What is reincarnation?

It is well known scientific fact that over time, the animal population including human population increases. How does a colloquially scientific religion explain reincarnation if populations increases?

First, I wouldn't state there are well-known scientific facts representing animal populations increasing. One counter example could be found in a correlational balance between prey and predator. When prey increases in population, than predators increase in population. Eventually the population of prey will be lower than the predator population leaving the remaining predators vulnerable to starvation thus dying.

Next, the question is actually very simple to resolve. Let us suppose you have three pencils. Each pencil is labeled independently giving them their own distinct letter (a, b, c). Therefore, (a ≠ b) (a ≠ c) ( b ≠ c) but all three pencils share a common essence known as "d". "d" in this case represents nothingness/emptiness. The formula can be read as (a = d) (b = d) (c = d) ( a + b = d) (a + c = d) (b + c = d) (a + b + c = d). Similar to pencils, which are objects dependent on nothingness, people too are objects dependent on nothingness. You can increase the number of pencils to aleph-naught (infinity) and find there is enough emptiness for them to exist in; and that they will all share a common essence known as "d". Therefor, "d" is sufficient in sustaining an infinite number of objects including animals and people. An increase in the population only shows a new manifestation of the same essence which is who we are. Let us consider our independent existence. For example, those who become enlightened are considered to move into "Buddha Land" while we live in the realm of Samsara. If the population in all six realms are low, than it becomes reasonable to assume people will have to wait their turn. Of course, this is only one hypothesis out of many.

Here is an article/statement from Dalai Lama, since the OP seems to be asking about Tibetan Buddhism.

PS: Basically, the point I am trying to make is read for yourself, especially on such topics like reincarnation where there is mostly only personal experience to help. Don't take somebody's interpretation of it. Its a very bleak idea. And eventually you will say, how can I believe this? The answer is that's how it is, and you cannot believe it until you practice meditation and experience it for yourself.

And like it or not, this is a copout answer. I can find everybody who will give you copouts because OBVIOUSLY, the probability that somebody knows it for sure, is very less.

It is well known scientific fact that over time, the animal population including human population increases.

Actually it's not a scientific fact. Species are known to go extinct and new species are known to evolve..

There is no count of the total beings on earth yet alone the universe. So it's impossible to say whether the sum total of living beings is increasing, decreasing or stagnant.

As for reincarnation the Buddha understood that not all people would accept it. However, he said that accepting it is a safer bet than not accepting it. See the Appannaka Sutta for more about the Buddha's safe bet. Link

A search on reincarnation will yield a plethora of results from the Buddhist to the non Buddhist (Edgar Cassey, Richard Weiss, etc). There are many youtube videos of young children recollecting past lives.

My opinion is that, if not for theist religions forcefully denouncing the theory of rebecoming, that would be the default theory in the modern world.

to evaluate population increases/decreases, it has to be evaluated in a close system (where the boundaries are clearly defined and nothing moving in or out of these boundaries). Buddhist cosmology says there are more than one universe.

Even with this universe we are in right now, we dont know where the boundaries are nor what exact numbers of the unknowns in it. It is very perplexing like quantum physics. Someone asked Buddha where the boundaries (edge of the universe) were. His answer was edge of the universe could not be reached by traveling.