#### Question

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

#### Solution

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*_{o} *e*^{rt}

Where,

*N*_{t}= Population density after time *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:

⇒ 2*x* = *x* *e*^{3}^{r}

⇒ 2 = *e*^{3}^{r}

Applying log on both sides:

⇒ log 2 = 3r log *e*

*⇒ `log2/(3loge)=r`*

*⇒ `log2/(3xx0.434)= r`*

*⇒ `0.301/1.302 = r`*

*⇒ 0.2311 = r*

Hence, the intrinsic rate of increase for the above illustrated population is 0.2311