For some reason, I find this hilarious — it's an exercise in applying the mathematics of population ecology to the dynamics of human-vampire interactions. It's the real deal, the actual kinds of math used by those wacky evolution and ecology nerds, all built around some estimates of the rates of vampire siring measured against the rates that Buffy-style vampire slayers take them out. Here's the kind of thing you'll see in the document:
I like it. In case you're wondering, Buffy's Sunnyvale reaches a stable equilibrium with a population of about 36,000 humans and 18 vampires.
(Hmm. I posted this in the "Life Science" channel of scienceblogs. Maybe I need to lobby for an "Undead" channel now.)
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Jennifer Ouellette is coming to campus this week to give a talk about her book The Physics of the Buffyverse. Having never been a Buffy fan, and not seen more than snippets of a few episodes here and there, I figured I should at least watch a few representative episodes before the talk, just to…
After yesterday's all-out frontal assault on a dubious scientific journal (which, by the way, you should still read if you haven't already), how about some lighter fare for today?
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This post by ScienceBlogling revere about the horrendous human cost of influenza is getting some serious exposure. This gives me an excuse to mention something I haven't in a long time:
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I couldn't say why, but I have never been very interested in stories about vampires. I have never read Dracula, I have no interest in Buffy the Vampire Slayer or True Blood, and I think Twilight is some of the worst literary and movie cheese to come out in a while, but despite my general apathy…
Are werewolves not accounted for? Tzzzz.
Yeah,all nice and good,but what if there is more than 1 vampire population,as there is,as we all know from "Underworld" ??
I'm trying to decide which would be weirder - that you set this particular thread up to be auto-posted at this time, or that you're awake and decided that vampire maths would be an appropriate topic for 4.51 in the morning.
Any calculation which ignores the effect of zombies is hopelessly unrealistic.
It is from the current issue of 'Blood'?
I come on here just before going out for a walk on this beautiful night, and you just HAD to post about vampires. Now how am I going to feel safe?!?
Zombies are only important if you assume they can run. Otherwise zombies are not of any significant influence untill there are really a lot of zombies and this is mathematically not possible unless you assume they can run.
I prefer the original source of vampire mathematics
http://images1.wikia.nocookie.net/muppet/images/9/90/CountCountsLP(2).j…
Hey Wowbagger some of us in, you know the rest of the world? have been up and aware for hours. It's 10:40am here in Eastern Scotland. You can tell because yet again the rain we were forecast for now is here represented, yet again, by a black cloud that is sliding north of us, again. You would think that the fact that we live in a rain shadow would have filtered down to the weather forecasters, wouldn't you? What is the point of offering place specific forecasts if they are nothing of the sort?
If you are in Forfar right now, how heavy is the rain?
Peter Ashby said:
I think they find it amusing, at the very least it keeps things interesting I suppose!
I'm in Edinburgh, so I can't help you with the rain situation in Forfar I'm afraid, but if it helps it was lovely and sunny when I showed up for work here this morning. :)
There were far more than 18 vampires in Sunnydale, though. Clearly the equations are flawed, they haven't taken the Hellmouth factor into consideration.
So, does this mean that vampires are classed as an endangered species? Imagine trying to do fundraising for them... 'Give More Than Blood!'
Fascinating. I wonder if anyone knows a Game of Like-style program to visualize the vampire/prey population interactions?
750 pm in Melbourne,pleasant 18 or so and clear sky.....What is this,scottish invasion?? This is the Aussie night shift here guys !!
Depends on the relative predation rates. One Slayer can screw up the whole thing. Lotka-Volterra didn't allow for two Slayers.
But I used to toy with a SF novel about the time when the Vampire virus suddenly mutated to 100% infection rates.
People criticize the Vampire Agenda, but sucking blood from the population of humans for sustenance is definitely a sustainable adaptation.
I wonder if they carry little bottles of Vodka and Worcestershire Sauce around with them.
Holy Crap, Rorschach what is up with your face?
Yikes, that's it, no more crack smoking for me, that is too weird.
PZ: It should be "Sunnydale".
Sorry, this is important; how else can I get Willow to notice me? Even if she doesn't drive stick anymore, she's still cute as all hell.
¨
Vampires still thriving at AIG, Goldman Sachs, etc. And they've learned how to roam the streets in Daylight, not good.
I wonder if we can train them to hunt pirates, are there aquatic Vampires, like Mermpires , anybody have any info on that?
Well... that ^ was odd.
But as I was trying to say, this reminds me of the paper on dragon migration me and a couple of others wrote for an ecology class once. The ecology and evolution of mythical beasts is a vastly underappreciated area of research.
11:20 am in Edinburgh now and it has just started to get overcast. I am busy trying to blow the clouds back Forfar way. The last two days have been gorgeous (by Scottish standards).
Although considering Scotland, how do the differences in daylight hours between winter and summer affect the vampire populations? Do they hibernate during our summer? Do they gorge themselves during our 26 hour nights? (there may be slight exaggeration in the last question - everyone knows we have at least 32 hour nights in december)
How about trolls? Where did they live before there were bridges to lurk under? and billy goats to eat? Enquiring minds wish to know.
@Rorschach
Two Scots constitute an invasion? Looks like our martial reputation is still live 'n' kicking. Yes!
There used to be, but I believe the Deep Ones hunted the last ones to extinction some time in the fifties.
scooter,
(as usual) what on earth are you talking about??And crack is not good for you,you should know.....
I wrote a dragon thesis for my psychopharmacology class, dwarves were involved
Yeah but you have a writhing sock on your face and you kill people
@Scooter
Of course dwarves were involved, dragons don't mine their own hoards you know.
The vampires can increase their number. For this, they would have to switch from hunting to farming. The problem is that it's difficult to keep humans herded and coralled. Of course, TV is a solution. It fixes humans, dissuades them from wandering and makes them obedient. But with massive TV consumption and its natural complements (beer, pizza, potato chips, not moving from the couch), another problem appears: the quality of their blood gets lower. And, who knows, blood cholesterol might be toxic to vampires.
Hey, perhaps the best solution is a "Matrix"-like world of life support pods coupled with virtual reality?
nah, it just gives them the shits
Peter Ashby said:
I think technically we need a bagpipe if its to be a true Scottish invasion force! ;)
Alan Worsley said:
They all live in Edinburgh - why else do you think we encourage innocent tourists down into the undercity throughout the summer months and then have a street party in the dead of winter?
Mate,I love that character.Had never even heard about it until I saw the movie...:-)
How can anyone even think of a sentence like that??
Upside-down Flintstones..
My head is plode
Ah yes..the tourists much change the "immigrant" portion in the differential equation.
Lilly you wouldn't be in King's Buildings would you? Would be rather amusing to find out one of the other commentators on Pharyngula was in the next office.
@Christophe
Except that Vampires can't cross a threshold uninvited, so all those couch potatoes are off limits and unavailable. The same cannot be said for the poor pizza delivery guys who provide the sustenance to them in the evenings . . .
Yes, but pineapple will make it sweeter...
I took psychopharmacology around 1974 at University, and hallucinated on the weekends to unwind.
Alan Worsley said:
Alas, no - I used to be based there before I graduated though.
Unless they've been buying up foreclosed/repossessed properties on the sly and actually own the building the couch potatoes are living in . . . .
This is why the sensible vampires invest in rental properties. It allows them to enter, but they have to provide notice in advance.
The ones with less funds available get jobs as building superintendents in apartment complexes.
@Lilly
Our youngest plays the pipes and is in residence (boomerang generation), does that count?
Peter Ashby said:
I think it could be made to at a pinch (especially as our invasion force has now swelled to 3).
If s/he could bring the boomerang for extra artillary, that would be great!
@Lilly
So that is why they opened Mary King's Close, hungry vamps. That explains a lot. it might also explain why so many Scots look so pale and wan . . . I'm coming out of my winter colours now the weather is warming and the days are longer. I ran in shorts and a vest last night and am tanning nice and gently.
I just fed the differential equations into my graph-plot crypto-keeper pixelmorte image generator and this image came up:
http://static.howstuffworks.com/gif/vampire-power-1.jpg
Holy fucking shit, that's Al Lewis, the guy was a socialist, somebody call Glenn Beck, it's worse than we thought
Peter Ashby said:
Aha, clearly you are not a true Scotsman . . . :)
I'll get my coat . . . .
I'm sensing some spleen venting about rental situations here. Pharyngula as therapy, nice. Let it out folks.
Just got back to Wales from sunny Scotland last night and, of course brought the wonderful weather home with me (sorry).
The only vampires in evidence on Cairngorm were the "poor old me, my bonus is only £1 million this year" skiers but the capercaillies were lecking at Loch Garten and the red squirrels were frolicking in the sunshine in the forest.
Thought I saw a werewolf but it was just an RSPB volunteer with spectacular red whiskers.
Looks like I was lucky I decided to forgo the pleasures of Edinburgh!
(by the way posted on top of Cairngorm on the weather board - "As good as it gets!!!" - the visibility was fantastic)
another Edinburghite here. lovely weather we're having now, eh? thought i'd have to go get my umbrella for a bit this morning but it's turned out nice again.
I live in Texas, so many bloodsucking rednecks so few wooden stake and mallet opportunities.
Damn That COMMUTE!!!
Kitty said:
I dunno, not content with a decent Rugby team you have to nick our weather as well . . . . ;)
Oddly, it is still freakishly nice here - I keep expecting a typhoon or something to make up for it!
This is a good start, but a more thorough model should take into account both vampiric sparkling and their irresistible attractiveness to unbearable teenage girls with no personality.
Another member of the Edinburgh invasion-force reporting for duty!
Still no rain, bad news for the seedlings and tatties up at the allotment.
We have a complete vampire protection phrase in Scotland, of course; "Frae ghosties an ghoulies an lang-leggit beasties, an things that go bump in the night: may the good FSM preserve us".
Works 100%, as I know because I've never been grabbed by the ghoulies....
Is it just me, or should that first expression be rH(K-H)/K -aHV, rather than rH(K-H)/(K-aHV)?
Why is it that it's perfectly possible to apply interesting science and mathematics to vampires, and get a reasonable outcome, whereas if you apply science and mathematics to Jesus, you have to starT WRITING IN CAPS AND IGNORING GRAMMAR AND SPELING AND YOU MAKE SENSE NO ALL AT.
1.20pm. Liverpool has just gone overcast from a nice sunny day. Looks like rain. Bollocks.
@Lilly
Ah so it's raining in Edinburgh, I knew it!
The interesting thing is that despite having spent a significant proportion of my life living and running in sub tropical Auckland, NZ I wear less when I run here than most people. Though there is one guy who was in shorts and a vest before I was but he runs much faster than me so he is working harder and generating more heat. Still I see so many people who are dressed for the season rather than what the temperature and weather forecast actually are. Either that or they are just soft. Maybe they're the immigrant Sassenachs.
I don't think the equation account for the organizing influences of elder vampires like the Master? plus as the alternate universe episode shows, the absence of a slayer in sunnydale throws this totally outta whack, scoobies alone just aren't good enough. in fact most scoobies get turned into sexy evil vampires, like sexy evil zander and sexy evil willow.
plus there should be a variable for how much the government knows about the situation, like as vampire population rises, the likelyhood of the Initiative taking... you know... initiative. vampires are tough, but when faced by teams of guys with holy water super-soakers or flame throwers they ain't that tough.
if you bless napalm, does the gel count as holy water?
A good vampire movie showing in (UK) cinemas at the moment is Let the Right One In. Not sure if it's out in the US yet.
Regarding the main topic it just occurred to me that this mathematical approach can also be used to explain why there are still monkeys although we evolved from monkeys: There is a certain rate at which monkeys evolve into men AND there is a certain rate at which men (preferably creationists) turn into monkeys again. As there are certainly more humans than monkeys, we can conclude that de-evolution is less likely than evolution, but again it cannot be ruled out entirely.
:-D
Anyone remember what the film is which is about the filming of Nosferatu, but the reason that the actor who plays the title role is so good at it is because he's a real vampire?
I half watched half of it, so would like to see it again properly, seeing as I must only know about a quarter of what happens.
All this Vampire stuff is old, it doesn't sell.
Bring me a Man bites Vampire story.
Geoffrey is right, the first equation is messed up.
Well if its any help, it was raining earlier, just for a short time, then bright and sunny again here in Dundee, now its back to grey cloudy skies and turning cold! No idea in which direction the wind is blowing, but will keep you edinburghites posted!!
#57
2000 – A Hollywood movie called Shadow of the Vampire told a secret history of the making of Nosferatu, imagining that actor Max Schreck (Willem Dafoe) was actually a genuine vampire, and that director F. W. Murnau (John Malkovich) was complicit in hiring the creature for the purposes of realism.
I wouldn't put that past Malkovich, the guy is pure slime
Roger Zelazny wrote a short story called "Dayblood" that touched on the ecological implications of vampirism - and proposed that there were predators that fed on vampires, keeping their population in check.
Colonel Molerat:
Shadow of the Vampire(2000) with John Malkovich and Willem Defoe?
Normally I don't subscribe to the idea that science "unweaves the rainbow". But I have a hard time believing that any fiction is enhanced by the addition of differential equations. Who watches Buffy and thinks "ok, it's good, but it needs more calculus"?
Haha, we did a nearly identical project in my Ecology class last semester. Same modeling, used humans as the prey and vampires as the predator. Though we ended up getting in big nerdy debates about how the modeling would actually work, since humans didn't just die, they would get turned into vampires. Then our TA said we were being too geeky and to just do the modeling =(
Wouldn't vampires fit all the requirements for life? Not sure why they'd need a new category...
@64: well, the authors of the paper, clearly.
And I find this quote from Buffy:
Xander Harris: Giles lived for school. He's actually still bitter that there were only twelve grades.
Buffy: He probably sat in math class thinking, "There should be more math. This could be mathier."
So. Giles, then.
Facehammer said:
Would that not be offset by the tendency for enraged literature profs to bludgeon them over the head with hardback editions of the complete works of Anne Rice before setting fire to them using copies of Twilight as kindling?
For those still following the Scottish weather situation, we had a brief shower here in Edinburgh during lunch, now back to blue skies.
Weird weather!
I tried to model whether or not the humans left over from Cylon attacks in Battlestar Galactica would be able to make it as a population...but trying to model the stochastic nature of further Cylon attacks was too much for me.
Hey, in Analog (I think) about 20 years ago (fewer, I hope, or wish), there was a short story about just this sort of thing. Some ecologist works out the vampire math, notes the similarities to actual reality, alerts the authorities... conflict ensues.
!SPOILER ALERT!
(No, I can't spoil things literally, just in case, but you've probably guessed the outcome already - I did as I was reading the story, and I think the author intends that to happen)
So anyway, without revealing exactly the outcome, I would just like to suggest that Brian Thomas be very, very careful about revealing his calculations. Oh, wait, now they're on the internet? Well, it was great hearing about you Brian.
Peter Ashby @ #34:
solution: vampiric pizza delivery.
It gets them into the houses, and they're up all night anyway so they're well suited to the job. Garlic might be an issue though.
@ 67 if i remember correctly young giles (known as ripper) was a punk rocker who dabbled in dark magic.
And the equation completely neglects the many, many different types of demon to be found in sunnydale (over 200 different species appeared and dozens more mentioned in series and canon comics). So the population of Sunnydale would have to be much larger since a majority of those demons were predators.
What about Aztec Mummies?
and Preying Mantis Substitute teachers?
and demons?
and demon/ROTC hybrids?
and Mayors?
Those vampires are being helped out by Wolfram And Hart. It only seems that they can move about under the daylight. Their buildings and vehicles have tempered glass to protect them from bursting into flames.
DaveX @ #66:
Vamps are considered to be immortal barring violent death by Slayer. I'm sure that's gonna throw their chances of being incorporated into normal life equations.
But this says nothing about plasmavores (http://www.rdwf.org.uk/images/comebacks/anneplas2.jpg) and how many rhinoceros-headed Judoon will show up.
See also the biochemical origins of vampire myths! -->http://www.tompainesghost.com/2008/10/biochemistry-of-halloween-install…
"That's one thing I always hated about Santa Carla, all the Goddamn Vampires! " -- Grampa, "The Lost Boys"
Best. Math. EVAR.
I need to share this with some of my fellow geeks.
Reminds me an awful lot of the epidemiological calculations used to calculate herd immunity...
I'm English. So that's a yes.
Seriously, what are all these Scotsmen doing here? Get back behind your wall!
Show it to Randall Munroe and you're sorted.
I tried modeling worldwide vampire populations once. It took into account the initial number of people, human population growth rate, initial number of vampires, how often vampires had to feed, how often someone fed on turned into a vampire vs. dying, how long after being bitten a vampire would become active, the percent of vampires killed by vampire hunters, and a threshold number of vampires needed before hunters would appear at all. It was very hard to get numbers that wouldn't result in either the vampires going extinct or wiping out all humans and then going extinct (the model assumed that if the number of humans was insufficient to feed all the vampires, any vampires that could not feed would die).
But all this math is useless if it fails to take into account the depredations of the werewolf and demon populations, and the rate-of-sacrifice on the part of Dark Mages. I don't think a stable population of 36,000 would be able to keep up! Fortunately, help is on the way in the form of undocumented illegal aliens (both "foreign" and "domestic"). Of course, if the presence of the Hellmouth feeds energy into the supernatural elements, all bets are off until and unless some means of accurately monitoring that energy and its effects on the various different types of supernatural fauna becomes available. Anybody care to write up a research proposal? We know that historically the military has been interested in funding projects in this general area of study.
________________________
silentsanta @ 17:
Then perhaps you would be interested in How I Met Your Mother; hereabouts, at least, it airs on Monday nights right after The Big Bang Theory, but some people miss it because it airs opposite Chuck.
According to the math (2000:1 human/vampire ratio), the world at large can support 3,000,000 vampires. Take THAT, Ann Rice!
Laurell K. Hamilton, in one of her earlier books about "Anita Blake, Vampire Executioner," had a Very Senior Vampire opposing legalization of vampires because he projected that vampires would eventually grow in numbers and exterminate humans. I think it was "Circus of the Damned."
Of course, before she went completely off the rails with All Sex, All the Time.
This is quite possibly the awesomest document ever produced by human beings!
How about ecology of Aliens vs. Predators on Earth?
"Maybe I need to lobby for an "Undead" channel now."
Won't the "Creationism" category do double service for that?
Larger cities might have an increased cause-of-death rate due to increased potential for fatal conflicts between vampires, much as human-human murder rates are usually higher in urban areas.
had a vampire next door, one moring we found him sleeping so we drove a stake tru his heart. As he awoke, he screamed that he was not a vampire, but as we all know, vampires lie.
A calculation in a similar light-hearted vein, for all of you particle physics followers:
http://minos.phy.tufts.edu/tagg/personal_old/eggsandwich
I did this about 10 years ago, and stand behind my flagrant estimatations!
#4: "Any calculation which ignores the effect of zombies is hopelessly unrealistic."
Hear, hear!
I may be a bit thick but are the equations really reasonable?
The first equation seems to imply that the human population grows faster the more vampires there are, as long as the humans are below carrying capacity.
I love this. And I love the fact that, even without this math, Whedon got it pretty close to right. (There were tons of vampires in Sunnydale throughout the course of the series, #11 -- but except for the obvious apocalypses at any one given time, it seemed like there were only a handful. And it was a smallish city/ largish town -- population 36,000 seems about right.)
And #83: You don't need to assume that vampires need to feed off humans to live. At least, not in the Buffyverse -- vampires can feed off non-human mammals as well. (Otter seems to be especially tasty.) Would that help stabilize your model?
Oops. That was supposed to be, "but except for the obvious apocalypses, at any one given time it seemed like there were only a handful." One little misplaced comma can ruin your whole day.
As a fledgling buffy nerd, if we did this in maths class and similar calculations people would pay a lot more attention. For example, when would a zombie population reach its peak, and when would the numbers start to decline? They're slow-moving creatures and they can't farm.
I agree with the other people here who complained about the simplistic model and absence of the demons, murderers etc who are par for the course in such an environment.
Also, why would humans immigrate to Sunnydale?
Here's the 20:45 evening update on the Edinburgh weather situation.
We now have a blue sky, with a tint of green, yellow and something vaguely pinkish to the northwest. One small cloud has been spotted hovering near the horizon in the direction of Musselburgh, but the chances of rain in the next 6 hours is at, or slightly below, 0%. The temperature is 11.3°C and the wind is hardly noticeable.
Geoffry @51:
Is it just me, or should that first expression be rH(K-H)/K -aHV, rather than rH(K-H)/(K-aHV)?
I think you are right; except that there should also be an additional factor 'b', so:
dH/dt = rH(K-H) - abHV.
The first term is the logistical growth equation, and the second is the loss in human population due to vampirism. It appears the the error you point out was a typo, as the population equilibria are correctly expressed in the following section; but missing that factor 'b'. Since b = 0.004, the vampire population is not 18, but 4500! Yikes! Get busy, Buffy!
Ah, that explains it. But dahduh I don't think the b factor should be there. It expresses the rate of victims that turn into vampires and doesn't belong in the first equation.
This equation also fails to account for the proportion of vampires who get Weatherwaxed when they get too troublesome.
"Except that Vampires can't cross a threshold uninvited"
Not a problem in Embra. They just need to perfect the phrase "I've had my tea".
Posted by: Podblack @ 12 "So, does this mean that vampires are classed as an endangered species? Imagine trying to do fundraising for them... 'Give More Than Blood!'"
No, they are a top predator. It takes 36 000 humans to support a stable community of 18 vampires. If there were more than 18 vampires in Sunnydale, and it was a stable community, that implies there were a lot more than 36 000 humans living in Sunnydale. Or, possibly it could be that Sunnydale's hellmouth was a source of vampires that moved out into other communities, so the local population looked larger at times, but many of them didn't stick around, or they fed from a wider population area than just Sunnydale itself.
@Carl Troein
No wind to speak of? There was a stiff SW wind here when I went for my run and when I went outside to try, unsuccessfully find the big dipper (youngest was missing the Southern Cross) it was still breezy. Grumble, grumble what's the point of being in a bleeding rain shadow if it's windy? No wonder I have to water the garden lots . . .
I think the assumption is that vampires kill humans while feeding, at a rate of a*HV (HV proportional to rate of vampire-human contacts), but only fraction b of those humans convert into vampires - hence why b is present in the vampire-side expression but not the human-side.
Geoffrey's got the correct first equation, given the assumptions. Fortunately for the residents of Sunnydale, however, the equillibrium calculations in the paper are perfectly correct...except...
What about handling time--clearly a vampire has a type II functional response?
I propose: dH/dt =rH(K-H)/K-(aHV)/(1-asH)
where s is some constant handling time.
man, i was gonna post "ooh ooh ooh now do the alien!" but i see someone already posted something to that effect.
Just wanted to highlight that the original paper is dated November, 2002. Worth reviving, though.
And as noted upthread, it's SunnyDALE not Sunnyvale. I'm pretty sensitive to that since I live just a few minutes from Sunnyvale. Sunnyvale has feral parrots, palm trees, the USAF's Blue Cube, but (to my knowledge) no more vampires per capita than elsewhere in the Golden State.
@#64: Maybe not math, but physics in the Buffyverse has already been done: http://tinyurl.com/df92ee
@#97: Xander said in 'Witch' (season 1, ep 3): "First vampires, now witches. No wonder you can still afford a house in Sunnydale." Clearly the housing prices in Sunnydale didn't see the astronomical ballooning that the rest of California did. Honestly, if I could get the Summers' house for less than 200k, I'd move there in a heartbeat.
!Spoiler for season 7 of Buffy!
Also, the above formulation obviously doesn't take into account that there are thousands of slayers now.
Uh, you'll get invited across the threshold if you ARE the pizza-delivery person. And everyone will assume you're eating pizza at work, so they won't wonder why you're never hungry.
I've seen feral parrots in Orange, CA.
scooter, April 21, 2009 6:45 AM
The dragons originally planned to domesticate Australopithecines to herd and tend cattle, sheep, and goats originally. Dwarfs entered the picture when a feral human population proved facile with mining, and discovered sparklies for dragons to drool over. Overall dragons tend to think that this was not such a hot idea in the long run.
BTW, if you ever get hired to nurse maid baby dragons, be sure your employers remember to maintain adequate protection for you. Dragon breath can do a lot of damage, and baby dragons do tend to "breathe" spontaneously.
I have seen feral parrots in Chicago.
Okay, regarding the definition of 'r':
If 'r' is supposed to be the human population of the earth, where are the immigrants coming from?
Inquiring mind wants to know.
If there is an Undead Channel on Scienceblogs, I would happily call Sciencblogs my home for my zombie-lore and -science themed webpage.
Re: The BeadKnitter
r is intrinsic growth rate: births - deaths + immigrants -emigrants. In the original paper, the model is parametrized for Sunnydale, and treats that as the population. If you were to do the whole earth, you'd just set the immigrants and emigrants to zero, but this would be unrealistic given the ecological definition of population. An ecological population is simply all individuals who have a non-zero probability of interacting with one another--ultimately in general a good bit smaller than the total number of humans on the entire earth.
Of course, if the vampires of Sunnydale would just turn to Mad Science for help, they could kick the crap out of the whole need for a base feeding population. (And we know they have Mad Science in Sunnydale. That Buffybot didn't build itself, ya know.)
With huge vats of cloned blood available, they could issue licenses with size/weight and number limits to maintain the viability of the local population. Of course, they would need to institute some sort of non-compliance penalties.
If anyone is in the mood for more mathematical pseudobiology, noted tard William Wallace is trying to model the insertion probability of ERVs over at ERV's.
"no more vampires per capita than elsewhere in the Golden State."
You mean eighteen per 36,000 humans?