If a population growing exponentially double in size in 3 years, what is the intrinsic rate of increase (r) of the population?
If a population growing exponentially double in size in 3 years, what is the intrinsic rate of increase (r) of the population?
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:
$N_{t}=N_{0} e^{r t}$
Where,
Nt= Population density after time t
NO= 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:
$\Rightarrow 2 x=x e^{3 r}$
$\Rightarrow 2=e^{3 r}$
Applying log on both sides:
$\Rightarrow \log 2=3 r \log e$
$\Rightarrow \frac{\log 2}{3 \log e}=r$
$\Rightarrow \frac{\log 2}{3 \times 0.434}=r$
$\Rightarrow \frac{0.301}{3 \times 0.434}=r$
$\Rightarrow \frac{0.301}{1.302}=r$
$\Rightarrow 0.2311=r$
Hence, the intrinsic rate of increase for the above illustrated population is 0.2311.