For many events in the world around you, growth will start out fast, but then flatten out. Logistic functions describe events like this. Logistic growth is often used to model animal populations, the growth of a tree, or the cost of a business. When you come across cases where the growth starts out exponential for a while but then flattens out, you will be happy to have access to a logistic model.

Theory

The logistic curve is the well-known “S-curve”. When $t\to \infty $, $N(t)$ goes towards $C$. The $N(t)$ function increases the fastest when $N(t)=\frac{C}{2}$.

This is what the graph of a logistic model looks like:

Example 1

**In 1940, 45 toads were introduced to a Pacific island. The population developed according to the logistic model **

$$B(x)=\frac{\text{}4\phantom{\rule{0.17em}{0ex}}500\phantom{\rule{0.17em}{0ex}}000\text{}}{1+\text{}112\phantom{\rule{0.17em}{0ex}}400\text{}\cdot {e}^{\text{}-1.14\text{}x}},$$ |

where $x$ is the number of years after 1940. According to the model, what value will the population stabilize at?

The graph of the logistic function will stabilize at $y=C$. That means the population will stabilize at $4\phantom{\rule{0.17em}{0ex}}500\phantom{\rule{0.17em}{0ex}}000$ toads.

When did the population grow the fastest?

The growth rate of $B(x)$ is at its greatest when ${B}^{\u2033}(x)=0$. This is most easily determined with a digital tool. You will get

$$x=10.2$$ |

The toad population grew the most $10.2$ years after $1940$, which was in $1950$.

There’s another way to find out when the growth rate was at its greatest as well. $B(x)$ grows the fastest when

$$y=\frac{C}{2}=\frac{4\phantom{\rule{0.17em}{0ex}}500\phantom{\rule{0.17em}{0ex}}000}{2}=2\phantom{\rule{0.17em}{0ex}}250\phantom{\rule{0.17em}{0ex}}000.$$ |

You set your function equal to this value and solve for $x$:

$$\frac{4\phantom{\rule{0.17em}{0ex}}500\phantom{\rule{0.17em}{0ex}}000}{1+112\phantom{\rule{0.17em}{0ex}}500\cdot {e}^{-1.14x}}=2\phantom{\rule{0.17em}{0ex}}250\phantom{\rule{0.17em}{0ex}}000$$ |

Insert this equation into a digital tool. That will give you

$$x=10.2$$ |

which is the same answer you got from the other method.