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.]
You'd better not leave it in his hands. He's already taken the hundreds of millions from the CFE lawsuit and placed our students in conditions like I described.
In case the mayor is reading, I'll spell it out. Reduce the population, place kids, rather than charters, in underutilized buildings, and reduce class sizes as you promised.
Reduce them so our kids can study under the same conditions you made sure your own had.
Thanks for the plug, POd!
News flash, Arthur: Charter schools house kids.
Ask the mayor to find some more millions to build more seats, because that's what we need in this city - not fewer charter schools.
If the charters disappeared the kids they serve would just go back to the district schools anyway, saving about 1 or 2 rooms per building for admin space. That isn't going to help the conditions you describe.
And let's face it, many charter schools wouldn't be open if the mayor wasn't giving them public school space to educate their public school children.
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