To model population growth, we can use an exponential function represented by the formula:

$P(t) = P_0 e^{rt}$

In this equation, $P(t)$ denotes the population at time $t$, while $P_0$ is the initial population size. The constant $e$ represents the base of the natural logarithm, approximately equal to $2.71828$. The variable $r$ indicates the growth rate, and $t$ signifies the time period over which the population is growing. This formula assumes that the population grows continuously and in proportion to its current size.

To illustrate this, let’s consider an initial population $P_0$ of $1000$ individuals, with a growth rate $r$ of $5\%$ per year. To convert this percentage into a decimal for use in the formula, we divide by $100$, resulting in $r = 0.05$. If we want to calculate the population after $3$ years, we can substitute these values into the formula:

$P(3) = 1000 \times e^{0.05 \times 3}$

Next, we can compute $e^{0.15}$ using a calculator, which yields approximately $1.1618$. Thus, we can further calculate the population:

$P(3) = 1000 \times 1.1618 = 1161.8$

This indicates that the population would be approximately $1162$ individuals after $3$ years.

Exponential growth is characterized by the principle that the rate of growth is proportional to the current population size. As a result, this leads to a rapid increase over time, which is why we refer to it as “exponential” growth. Understanding this concept is essential for analyzing real-world scenarios, such as population dynamics, where limitations in resources and space can have significant impacts.