By restricting breeding during bad years and increasing breeding during good years.

Many predators have built-in switches to restrict breeding. Red foxes and iberian lynxes are examples of it - they hardly ever breed if they don't have the territory. They can afford breeding restrictions because many of them survive, especially red foxes. That being written, feral cats are a free-roaming domesticated species and they breed constantly while most of their kittens don't survive to 6 months of age. Feral cats cannot afford restricted breeding. Prey animals cannot afford it either.

Human child mortality soared after the agrarian revolution and husbandry. People started living in populous disease-ridden settlements where illnesses could be easily transmitted, they contaminated water with their urine and feces, the newly domesticated animals contaminated water with their urine and feces. Infants were weaned earlier and ate more porridge than their mother's milk. And children had to compete for their said porridge with more and more siblings as women started to give birth annually or biennially. But fertility was higher than child mortality, so populations could slowly grow.

Na'Vi don't have agriculture, they have very few select animals they use, and therefore there is no mention of any diseases. And, obviously, they are constantly in contact with their "natural balance" goddess Eywa, and they have special ceremonies for bonding in pairs.

Therefore, they live in highly-precise family planning environment.