For any particular species in a given environment, the carrying capacity is the maximum sustainable population. That is, it is the largest population the environment can support for extended periods of time.
A logistic model assumes that the growth rate automatically adjusts as the population approaches the carrying capacity. However, because of the astonishing rate of exponential growth, the real population often increases beyond the carrying capacity in a relatively short time. This is called overshoot.
When a population overshoots the carrying capacity of its environment, a decrease in the population is inevitable. If the overshoot is substantial, the decrease can be rapid and severe--a phenomenon known as collapse.
(source: Using and Understanding Mathematics by Jeffrey Bennett and William Biggs)
Logistic problems are quite complex to solve.
Here is a simpler problem (not a logistic growth) that deals with exponential growth. It was inspired by Arthur Goldstein's article, over at Gotham Schools, and I am dedicating it to Mayor Bloomberg.
1. Let P(t) represent the number of students attending Packemin HS at a time t years, when t is great than or equal to 0. The population, P(t) is increasing at a rate directly proportional to 800 - P(t), where the constant of proportionality is k. We know that initially the population is 500 and two years later, the population is 700.
a) Find a function to represent P(t), the population of Packemin HS at any time t.
[solution: P(t) = 800 - 800e^(-.5t ln 3) ]
b) What is the maximum capacity of the school as time approaches infinity?
[solution: 800 students]
c) What will happen to the school if it is possible for the DOE to keep adding students beyond the limiting capacity?
[solution: I'll leave this one to the mayor to figure out.]