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 in the threshold logistic equation P′(t)=rP(1−PK)(1−TP)?

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Usamah Prosser · Accepted Answer

The maximum population can be found by solving 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 P0 for P as followsPddt(rP(1−Pk)(1−TP))=0⇒ddt(P(1−Pk)(1−TP))=0⇒(P(1−Pk)(1−TP))ddt(ln|P(1−Pk)(1−TP)|)=0 (logarithmic diff.)⇒(P(1−Pk)(1−TP))ddt(ln⁡P+ln⁡(1−Pk)+ln⁡(1−TP))=0⇒(P(1−Pk)(1−TP))[P′P−P′k1−Pk+TP′P21−TP]=0 (chain rule)⇒(P(1−Pk)(1−TP))[1P−1k1−Pk+TP21−TP]=0 (dividing by P&#039;)⇒(1−Pk)(1−TP)−Pk(1−TP)+TP2P(1−Pk)=0

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.

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