The logistic equation is a simple model of population growth:

${d x \over d t} = r x\left(1-\frac{x}{K}\right)$

Here $x$ is the population, a function of time $t$. $K$ is the equilibrium population, and $r$ is the growth rate.

In the limit $K \to \infty$, we get an even simpler model:

${d x \over d t} = r x$

which describes exponential population growth:

$x(t) = x_0 e^{r t}$

For finite positive values of the equilibrium population $K$, the logistic equation describes population growth that is approximately exponential when the population is much less than $K$, but levels off as the population approaches $K$. If the population is larger than $K$, it will decrease. Every solution has