Logistic Growth: Fastest Growth and the Inflection Point

Logistic growth follows dP/dt = kP(1-P/K), where K is the carrying capacity. This rate is a function of P, and it's not constant: it's small when P is small, grows as P grows, then shrinks back down as P approaches K. That's because two forces fight each other. The factor P pushes the rate up as the population grows. The factor (1-P/K) pushes the rate down as the population nears capacity, since resources get scarce. Somewhere in between, the product is as large as it can get.

To find that peak, treat dP/dt = kP - (k/K)P² as a function of P and differentiate with respect to P. Setting d/dP(dP/dt) = k - (2k/K)P equal to zero gives P = K/2. That's the population size where the growth rate itself is maximized. In terms of the P-vs-t graph, this is the inflection point: the S-curve is concave up while P < K/2 (growth is accelerating) and concave down once P > K/2 (growth is decelerating), exactly matching concavity ideas from Unit 5, where sign changes in the second derivative mark inflection points.

This connects logistic growth back to everything learned about using derivatives to analyze graphs. The population curve itself never has a max in the middle - it keeps climbing toward K - but its slope does. Recognizing that 'fastest growth' means 'maximum dP/dt', not 'maximum P', is what ties this topic to the broader toolkit of optimizing a function by studying its derivative.