If you're feeling a little blue about derivatives, infinite series, and things that rhyme with shmalculus, check out the Shmoop guide to AP Calculus BC for all your calculating needs.
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Exponential growth is a nice model to work with mathematically (well, nice-ish), but its big flaw is that it's not completely realistic. If a population of bacteria kept growing exponentially, the world would be overrun with bacteria pretty quickly. Nobody wants that.
There's a model that accounts for the fact that resources and space are not infinite and that a population has a "carrying capacity" that it can't exceed in the long run. The model is called logistic growth, and it takes the form of a differential equation. In words, the logistic growth formula says, "The rate of change of a population is jointly proportional to the size of the population and the difference between the population and the carrying capacity."
Uh, let's translate that into math. The differential equation for logistic growth is this guy right here.
y '(t) = ky(a – y)
In this equation, k and a are constants, and a is the carrying capacity. The equation might not always appear in this form, so to find the carrying capacity, set the equation equal to 0 and solve for y. One solution is y = 0, and the other solution is y = carrying capacity.
The carrying capacity is a big deal for a couple of reasons. If the initial value y(0) > a, the population will decrease over time until it levels out to y = a, and if y(0) < a, then it will increase until it reaches y = a. The greatest rate of change of the population occurs when the population is at exactly half the value of the carrying capacity.
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