Question

If a population growing exponentially double in size in 3 years, what is the intrinsic rate of increase (r) of the population?

Answer 1

A population grows exponentially if sufficient food resources are available to the individual. The following integral form of the exponential growth equation can calculate the exponential growth:

N_t = N₀e^rt

Where,

N_t = Population density after time (t)

N₀ = Population density at time zero

r = Intrinsic rate of natural increase

e = Base of natural logarithms (2.71828)

From the above equation, we can calculate a population’s intrinsic rate of increase (r).

Now, as per the question,

Present population density = x

Then, population density after two years = 2x

t = 3 years

Substituting these values in the formula, we get:

⇒ 2x = xe^3r

Dividing both sides by x,

⇒ 2 = e^3r

Applying log on both sides:

⇒ log 2 = 3r log e

⇒ `log 2/(3 log e) = r`

⇒ `log 2/(3 xx 0.434) = r`

⇒ `0.301/1.302 = r`

⇒ 0.2311 = r

Explanation: The intrinsic rate of increase (r) is a population’s maximum potential per capita growth rate under ideal conditions. When a population doubles exponentially in size over a time period (t), (r) can be found by the time it takes for the population to double using the above formula.

Hence, the population’s intrinsic rate of increase (r) is approximately 0.231 (per year).