The 10% conversion efficiency mentioned in other answers is a decent rule of thumb — there's a lot of variation in the real world, but if you assume that the total prey biomass equals somewhere around 10 times the total predator biomass, you'll get a fairly plausible-looking food chain.
Tim B makes an excellent point in the comments, though: generally, even apex predators mostly hunt herbivores, simply because they're usually the easiest and most abundant food source around. So a 100 kg apex predator does not need 1,000 kg of lower predators to support it — it just needs 1,000 kg of some kind of prey, which may include both herbivores and other carnivores.
In fact, in real life, many apex predators (such as bears and, indeed, humans) are even omnivorous to some extent, consuming some plants (usually parts with high nutritive value, like fruits, nuts and berries) to supplement their hunting. Indeed, one major advantage of a flexible omnivorous diet, for species high up in the food chain, is that it helps guarantee a steady food supply, minimizing the risk of mass starvation (from which apex predators, with their small population sizes and long generation times, have a hard time recovering from) due to fluctuations lower down in the food chain. Conversely, since apex predators, by definition, have relatively little competition, they don't suffer such a strong pressure to specialize as species lower in the food chain, and can thus afford to maintain a generalist diet.
Actually, the only reason everything in nature isn't omnivorous is that different nutrition sources sometimes require incompatible adaptations. For example, the reason why autotrophs (plants) and heterotrophs (animals) are mostly distinct is because efficient autotrophy requires some adaptations (like a low-energy sessile lifestyle) that are incompatible with those needed for efficient heterotrophy (in particular, mobility for grazing/hunting). Similarly, the distinction between primary consumers (herbivores) and secondary consumers (carnivores / omnivores) is a follow-on effect to this: efficient grazing on such a low-density nutrition source as most plant tissue requires behavioral traits and digestive adaptations that are not well suited for hunting, and vice versa, so while a predator may occasionally eat plants, it is unlikely to be able to survive well on plants alone.
However, on higher levels of the food chain, this specialization starts to break down: the adaptations needed to hunt songbirds are not that different from those needed to hunt falcons, so an apex predator that can do one will most likely be capable of both. They're still most likely to hunt mostly songbirds, though, simply because there will be a lot more songbirds around than falcons (and also because songbirds will likely be easier to catch, and less likely to fight back, than falcons).
(In fact, IRL, falcons would generally be considered apex predators themselves. While there are species that may, occasionally, kill and eat falcons, none of them really do so routinely or to such an extent as to put any significant predation pressure on the falcons. To a first approximation, nothing eats falcons.)
So, with that out of the way, how should you figure out the biomass of different species in your ecosystem? Well, the first step would be to roughly sketch out the food web for the ecosystem. For example, a quick sketch might look something like this:
Large apex predator (100 kg), large carnivore:
- Mainly eats mountain goats (80%), supplemented by some lemmings (15%) and songbirds (5%).
- May opportunistically eat falcons, but not very often (< 1%).
- Does not usually eat insects (too small to hunt efficiently) or plants (not easily digestible).
Mountain goat (50 kg), large herbivore:
- Mainly eats plants (> 99%); can eat almost any plant, even those inedible to most other herbivores.
Falcon (0.2 kg), small carnivore:
- Mainly eats songbirds (25%) and lemmings (75%).
- May opportunistically scavenge mountain goat remains left by apex predators (< 5%).
Lemming (0.1 kg), small herbivore / omnivore:
- Mainly eats plants (90%; shoots, leaves, roots and seeds / berries) and some insects (10%).
- May occasionally eat eggs (< 5%) when available.
Songbird (0.02 kg), small herbivore / insectivore:
- Diet consists mainly of insects (50%) and seeds (50%); proportion varies by season (mostly insects in spring / summer, seeds in autumn / winter).
Insects and arachnids:
- Broad group subsuming a complex sub-ecosystem of herbivorous, predatory, scavenging, symbiotic and parasitic species.
- Predatory insects and arachnids mainly hunt other insects; thus, overall, the group may be considered mainly herbivorous (> 95%).
- Some parasitic species, such as ticks and mosquitoes, derive a significant part of their nutrition from birds and mammals (< 5% overall).
Plants and fungi:
- Autotrophs / detritivores, obtain their energy and nutrients from sunlight and/or from waste and remains of other organisms.
- A few species in nutrient-scarce habitats may catch insects for extra nutrients (< 1%).
Note that I've added a few land herbivores to your ecosystem, since it didn't seem realistic to me not to have any. In particular, if you want large apex predators, you really do need some large prey that they can hunt efficiently; without those, there probably would not be any niche for predators much larger than your falcons.
Now, since you've already decided how many apex predators you want, you can start from the top and work out how much food they need. So, for example, three apex predators (3 × 100 kg = 300 kg) will, by the 10% rule, need around 3,000 kg of prey. Around 80% of that will be mountain goats, so that's 2,400 kg / 50 kg = 48 goats; let's round that up to 50. (Nothing else really eats goats in this ecosystem, so we don't need to account for other predators.) That's not a huge lot of goats, but then, three apex predators is quite a small population in itself.
The apex predators also eat some lemmings; the 10% rule says we need 450 kg / 0.1 kg = 4,500 lemmings to satisfy their craving for small furry snacks. However, the lemming population is also harvested by falcons; we haven't yet decided how many falcons there should be, since falcons are not a major food source for anything, but let's say there are 100 falcons, making their total biomass 20 kg. They'll thus need 200 kg of prey, of which 75% will be lemmings, giving us a total lemming biomass of 450 kg + 150 kg = 600 kg, and thus a typical population of 6,000 lemmings.
(Of course, if these are anything like real lemmings, their population size will be cyclic, growing over a few years to a peak and then crashing. This will likely induce a similar cycle in the falcon population, or at least in their offspring production rate. During peak years, the apex predators may also consume a significantly higher proportion of lemmings, since they'll be plentiful and easy to catch.)
The apex predators and falcons will also require a songbird biomass of 150 kg + 50 kg = 200 kg, giving us a population of around 10,000 songbirds. Half of the songbirds' food will be insects, which means they'll need around 1 tonne of insects to support them; however, the lemmings also eat some insects, pushing the total insect biomass needed to support both populations to around 1.6 tonnes. (In practice, the real insect biomass should almost certainly be higher, since some of it will be consumed by other insects and arachnids. I don't have a good conversion factor for that, so let's just arbitrarily call it 2 tonnes.)
The 2.5 tonnes of mountain goats will, by the 10% rule, need 25 tonnes of plants to support them; the lemmings will need about 5.5 tonnes, and the songbirds will need about 1 tonne of fruits and seeds. Treating the 2 tonnes of insect biomass as roughly 100% herbivores means they'll need 20 tonnes of plants to support them (and everything that depends on them), for a total plant biomass of around 50 tonnes. (This figure does not generally include things like tree trunks, which are not easily consumed by herbivores.)
We'll thus get the following rough biomass / population figures:
- Apex predators: 3 × 100 kg = 300 kg
- Mountain goats: 50 × 50 kg = 2,500 kg
- Falcons: 100 × 0.2 kg = 20 kg
- Lemmings: 6,000 × 0.1 kg = 600 kg (typical)
- Songbirds: 10,000 × 0.02 kg = 200 kg
- Insects: 2 tonnes
- Plants: 50 tonnes (live tissue; not including tree trunks etc.)
As noted above, if the lemming population is anything like in the real world, it may cycle strongly, from, say, 600 to 60,000 individuals. This oscillation is likely to be reflected, to varying degrees (and with varying delays) in the other populations as well.
These cyclic interactions can get quite complex. For example, during peak lemming years, the apex predators may hunt less goats, which will allow the goat population to rise next year; however, the lemmings will also compete with the goats for food, which may somewhat moderate the rise. If the lemmings deplete the plant resources considerably on peak years, this may cause the goat population to first peak (due to reduced predation) on the next year, and then crash (due to lack of food) afterwards. Similarly, a lot of lemmings means that falcons will hunt fewer songbirds this year, but also that there will be more falcons next year.
In any case, all of this is assuming an essentially closed ecosystem. However, in nature, few ecosystems are totally isolated from their surroundings, so there will likely be migration and other interactions with surrounding areas. Fortunately, these interactions often tend to be stabilizing: for example, if there aren't enough songbirds and lemmings around, the falcons can fly off the mountain and look for prey elsewhere.
In particular, a population of three apex predators is not anywhere near stable in isolation; unless rescued by immigration from elsewhere, it will almost surely go extinct within a few generations (and even if it did not, it would suffer greatly from inbreeding). However, a population of three large predators can live just fine on a mountain, as long as there are other populations nearby from which new individuals can occasionally immigrate, and to which the offspring of the current population can emigrate if there's not enough local prey to support them.