Exponential Growth

Exponential growth refers to a pattern of growth where the population size increases at a constant rate over a specific period of time. In this scenario, the population size is doubling every 3 years. To determine the intrinsic rate of increase (r) of the population, we can use the exponential growth formula:

N(t) = N(0) * e^(rt)

Where:
N(t) = population size at time t
N(0) = initial population size
e = base of the natural logarithm (approximately 2.71828)
r = intrinsic rate of increase
t = time

Population Doubling Time

Since the population is doubling every 3 years, we can use this information to find the value of r. Let's assume the initial population size is N(0) = 1.

After 3 years, the population size becomes N(3) = 2 * N(0) = 2.
After 6 years, the population size becomes N(6) = 2 * N(3) = 4.
After 9 years, the population size becomes N(9) = 2 * N(6) = 8.

Calculating the Intrinsic Rate of Increase (r)

To find the intrinsic rate of increase (r), we can rearrange the exponential growth formula:

N(t) = N(0) * e^(rt)

Taking the natural logarithm of both sides:

ln(N(t)) = ln(N(0)) + rt * ln(e)

Since N(t) = 2 * N(0) and t = 3, we can substitute these values into the equation:

ln(2 * N(0)) = ln(N(0)) + 3r * ln(e)

Simplifying the equation:

ln(2) + ln(N(0)) = ln(N(0)) + 3r

The ln(N(0)) terms cancel out, and we are left with:

ln(2) = 3r

Solving for r:

r = ln(2) / 3

Using a calculator, we can find the value of r to be approximately 0.231.

Therefore, the intrinsic rate of increase (r) of the population is approximately 0.231, indicating that the population is growing at a rate of 23.1% per year.