It's widely-believed that a "replacement-rate" of population (of 2 children per couple, i.e. 1 child per person) results in a stable total population. However, this isn't true in a population whose life expectancy is increasing, because each successive generation will - after having children - be around for longer than their predecessors.
This is a JavaScript simulation to demonstrate this phenomenon.
Open index.html in a web browser of your choice. A random population of 100 individuals, with ages spread across a skewed Gaussian distribution, will be generated (you can press Reset to get a new one at any time). Press Play/pause to begin the simulation.
The population will age. Each person will spawn exactly one child when the parent reaches a randomly-selected age (with a probability represented as a bell curve between the ages of 20 and 30). Each person will die exactly when they exceed the life expectancy, which begins at 70.
Let the simulation run for a while: you can change the speed using the Faster and Slower buttons. To begin with, you will see generational "waves", but random factors will flatten this out over time. The "Highest population" and "Lowest population" will stabilise.
Now, try hitting the Increase life expectancy button a few times so that people live longer. You'll probably see the population increase, pushing up the "Highest population" to a never-before-seen level (if you don't, it's probably a rare random quirk of your population: reset and try again!). The population has increased despite the birth rate still being at "replacement" levels, on account of increasing life expectancy.