Chapter 4.4, Problem 81E

Some populations initally grow exponentially but eventually level off. Equations of the form

$P(t)=\frac{M}{1+A{e}^{-kt}}$

Where M, A, and k are positive constants, are called logistic equations and are often used to model such populations. (We will investigate these in detail in Chapter 9.) Here M is called the carrying capacity and represents the maximum population size that can be supported, and $A=\frac{M-P_0}{P_0}$, where P₀ is the initial population.

(a) Compute ${\lim }_{t\to \infty }P(t)$. Explain why your answer is to be expected.

(b) Compute ${\lim }_{M\to \infty }P(t)$. (Note that A is defined in terms of M.) What kind of function is your result?