Exponential Growth:

Integrating the population growth formula.

The exponential growth of a population is a fundamental concept in biology and ecology. It describes a situation where the rate of growth is directly proportional to the size of the population. This idea is first expressed using a **differential equation**. Don't worry if that sounds complicated—it simply means a formula that describes a rate of change.

Step 1: The Differential Equation

The starting point is the differential equation that models the rate of change of population ($N$) over time ($t$).

$$ \frac{dN}{dt} = rN $$

Here, $N$ is the population size, $t$ is time, and $r$ is the constant growth rate. The term $\frac{dN}{dt}$ means "the rate of change of population with respect to time."

Step 2: Separation of Variables

To solve this equation, we need to get all the terms involving $N$ on one side and all the terms involving $t$ on the other. This is called **separation of variables**.

$$ \frac{dN}{N} = r dt $$

Step 3: Integration

Now, we integrate both sides of the equation. This is the process of finding the original function from its rate of change.

$$ \int \frac{1}{N} dN = \int r dt $$

The integral of $\frac{1}{N}$ is $\ln(N)$ (the natural logarithm of N), and the integral of a constant $r$ is simply $rt$. We also add a constant of integration, $C$, because the derivative of any constant is zero. Since we're undoing a derivative, we have to account for any constant that might have been there originally.

$$ \ln(N) = rt + C $$

Step 4: Solving for N with the Exponential Function

To get $N$ by itself, we need to undo the natural logarithm ($\ln$). The inverse operation of a natural logarithm is the exponential function, $e^x$. When these two operations are combined, they cancel each other out because the natural logarithm asks, "What power do I need to raise $e$ to in order to get this number?". So, $e^{\ln(N)}$ just gives you $N$ back.

$$ e^{\ln(N)} = N $$

So, we raise both sides of our equation to the power of $e$ to isolate $N$.

$$ N = e^{rt + C} $$

Using the rules of exponents ($e^{a+b} = e^a \cdot e^b$), we can separate the terms:

$$ N = e^{rt} \cdot e^C $$

Step 5: Defining the Initial Population ($N_0$)

The term $e^C$ is just a constant number. We can define this constant by considering the population at time $t=0$. The initial population is represented as $N_0$.

Let's substitute $t=0$ into our separated equation:

$$ N_0 = e^{r(0)} \cdot e^C $$

Since any number raised to the power of 0 is 1 ($e^0 = 1$), the equation simplifies to:

$$ N_0 = 1 \cdot e^C $$ $$ N_0 = e^C $$

By substituting this back into our equation, we arrive at the final, well-known formula for exponential growth.

The Final Formula

$$ N_t = N_0 e^{rt} $$

Where $N_t$ is the population at time $t$, $N_0$ is the initial population, $r$ is the growth rate, and $e$ is Euler's number (approximately 2.718).