Why does reproduction fall off around ten years in?

I've played twice now about thirteen years of the game. It feels like my people are doing OK -- I can keep them at 4+ stars and 4+ hearts and they seem to start the game pregnant, so babies come up at first. But then, after a while; certainly by year 10, no more babies are born and the villagers have started dying of old age and my population crashes down below 18 or so which feels like a tipping-point after which I can't generate all the stuff I need to -- not enough people to both mine iron and make tools, and I have to start giving up nutritional variety, etc.

And it feels like there aren't really that many more knobs to twiddle. I'm pretty patient with games, trying again and again with different strategies; I just don't see what I might do differently.

So, are there factors that go into reproduction that I'm missing?

Villagers will only marry and produce children if there is an empty home for them to move into. That is, there can only be one pair of villagers producing children per house. As homes start to become available as your population dies of you will begin to see a growth in your population, but it'll eventually stop growing again since the villagers will just run out of empty homes again. If you want your population to continue to grow you'll need to continue to build new homes for villagers to pair off into and begin families. Be careful, however, since building too many homes can lead to your population growing faster than you can produce food and other resources to support it.

Well the game says that once you have a town hall, trading post, and market then occasionally people will start to come ask to be a citizen in your town. it is therefore a great way to boost your population quickly. But also a disease outbreak may occur so be prepared.

If you're really advanced and have built your town hall you can use the statistics page there to see the number of houses and number of families. Always try to have 1-2 more houses than families, so the younger ones will move out when coming of age.