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^(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:

`rArr 2x= x e^(3r)`

`rArr 2 = e^(3r)`

Applying log on both sides:

`rArr log 2 = 3r log e`

`rArr (log2)/(3loge) = r`

`rArr (log2)/(3xx 0.434) = r`

`rArr(0.301)/(3xx0.434)=r`

`rArr (0.301)/(1.302) =r`

`rArr 0.2311 = r`

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