It almost certainly isn't. The extremely strong propensity for rich countries to have fewer-than-replacement children kills this model. Animal models have no equivalent to this.
We're headed into uncharted territory with human population. Usually this is supposed to be a sort of implicitly scary thing to say, but since the charted territory here is "guaranteed major ecological catastrophe", it's net good news. Disaster is not assured! (Of course, it's still on the table. It's always on the table.)
On the flip side, we're in thoroughly uncharted territory in terms of carrying capacity as well. On the relevant timeframes (decades, minimum), nobody can correctly predict what our technological carrying capacity will be. There's no guarantee it will continue going up, but contrary to the doommongers there's no guarantee it will stay static or go down significantly either; if in 40 years we've licked nuclear energy and cheap robots we could well be growing huge amounts of food in robot-tended greenhouses stacked into skyscrapers, powered by the nuclear energy, or growing meat in vats with chemical energy more efficiently than going through the full biological cycle. And while that may sound like sci-fi, those actually feel like conservative extrapolations of existing trends. (Just about the only prediction I won't buy is the one where we make no further progress and everything stays the same but we keep consuming resources at the same rate, but that's the one you hear most often.)
Probably depends on the quality of life people are willing to put up with.
Of course, by this time it would be energy-cheap to expand beyond Earth.
As far as the absolute limits, it's unlikely we're anywhere near them. If humans switched to becoming significantly more vegetarian in their diet we could support quite a lot more people with current levels of food production. Yet food production is not even remotely running up against fundamental limits at current levels.