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 amounts of food resources are available to the individual. Its exponential growth can be calculated by the following integral form of the exponential growth equation:
N = N_o ^rt
Where,
N_t  Population density after lime t 

N_o 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 the intrinsic rate of increase (r) of a population. 
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 = x e^3r
⇒2 = e^3r
Applying log on both sides:
⇒ log2 = 3r loge
\(\frac{log 2}{3loge} = r\)
\(\frac{log 2}{3x0.434} = r\)
\(\frac{0. 301}{3x0.434} =r\)
\(\frac{0.301}{1.302}=r\)
\(⇒ 0. 2311=r\)
Hence the intrinsic rate of increase for the above illustrated population is 0. 2311.