# 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 $\frac{d}{dx} P(t) = P'(t)$.

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

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

(1)Therefore, at $t = 100$, $P'(100) = \frac{10000}{120} = 83.33....$, which is the rate of change of the population.

Furthermore, we can also look at some important details regarding the first derivative of $P$, that is:

- If $P(t) > 0$ for any $t$ of interest, then the population is growing.

- If $P(t) < 0$ for any $t$ of interest, then the population is decaying.

- If $\Delta P \to 0$ (the change of the population) and $P(t) > 0$, then the population is plateauing in growth.

- If $\Delta P \to 0$ and $P(t) < 0$, then the population is plateauing in decay.

We can see from our example above that $\Delta P \to 0$ as $t \to \infty$, implying that our population is plateauing. Of course, there are many biological reasons to why a population plateaus such as reaching a sustainable carrying capacity, but we will not dive into these topics.