Exponential growth

For unrestrained population growth, the rate of change of population is directly proportional to the population.So that means dP/dT = kP. So this year we're going to be adding in growth restraints because infinite growth is inappropriate.

Logistics

One assumption for restrictive growth is that there is a certain maximum size, m, for the population. (For example, in ours, there was a maximum number of people that could contract the disease (117).) And the rate goes to zero (the slope approaches zero). So we started at zero and we ended at zero, so there were two different values that made our derivative zero, there were those two different values. That's what we are going to end up seeing. The rate goes to zero as the population approaches that size m. The rate of change of population is jointly proportional (we are going to have an = k(one)(two)) to the population and to the difference between the maximum and current populations.

dP/dT = kP(m-P)

So for any logistic graph, there's going to be a high horizontal asymptote at whatever x=m value is.

Let's solve for P.

Wikipedia - exponential growth

Meadows, Donella H., Dennis L. Meadows, Jørgen Randers, and William W. Behrens III. (1972) The Limits to Growth. New York: University Books. ISBN 0-87663-165-0
--- Good (dead) author, appears in a few other places.
Exponential and logistic growths
A neat visualization of exponential growth
Compound interest and exponential growth
Kurzweil's law of accelerating returns
Human population growth (Kimball's online biology book)