follows
Pd dt (αln(kP)P)=0
d dt (ln(kP)P)=0
d dt (PlnkPlnP)=0
PlnkPlnP¬{P}P¬{P}=0 (product rule and chain rule)
PlnkPlnPP=0
lnklnP1=0 (dividing by P')
lnP=lnk1
lnP=lnklne (logatithmic properties)
lnP=ln(ke) (logatithmic properties)
P=ke

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