Chapter D1.1, Problem 53E

Logistic population growth Widely used models for population growth involve the logistic equation $P^{\prime }\left(t\right)=rP\left(1-\frac{P}{K}\right)$, where P(t) is the population, for t ≥ 0, and r > 0 and K > 0 are given constants.

a.    Verify by substitution that the general solution of the equation is $P\left(t\right)=\frac{K}{1+C{e}^{-rt}},$ where C is an arbitrary constant.

b.    Find that value of C that corresponds to the initial condition P(0) = 50.

c.    Graph the solution for P(0) = 50, r = 0.1, and K = 300.

d.    Find $\underset{t\to \infty }{\lim }P\left(t\right)$ and check that the result is consistent with the graph in part (c).