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

The intrinsic rate of increase (r), can be calculated by the following exponential growth equation:

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)

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).