# Value of r (intrinsic rate of natural increase)

What is the current value of 'r' (instrinsic rate of natural increase) in India? How do we calculate it?

My book says that In 1981, the r value for human population in India was 0.0205. But this link (https://www.google.co.in/amp/knoema.com/atlas/India/topics/Demographics/Population/Rate-of-natural-increase%3fmode=amp) says it was around 22.50 in the year 1981. Such big discrepancies are a query.

• Perhaps you should check the census report. You should add details to your question and be clear about what you are asking. Edit the question to add all necessary details. Comments should be used for asking clarification. Jan 2, 2018 at 13:30
• About the value of 22.5 (in your question); that is referring to a growth of 22.5 persons per one thousand population. So a growth of 2,3% (22.5/1000= 0.0225). Mar 1, 2018 at 10:09

NCERT shows per capita value is divided by 1000, not hundred (%). So the data you see is 22.50 per thousand! So, the per capita rate is 22.50/1000 = 0.02250 that is almost equal to 0.0205 given in book!

To make you understand better, remember that GDP and per capita GDP is different, the same way birth rate or death rate is calculated per 1000 but when we ask per capita we have to actually divide it by 1000.

Now in 2016 , it is 0.01019 i.e. 0.0102

I hope you understand!!

From the UN database (this info is also reported on wikipedia), the growth rate in India in 2016 was $r=1.019$.

It is computed as $r = \frac{N_{2017}}{N_{2016}}$, where $N_{y}$ is the population size at year $y$. This estimate is based on the total number of residents (regardless of their legal status).

More explanations

$r$ is simply defined as $r = \frac{N_{2017}}{N_{2016}}$. Often, you will read statistics in the sense of percentage of increase. Let's call this value $x$. $x$ is then defined as $1+x = r$.

It is just like when you put money in a bank account. If your interest rate is $x = 2\% = 0.02$, then if you had $N_{2010} = 1000$ dollars in 2010, in 2011, you'll have $$N_{2011} = 1000 + 1000 * 0.02 = (1+0.02) * 1000$$. Note that $1+0.02 = 1 + x = r$. Over $k$ years, you'll have $$N_{2010+k} = N_{2010} (1+0.02)^k$$.

• My book says that r is calculated as birth rate minus death rate, and calculating it through the values mentioned on the internet, it shows big discrepancies! Jan 2, 2018 at 17:48
• en.m.wikipedia.org/wiki/Demographics_of_India Considering Wikipedia, the birth rate minus death rates comes out to be 11. Jan 2, 2018 at 18:10
• Per 1000 individuals, yes. Divide 11 by 1000 and you get $0.011$, which means $N_{2016} = N_{2017} + N_{2017} * 0.011$, which can be rewritten $N_{2016} = (1 + 0.011) \cdot N_{2017}$, hence $r = 1.011$, which is pretty close to $r=1.019$. Does it make sense to you? Jan 2, 2018 at 18:29
• Do we always take the original population size as 1 for a year, while calculating this rate? Jan 3, 2018 at 11:21
• I don't really understand this follow-up question. But please see edit. Jan 3, 2018 at 14:06

It the most basic form, the intrinsic rate of increase, $r$, is defined as:

$\frac{dN}{dt} = rN = (a-b)N$

where $a$ is the birth rate per unit time and $b$ is the death rate per unit time. So $r$ is the birth rate minus the death rate. The modell can naturally also be extended to e.g. include carrying capacity or to make $r$ (or underlying parameters) functions of time or environmental variables.

However, from census data (taken at fixed points in time), you can calculate lambda ($\lambda$), the discrete-time per capita growth rate (also called the finite rate of increase), as:

$\frac{N_{t+1}}{N_t} = \lambda$

This shows the growth across a fixed, discrete time intervall. $\lambda$ can be converted to an instantaneous instrinsic rate of increase as $r = log(\lambda)$. Using the values in Remi.b's answer (from https://esa.un.org/unpd/wpp/DVD/), this yields and $r$ of 0.0188 (i.e. log(1.019)). For a bit more background see https://en.wikipedia.org/wiki/Population_dynamics and http://www.zo.utexas.edu/courses/thoc/PopGrowth.html (or any textbook on population dynamics).

Population growth curved à when responses are not limitting the growth , plot is exponential , b when response are limitting the growth , plot is logistic , k carrying capacity