Biology Application - Population Growth Curves

Population Growth Curves

In the wild, the population of a certain group varies as time passes. We denote a function $P$ to represent the population of a group at time $t$, and we denote the rate of growth to be $\frac{d}{dx} P(x) = P'(x)$.

For example, consider a group of rabbits whose population is estimated by the growth curve $P(x) = - \frac{10000}{x + 20} + 500$ as graphed below.

Now suppose that we want to find how fast the population is growth at time $x = 100$. In order to figure this out, we could use calculus to differentiate $P$, that is:

(1)

\begin{align} P(x) = - \frac{10000}{x + 20} + 500 \\ P(x) = -10000 \cdot (x + 20)^{-1} + 500 \\ P'(x) = 10000 \cdot (x + 20)^{-2}\cdot 1 \\ P'(x) = \frac{10000}{(x + 20)^2} \end{align}

Therefore, at $x = 100$, $P'(100) = \frac{10000}{120^2} = 0.6944....$, which is the rate of change of the population.