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My hope is that this will clarify some of the problems of balancing populations (see the 'balancing slow rates' thread) as well as just generally making it believable. I hope that what Enarwen and the others are thinking of is something along these lines, and if it isn't, I hope they will think of it now that I'm putting it down.
Population change = Birth rate - Death rate + Immigration rate - Emmigration rate. (Eq. 1)
This is a simple enough equation; if all of these numbers are zero, then you're in stasis. If the numbers are not zero, but they add up to zero, then you're in equilibrium. I will now go into each of the terms that result in population size change.
Immigration and emmigration are the most complicated of those terms because they are affected by many social factors, not the least of which being the existence of other civilizations of appropriate race available for population to immigrate from or to emmigrate to. If you stand alone, then there will be no immigration or emmigration. Of course, if your citizens are emmigrating, then the population that they are emmigrating to will see them as immigrants, and will get bigger, so there is a conservation effect here; the only real way to add people is by birth, and the only real way to remove them is by death.
Birth rate is normally and simply represented by the following logistic equation:
Birth rate = Rate constant * Current population * (1 - (Current population / Carrying capacity)), Eq. 2
where your carrying capacity is the number of people that your civilization can sustainably support. Ordinarily, this would be your food income, although if your race or civilization otherwise depends upon some other resource for living - such as housing or other living space, or even something essoteric such as the Light of Eru or the Darkness of Morgoth - then it might be influenced by other factors. As your population approaches this limit, your birth rate will fall to zero.
The rate constant is a proportionality constant peculiar to the reproductive biology and behavior of your society. Presumably, Elves, who take fifty years (according to Laws and Customs Among the Eldar) to mature, would have a proportionally lower rate constant than most Men, who take about twenty years to mature (although Edain, Dúnedain, and Hobbits all take closer to thirty). Orcs would have an even higher rate constant. Dwarves are probably similar to Elves. This rate constant could concievably also be not really constant, if it depends upon various societal factors, for instance the tendency of the Men of Gondor and seemingly most of the Elves of Middle Earth to stop reproducing in the latter third of the Third Age.
Now, if a population is changing only by its birth rate, it will eventually and asymptotically approach its carrying capacity, provided that the rate constant is non-zero. However, there almost always is death involved as well:
Death rate = (Current population * Scenesence rate) + Accidental deaths + Competitive deaths. (Eq. 3)
The scenesence rate is the rate at which people die of natural causes, such as old age. This is of course zero for anything immortal, such as Elves, but it does not affect the other two terms in the equation: accidental death and competitive death. Using the birth rate equation I gave above, one would calculate a negative birth rate if the population ever were to exceed the carrying capacity, or, more realistically, if the carrying capacity were ever to drop below the population due to a change in the economic situation of a civilization. These deaths, probably starvation, could be treated as accidental deaths. Even if there is no starvation, however, accidental deaths will always occur: they are inevitable. Agile and sure-footed as Elves may be, they will occasionally slip and fall and break their necks, or get caught in particularly bad weather and freeze to death.
Then there is the competitive death term. This is what accounts for deaths in warfare, both of soldiers who go out to war and fight and die, as well as people killed by enemy soldiers when they raid or otherwise destroy a civilization's settlements. How to handle this requires thought given to how many people are in a civilization's military, e.g. how much of the total population a single military unit contains. One would normally assume that, in a game like TLA, a civilization's buildings would be associated with some of the population, a civilization's units with the rest.
Sex and demographics
It is worth considering the question of sex when thinking about military deaths. Tolkien clearly did not envision the women of civilizations, by and large, being warriors. This means that the deaths in warfare are going to be disproportionately male. How does this affect the population as a whole? The answer depends upon the "mating system" of the species.
If the civilization either is or otherwise can be highly polygenous, i.e. a single male is capable of fertilizing a large number of females, then the first 'current population' term in the birth rate equation (not the second!) should account only for the population of women, and should ignore the men. Obviously, men should not be ignored for the other places in which current population is important: carrying capacity and scenesence rate. In order to do this realistically, you would need two population numbers for the race in question, one for women and the other for men. You could, however, simplify this by saying that the deaths of males do not count as deaths for the result of the overall population.
If the civilization is obligatorily monogamous, however, then the limiting factor in a population's growth rate is the rarer sex, whatever that happens to be - most likely males, if the civilization has been doing much fighting and males are the ones who become soldiers. Once again, in order to do this truly realistically, you need both male and female populations, in which the first current population in the birth rate being the rarer sex. You could, alternatively, make it simpler by just saying that if a man dies in battle, his wife is dead, too, since she needs her husband to reproduce.
For religious reasons, Tolkien envisioned most of his societies in the 'obligatorily monogamous', I think. Certainly he thought of the Elves this way, although I don't know what he would have said if you made him read up on how population dynamics actually works. It isn't clear what he thought of the Men in his worlds; obviously he realized that humans are capable of taking multiple spouses, but believed that it was wrong. The highly polygenous model, however, might actually work fairly well for Orcs - we really know almost nothing about their reproductive biology. If we draw analogies to real-world human societies, the Drúedain might also be potentially polygenous, although of course if enough males are available each woman will have her own husband.
It is also reasonable, and this is true both in real human societies and probably also in Tolkien, that if the sex ratio gets highly skewed - for example, if all of the men are killed in battle, that the women who remain will begin to join the military themselves. This frequently happens in real human societies modern and historical, and it is certainly reasonable for some of Tolkien's races, the Elves and Dwarves in particular.
Dwarves of course have twice as many men as women, so their populations are naturally skewed in the other direction. There is no real reproductive advantage to having multiple men to one woman, so the population growth rate of Dwarves would be limited by the number of women they have, regardless of whether they are polygenous or monogamous (although I suppose we can probably assume they are monogamous).
Metapopulation Dynamics
Metapopulations are populations that are composed of many sub-populations, of which the sub-populations occasionally become extinct (i.e. population size falls to zero), but are then recolonized by other metapopulations. This seems quite common in Middle-Earth for several of the races or civilizations, particularly the Elves and Edain/Dúnedain. Recall that there are those immigration and emmigration rate terms in the population change rate equation. If a population becomes extinct, its birth rate falls to zero, because birth rate is always proportional to current population. Likewise, the death rate also must be zero because there is nobody left to die. However, immigration is not necessarily zero. This means that extinction is not forever, at least as long as the metapopulation survives!
It is reasonable to suggest that populations that are well below their carrying capacities will recieve immigration, these immigrants being able to gain the reproductive advantage of having plenty of resources and thus a larger number of children. Populations near or above their carrying capacities are likely to experience emmigration, because people leave to other populations where their prospects are more promising. If the only causes of death are accidental or otherwise random, i.e. competition or other danger is spread evenly over the entire metapopulation, then immigration and emmigration will follow this relationship between population density and carrying capacity. Of course, this is not necessarily the case if the danger is not evenly distributed!
Immigrants will go into danger if they figure there is something to be gained for it. There are a lot of reasons people would do this in Middle-Earth, from opportunity to revenge. Immigrants can of course also be curtailed by societies: there is such as a thing as competition within a population, as well as emnity between species. How this works will of course depend on the scarity of the limiting resources and the tendency of the races. It is unlikely that Elves or Hobbits will slay others of their own kind unless they are desperate, but Orcs will do it routinely. This should be kept in mind when considering population dynamics: it is theoretically possible for races like Hobbits to overpopulate, and possibly practically possible. This can have bad diplomatic consequences. Elves have a slow enough population growth rate and otherwise are usually stealthy enough to avoid such conflicts even if they do have to expand as a result of population pressures.
Population pressure, of course, will make it more or less difficult to create your own metapopulation. Hobbiton was founded by overflow from Bree, and Buckland by overflow from Hobbiton. If your population is near your carrying capacity, then expanding outward should be fairly easy, even for a non-adventurous race like the Hobbits. The Elves on the other hand have problems: by the land and resources, Imladris should have reclaimed most of Eriador after the fall of Fornost, but they didn't. The Noldor were few, and they have seemingly forgotten how to procreate!
Population pressure, of course, is the primary reason why warfare exists in the first place. Orcs tend to be raiders, so they will always be on bad terms with other peoples. However, most races will tolerate each-others' presence when times are plentiful, i.e. populations are well below carrying capacity. Even Elves will probably fight amongst themselves when they really are overpopulated, however.
There is, in metapopulation dynamics, the question of switching civilizations. The Noldor and Sindar, for instance, are both Elves, and it is probably possible for a citizen of one to emmigrate to the other and mesh in fairly easily. This would not work if a Noldo or a Sinda were to try and become a Dunlanding. Some consideration of race and kind should be kept in mind when considering immigration and emmigration, for races that have multiple civilizations within them. Elves can obviously intermingle with Elves. Lesser Men can intermingle with other lesser Men. Dúnedain can intermingle with other Men with some passability, but it's difficult. They can also intermingle with Elves, although with more difficulty. Presumably mountain Orcs can join Mordor if they wish also. Still, there will be some problems. There aren't very many Elf-mortal marraiges, after all, and all of them are to one of the 'High Men'!
) as well as take into account natural deaths.
