Twice in the last couple of months I have been faced with the claim that large population growth can continue to occur even though the birth rate remains at 2.0 or below. The claim is on its face not possible (with some notable caveats I mention below) 
Let me explain what the chart below cannot do. It cannot predict what will happen in the real world, but neither can my friends. It is a closed chart, meaning it requires no fudging of the numbers. The reason for this is simple. I am dealing with hypothetical numbers and rates, not real people. This is a math problem, not a people problem.
The specifics of the chart.
First, in order to make things simple, I will do this based upon the idea that the number of children needed to replace two parents is TWO(!). The reason that 2.1 or 2.2 is often used is because the figure is trying to represent the number of children needed, who survive into adulthood. This itself is an unnecessary and arbitrary fudge because the average I give already takes into consideration children who have no children, regardless of the reason (they die in childhood, they are infertile or they remain virgins, etc.). In other words, to present the information in an understandable format, it is necessary that each family have two and only two children. In the real world, one family could have 8 children, and three other couples could have none and the results would be the same, but using this in the chart would be unfollowable(!) and as big a distraction as this long introduction.
Second, even though the chart mentions father and mother, I recognize we are really only dealing with the number of children particular mothers have. This is necessary for the chart to have any meaning whatsoever. In the real world, mothers may have children by more than one man for various reasons, but mathematically it only confuses things.
Third, the chart only deals with population growth in a closed setting. That is, one world, one nation, or one town. Immigration does not exist. Of course, a nation with a low birth rate could still grow, if you allow for immigration.
Fourth, it is necessary to stretch out the chart to at least as long as the length of life of the first generation. The chart assumes no change in death rate, for this reason: Even if people lived to be an average of 120 years old, after six generations, the population growth would cease, if each couple only had two children.
To make it simple, each couple has a pair of twins at age 20, each of their children have twins at age 20, and so on, until the death of the original progenitors, all which die at age 81. We start the chart with 8 newlywed couples, all aged 19 in the year 2999.
World Population: 16 in the year 2999.
8 Sets of Twins Born in 3000, World Population: 32.
8 Sets of Twins Born in 3020, World population: 48
8 Sets of Twins Born in 3040, World Population: 64
8 sets of twins born 3060, World population: 80
First set of parents die in 3061 at age 81, world population 64.
8 sets of twins born in 3080, world population: 80
Second set of parents die in 3081, world population 64.
From the fourth generation on, the population will be only 80 and no higher. From this chart it is easy to see that with a birthrate of 2 children per household, the population can only grow if the death rate decreases. Even then, it won’t be by much. If people lived to be a 100, in 3100 A.D. the population would top out at 112 people. Suppose each person lived to be 200. It would only reach 160 TOTAL and then it would stop. Suppose everybody lived to be 1000 years old.
Let’s revisit our original chart with nobody dying for a 1000 years. Wouldn’t the world population explode? No.
Year and Population
3000, 32.
3020, 48
3040, 64
3060, 80
3080, 96
3100= 112
3200=192.
3300=272
3400=352
3500=432
3600=512
3700=592
3800-672
3900=752
4000 A.D. Population 832.
In other words, only 80 people are added every 100 years. After 1000 years—and nobody has ever lived that long—there would be only 800 people added, making a grand total of 832! That is the same population as that booming metropolis, Alma, Kansas, where hunger appears to be running rampant.