If P(t) satisfies the logistic equation, use the chain rule to show thatConclude that Pn &...

If P(t) satisfies the logistic equation, use the chain rule to show that

Conclude that Pn > 0 if 0 < P < 1/2M;Pn = 0 P = 1/2M; Pn < 0 if 1/2M < P < M; and Pn > 0 if P > M. In particular, it follows that any solution curve that crosses the line P = 1/2M has an inflection point where it crosses that line, and therefore resembles one of the lower S-shaped curves in Fig.