Some populations initally grow exponentially but eventually level off. Equations of the form where M, A, and

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

P(t) 1 + Ae -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 = M - P0/P0, where P0 is the initial population.

(a) Compute lim t →∞ P(t). Explain why your answer is to be expected.
(b) Compute lim M →∞ P(t). (Note that A is defined in terms of M.) What kind of function is your result?

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