In an epidemic, whether of the flu, AIDS, or a zombie apocalype, people become sick from contact between sick and healthy people. This means that $\frac{dy}{dt}$ is proportional to the product of the number of sick and healthy people. The standard form of the equation, which is called logistic, is

$$\frac{dy}{dt} = k y \left ( 1 - \frac{y}{M} \right )$$

where $M$ is the total number of people who could get sick. Being a separable equation, we can solve it by separation of variables. This gives us the integral $$\int \frac{M\,dy}{y(M-y)},$$ which can then be solved by partial fractions. The end result is an S-shaped curve that starts by growing exponentially and then levels off, approaching $M$ exponentially.

The same equation can be used to model the dynamics of a population with limited resources. Here $M$ represents the "carrying capacity" of the ecosystem, that is the maximum population that can live sustainably.