What is a realistic size and population for a predator that eats humans?

its favored prey is humans

despite how much some of us brag about humans being the apex of evolution, we make for a very poor prey: when raised free ranging we are very lean, and fat, energy rich tissues are the most prized tissues for carnivores feasting on their preys. Compare the fat content of a seal with a surfer, and you understand why often sharks bite and drop.

On top of that, we have this nasty habit of hunting into extinction any predator who feels too eager in having this weird two legged tofu sample on their menu.

That said, solitary predators, like most felids, need to be larger than their preys to have decent chances of overpowering them. Compare tigers with wolves: wolves, being able to rely on group hunt, can afford being smaller.

Big cats (tigers, mountain lions, panthers, cougars, lions...) are your reference here for a ballpark reference on the body size. Anything smaller than them would have a hard time in efficiently overpowering an adult. Population size-wise, still look at them.

What I said about the nasty habit we have also becomes the weak point of your setting: a true human like population would quickly act to hunt down the predator population, while for a population without that trait I would argue that it cannot be classified as human. Grazing herbivores do what you want, not hunter gatherers.

As L.Dutch has said, it hardly matters how big these predators get unless they're so tough and heavily armoured that hunter-gatherer humans basically have no chance at all of killing them.

Humans are very good at hunting things, no matter their size. These predators would need to be so heavily armoured that a spear or an arrow could do no damage, and even setting a spear against their charge would result in a broken spear. The predators would have to be awfully suspicious of poisons too, otherwise they might eat a poisoned prey and die.

However, assuming that these predators are able to usually survive any attempts by their prey to reduce their numbers, the populations of predators and prey are mathematically related, according to the Lotka-Volterra equations:

$\frac{dx}{dt}=\alpha x - \beta xy$

$\frac{dy}{dt}=-\gamma y+\delta xy$

Where:

• $x$ is the population density of prey (for example, the number of rabbits per square kilometre);
• $y$ is the population density of some predator (for example, the number of foxes per square kilometre);
• $\frac{dy}{dt}$ and $\frac{dx}{dt}$ represent the instantaneous growth rates of the two populations;
• $t$ represents time;
• The prey's parameters, $α$ and $β$, describe, respectively, the maximum prey per capita growth rate, and the effect of the presence of predators on the prey death rate.
• The predator's parameters, $γ$, $δ$, respectively describe the predator's per capita death rate, and the effect of the presence of prey on the predator's growth rate.

All these variables are positive real numbers.

For warm-blooded predators and prey, this typically works out to the prey having a biomass around 100 times that of the predators, while for cold-blooded predators, the prey might have a biomass of around 10 times that of the predators. As the wikipedia article shows, the population changes of the predators is similar to but lags that of the prey populations somewhat.

So, knowing this, you should be able to calculate the relative populations of hunter-gatherer humans and these mythical predators. Just be sure to assign reasonable values to the variables.

As mentioned by others, the actual question is: How can I prevent humans from hunting this thing to extinction so that it can survive to continue its predation?

In an Earth-like world, you can't really do that. Humans are just to good at killing both their predators and their prey.

However, with its magical origins, and depending on how much you want to lean into magical reasons, you do have options.

For example, these creatures are supernaturally stealthy, living primarily out of phase with the rest of the world in some kind of spirit realm. They could 'phase-in' when pouncing on prey and otherwise be undetectable. This would prevent humans from being able to track and hunt it, and allow your beast to strike solitary, or small groups of, humans at its leisure. Depending on whether (or how quickly) this creature could drag its prey back to the spirit realm, it could even strike in the middle of a village at any time of day.

Once your beast is able to hunt at will, you will likely have to place arbitrary limits on their populations, such as driving off and killing any competitor beasts from their hunting grounds. This would likely keep your population of beasts significantly below that which would cause the extinction of humans.

Let's start by assuming this thing is basically a tiger, except that it mostly eats humans, and then we can go from there.

I couldn't find any good references on how much a tiger needs to eat, but lots of web sites say they eat about one deer-sized animal a week, so let's go with that. I don't know how much meat is on a human compared to a deer, but let's say your animal needs to eat about one human every two weeks in order to survive, or about 25 per year.

Maybe that should be one per week, or maybe you can get it down to one per month if you assume a magically slow metabolism, but either way, the point is that for every individual of this species exists, humans will have to be killed at a certain rate. If there are 10 such animals in your country then they will eat 250 people every year, and if there are 1000 such animals they will eat 25,000 people per year.

This is quite different from the dangerous wild animals we have, which might eat you if you're not careful but which can be driven off or avoided with the right techniques. These animals must eat that many humans in order to survive, and the fact that they haven't gone extinct means they do eat that many humans - for the world to be consistent, humans can't have any effective way to prevent this. They will be one of the biggest causes of death in your society, probably the biggest.

So then all we have to do is figure out how many people your settlements can lose per year before they stop being viable settlements, and divide by 25 - that's how many tiger-sized human-eating animals your world can support.

To work that out, let's look at human birth rates. Wikipedia says that a birth rate of 50 births per 1000 people per year is "considered high", so let's assume that birth rate for your society. (I assume birth rates would be high to offset the high mortality rate due to tigerdragons.) If 1000 people produce 50 people per year and a tigerdragon eats 25 people per year, then a population of 1000 people could support up to two tigerdragons. This assumes that tigerdragons are the only cause of mortality though, so it's probably better to say one tigerdragon per 1000 people, max. Every child born in such a society has a 50% chance of being eaten by a tigerdragon eventually, as opposed to dying through some other means.

Then it just comes down to how big your population is. If your settlements have half a million people all together there could be 500 tigerdragons, which is generally considered a viable population, although half a million people might be a lot for your setting.

If those numbers are too extreme you can probably find ways to tone them down. The most obvious way is to give tigerdragons other prey that they eat as well as humans, so they don't need to eat as many humans. If they're magical creatures you also have a lot of leeway, e.g. to make them not need to eat much at all, or be able to survive with an unrealistically low population.

Your other question is how big you could make them. For this we can use allometric scaling. The relevant formula is $$ \text{metabolic rate} \propto \text{body size}^{0.75}. $$ This means that if you make one twice as big it will eat about $25\times 2^{0.75} \approx 42$ people per year, less than the 50 that two tiger-sized ones would eat. One 20 times the size of a tiger will eat about 236 people a year, about half what 20 tiger-sized ones would eat. So having a few very large ones is possible.

I'm envisioning a creature with the base morphology of the Chinese mythological shenlong ('god dragon'), albeit mammalian, with bat-like features: soft, dark brown fur, consecutive sets of leather wings that continue to grow out from down the length of its spine as it ages over the course of its natural approximately 400 year lifespan, ultimately looking like some nightmarish ferret with a thin, tubular body (up to 12-15 feet long? 250-400 pounds?), the ability to gracefully tuck its wings back (a la hippogriff), and a set of dexterous, raccoon-like forelegs which allow it to slither-sprint for short bursts along the ground in addition to being capable of longer-range flight.

The singular most horrifying thing about the creature is its mouth - an biomechanical abomination of evolution featuring 3 interlocking "teeth": prehensile cartilage-lubricated dental columns which lock together to form a cephalic jaw when not in use but split apart to become a set of razor-coated stalactites which it rotates along the roll axis of its face to quickly liquify prey (bones and all) into a yummy, bloody, high-protein slop paste that evenly distributes along the length of its gut, allowing it to get back in the air almost immediately after feeding with the additional weight evenly distributed.

Like a bat, it hunts at night, primarily during the long nights of the winter season, but employs a stealth-based ambush hunting strategy closer to that of an owl, plunging from the sky to smash its prey to the ground and disable it so it can begin to feed. Because it hunts at night, is highly intelligent, and hibernates in the summer, few humans have ever even seen one and lived to tell the tale, and its unknown how many of the creatures exist or where they nest (deep in the mountains) - there could just be 3-4 really hungry ones or there could be hundreds up there, but who really wants to go outside and check?

This provides an underlying rationale for why and how a moderately-pissed humanity hasn't collectively teamed up to exterminate these things into a line of high-end designer rugs yet - even when it attacks a group of brave (stupid) night-venturing humans and carries one off, nobody has much time to even react, much less get into a finer zoological study of the creature. The relative fragility of the wings can provide you with a softball opening for human-demonferret interaction that isn't immediately and inevitably lethal should your plot call for it.

You'll want to have the thing be evolutionarily blessed with a few other quirks - a slow metabolism (one adult human man-meal should satiate it for 2-3 weeks), stomach acid with an extremely low pH capable of digesting the bone it pulverizes and consumes when feeding in order to grow its wings, have probably some notably weird, canonically non-mammalian way of eliminating waste (e.g. sweating everything out in lieu of a single-stream ability to piss), a notable capability for anerobic respiration (fuzzy cuddle snake no need breathe right now), and a relatively late-stage (150+ years old) sexual maturity culminating in a almost-not-mammalian birth (think platypus).

The concept's pretty versatile in terms of how you want to frame the overall tone around it - is it a total nightmare scourge that adult men wet the bed over every night? It's smart; is it empathetic, albeit naive with a childlike ignorance to the horror it inspires? Do its big, long, soft ears flap back and forth in an obnoxious, almost sort of cute way while it's disintegrating your legs with its garbage disposal face? Is it curious? Does it hunt only out of necessity to feed itself and its young or does it seriously just enjoy watching the light leave the eyes of whatever/whoever it snatches? Can its horror-fangs be contorted into what's unmistakably a genuine, benevolent smile? Does it pair bond? Do males and females both hunt? Does it make cute noises to communicate with its own kind, or does it make awful, guttural nightmare sounds indistinguishable from the screams of its prey being fed on? Does it have the capacity to befriend humans instead of turning us into an endless supply soylent green, and if it does, would it ever choose to?

These questions are yours to answer, though it seems a common theme to cast even the most awful and terrifying of predatory mammals as having some semblance of redemptive qualities, as the casting would usually dictate something more reptilian and explicitly coded as unfeeling if its purpose is confined to merely being a backgrounded horror, antagonist, or controlling presence.

Hope this gives you some ideas either way. Good luck!