When would the Undying Lands become over-populated?

Question

The Undying Lands were a realm inhabited by Ainur and the Elves.

The Ainur are effectively gods, who never die. The Elves are naturally immortal, and never die of old age. However they can have children - which leads me to believe that the population of the Undying Lands is constantly growing (possibly exponentially?).

Assuming that the Ainur don't alter the landscape or size of the Undying Lands, or use their godly powers to otherwise circumvent the inevitable:

At what point would over-population start affecting the inhabitants of the Undying Lands?

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To answer this we would need to know the current population of the Undying Lands, which is a number I am struggling to find.

I am hoping to end up at a year, for example "roughly 5,000 years into the Fourth Age" or something similar.

For the purposes of "over population" I would consider any situation where there are more urbanized areas than rural areas (speaking about Elvish settlements as "urban" feels wrong, but it'll have to do right now). I am not after a date where the entirety of Aman is covered in Elves (although that could be interesting to work out), but I am more after a date where the population of the Elves has got to the point where some of them would realistically consider leaving because living space was becoming a serious issue.

In my question I am also ignoring any implications of the Ainur separating the Undying Lands from Middle Earth. For the purposes of this question it would be best to consider that the Undying Lands was still attached to Middle Earth, as it was just prior to the War of Wrath.

Answer 1

First, I'd note that demanding that a supernatural land -- it's Undying, after all -- not be supernatural in any other way seems rather contrived. But accepting that limitation...

What we know about Elvish reproduction is that some elves reproduce while young and then never again. We have no statistics about how many children they have, but we do have some information. First, elves believed that some of the parents' vitality went into their children, limiting the number they could potentially have. Feanor had enormous vitality and was unique in having seven children. Supporting this, we have from the Silmarillion Feanor's mother saying

'Never again shall I bear child; for strength that would have nourished the life of many has gone forth into Feanor.'

which supports the belief that they can have only limited numbers of children, but suggests that having more than one child is common.

On the debit side of the ledger, some elves decline and go into Mandos (Feanor's own mother did) but this seems to be the exception. It seems clear that the "life cycle" of elves is not to disappear forever into Mandos:

Whereas the Elves remain [alive] until the end of days, and their love of the Earth and all the world is more single and more poignant therefore, and as the years lengthen ever more sorrowful. For the Elves die not till the world dies, unless they are slain or waste in grief (and to both these seeming deaths they are subject); neither does age subdue their strength, unless one grow weary of ten thousand centuries; and dying they are gathered to the halls of Mandos in Valinor, whence they may in time return.

So elves reproduce (some) and die (some) but (some) return. If you assume that no elves die and that reproduction is only done by newly born elves (say at an average of 100 years old) then the only way that the elvish population can grow without limit is if on average each elf has one child. (If they have fewer then each generation of "newborns" will be smaller than the last and eventually there will be none and the population will be stable.)

So it turns out that because elvish reproduction is an affair of their youth, their immortality doesn't actually play a role in the question of whether the population expands without limit or not! (It's easy to construct a spreadsheet which demonstrates this.)

So we can ignore what happens to elves after their 200th year: If elves average 1 child each, their the number born each year will grow steadily, but linearly. More than one child/elf on average and it will grow exponentially. Fewer and it will halt altogether and the population as a whole will halt at a finite number.

For example, if the average elf has two children, then the number born doubles roughly every 100-200 years.

The actual numbers for elves in Valinor, unfortunately, are not available, but there's no compelling reason to assume that on average every elf reproduces, and thus no compelling reason to assume an ever-growing population of elves.