>> Amy Draves: Thank you for coming there my name is Amy Draves and I'm delighted to
welcome Colin Adams to the Microsoft Research Visiting Speaker Series. He is here to discuss
his book Zombies and Calculus. This is a novel that provides clues on how to manage a zombie
apocalypse using math to calculate the best way to survive. Colin is the Thomas T. Read
Professor of Mathematics at Williams College. He was honored with the Haimo Distinguished
Teaching Award, the highest teaching award given by the Math Association of America. He is
also the author of six previous books and a humor columnist for the Mathematical Intelligencer.
Please join me in giving him a very warm welcome. [applause]
>> Colin Adams: Thank you. Thank you very much. Thank you very much for being here. It
couldn't have been easy getting here given all the zombies that are out there right now.
Congratulations on still being alive. Me, it took me seven weeks to get here, seven grueling
weeks. I lost my dog. I lost my best friend. Do you know how long we have been best friends?
We have been best friends for three weeks. This whole thing has really changed the nature of
relationships, hasn't it? But the loss of life that was caused by my trip here is going to be offset
today by the number of people who are going to be saved by what I'm going to tell you today,
because I am going to tell you how to survive the zombie apocalypse. We are currently living
through one of the most horrific periods in the history of human time. Suddenly, it makes Paris
Hilton seem unimportant, doesn't it? She's gone, by the way. It's been awhile now; it's been a
few years. There's not a lot of us left. We're doing the best we can under the circumstances.
Me, I used to be a professor at Williams College in western Massachusetts, a math professor
before the zombie apocalypse descended upon the college. The college is gone now, so much
for tenure. But on the other hand, a lot of other institutions have disappeared as well. No
more police, no fire department, no hospitals, no super Stop & Shop, no crossing guards, no
cappuccino latte, no more organic kale. It's all gone. This very building that we're in right now,
this building used to house one of the most important corporations in America. What is it now?
It's just a place to protect yourself from zombies. It's really sad. Now some of you may be
sitting there saying to yourselves, okay. This guy has survived this long, but so have we. Why
should we listen to him? What does he have to tell us? Why does he know so much? But in
fact, it is true that I used to be an advisor on zombie affairs to the president. That's right, the
president of New Western Massachusetts. Or at least I was until he was eaten by the vice
president. Now I'm going to tell you how to survive the zombie apocalypse. The first thing I'm
going to explain is a lot of misconceptions that people have about zombies. Let's talk about
some misconceptions about zombies. Number one, they are the dead that have risen. Now
that's just patently false. They are actually people who have been infected by a virus. They get
bitten. The virus gets into the bloodstream, goes up through the blood, through the blood
brain barrier and then it gets to the brain where it liquefies most of the brain. In fact, all of the
higher brain functions reaching the point to where it now people devolve into an earlier
evolutionary state, the so-called reptile brain is all that's left. Those people then are, it seems
as if they are the dead that have risen when in fact they are not. They are actually functioning
beings at that point. Number two, zombies can live forever without sustenance. That also is
silly. There are no living creatures that can live forever without sustenance. If it were true that
zombies could live forever without sustenance, we would just take gyms all over the United
States. We would put treadmills in them. We would put zombies on the treadmills. We'd have
volunteer sitting in front of them. I forgot one thing here. And then we would generate as
much power as we ever needed. It's silly. Zombies need sustenance just like everybody else
does. Number three, zombies can be cured. Now as I've said, the zombie brain has been
liquefied and you cannot cure zombies. If you see your mother coming towards you looking like
she's just going to give you a big kiss, that's not what is going on and if you hesitate for a
moment it could be your last moment. Zombies cannot be cured. A lot of people believed they
could. That was the zombie rights movement. They're gone now. There are no zombie rights
people left anymore. They shouldn't have opened up the membership to the zombies. I want
to talk a little bit about how to survive the zombie apocalypse. I see two people here who are
dressed appropriately. You're dressed appropriately, right? A zombie can't bite through that
jacket. That's smart. Things like this are great. Hockey equipment like this, lacrosse equipment
like this, the best thing to do is go to the nearest high school, go in there. It's going to smell
pretty bad in the locker room, but look for stuff like this, great protection. The other thing you
have to worry about is long hair. If you have long hair you are running a huge risk. I did bring a
couple of these along here, anybody who needs them, bath caps. Very important, take a bath
cap, here you go. Bath cap, anybody else want a bath cap, bath cap? There you go. Number
three, now this is the thing that is really interesting and may be one of the most important
things to understand about zombies. That is zombies are not smart. In fact there are dogs that
are smarter than zombies. That looks so good. That's really perfect. [laughter]. Some dogs are
smarter than zombies, and I'm going to give you an example. This is a dog named Elvis. Elvis is
a dog who belong to a guy named Tim Pennings and he wrote this article Do Dogs Know
Calculus. It appeared at the College Math Journal, and this dog loves to fetch. Elvis just loves to
fetch and Tim Pennings lives right on the shore of Lake Michigan and so what he would do is he
would get on the shore and he would throw a ball into the water, diagonally, into the water and
Elvis would take off after it. The thing you have to know about Elvis is, here is a picture of
what's going on, Elvis can run seven times as fast on the beach as Elvis can swim in the water.
And so what Tim Pennings did is he would throw the ball in at various diagonals and various
distances and Elvis would take off and what was truly remarkable is that Elvis seemed to be
able to find that optimal point, the best possible place that you could go into the water to
minimize the time that it took him to get to the ball. Now that's something you can figure out
with calculus, where that point is, and this was as if Elvis could intuit where that point was. It's
kind of an amazing thing. Let's think about what would happen if you did the same thing with
the zombie. Here we are on the shore of Lake Michigan and, of course, if you throw a ball in
the water of zombie is not going to go after a ball in the water. That's not going to work, so
we're going to throw in Charlie Sheen. We throw in Charlie Sheen. Here's the zombie and the
zombie, what's the zombie going to do? The zombie should go along the shoreline a certain
distance and then cut into the water, but no. Every single time the zombie heads straight for
Charlie Sheen. No matter how many times you throw him in the same thing happens over and
over and over again. The zombie always heads for where the person actually is. I actually have
a video of this and I'm going to show you right now. This happened on the campus of Williams
College, to demonstrate what happens. [music] You see the zombies down from there.
Typical, typical zombie behavior. Instead of running across the front and up the stairs like any
rational human being would do, they don't have the brains in their heads to do it that way.
Instead they have to come straight for me, straight up the chairs and it's going to take them a
half hour to get here. It's very sad. Those are professors Garrity and Morgan from the campus.
I want to show you another incident that occurred, actually, very early in this whole process on
the campus of Williams College where there was a scene I was watching from a window up
above and there was a Dean trying to escape from a zombie and trying to get to the Bronfman
Science Center. Here you see the Dean and here you see the zombie. If the zombie were
smart, what the zombie would do would be to cut off the Dean because the zombie is actually
moving faster than the Dean. This happens to be an old Dean, not moving very fast and the
zombie is moving faster and if the zombie when straight for the Dean the zombie would
succeed in catching the Dean just like we're going to see right here. The Dean's moving like
that. The zombie’s moving a little bit faster and right there you see the zombie catches the
Dean. All right? Unfortunately, for the zombie’s sake that's not what happens because the
zombie always tries to head for right where the Dean is at any given moment. In other words,
it's direction vector, the zombies’ direction vector is always headed towards the Dean. The
Dean's direction vector is headed towards the science center where he's going to get safe. Let's
see what happens when the zombie is chasing the Dean and what happens as we go along. The
zombies’ vector is always pointed straight at the Dean, the zombies’ arrow. And we go along
and the Dean makes it to the science center because of the fact that the zombie is always
heading to where he was. Let's watch that in real time. The Dean is trying to get away. The
zombie is still moving faster than the Dean, but because the zombie is headed for where the
Dean was, the zombie never succeeds in catching the Dean as the Dean tries to escape to the
Bronfman Science Center. We would like to try to analyze this situation a little bit more
carefully. We would like to figure out what is that curve that the zombie went along as the
zombie was trying to chase the Dean. Let's take a look at that and then we can determine
depending on the relative speeds of the zombie and the Dean whether or not the zombie is
going to succeed in catching the Dean. We're going to look at this curve right here. We're
going to assume that the Dean is starting over here, a distance of x0 from the zombie and then
the Dean is always headed straight for the Bronfman Science Center and the zombie is always
having its direction vector pointing straight to where the Dean is. Let's move things forward a
little bit in time. We'll call the speed of the Dean S sub D and the speed of the zombie S sub Z.
Let's move things forward a little bit in time. Here is the position of the zombie at this
particular instance in time when a certain amount of time T has already passed. And you'll
notice still the zombie has its direction vector pointed straight at the Dean. We're going to call
the coordinates of the zombie x,y and that's what we want to determine is what x,y is as time
progresses. We'll first look at the coordinates of the Dean. You'll notice that the x coordinate
of the Dean has to stay at x0 because that's a vertical line right there. The y coordinate of the
Dean will just be the Dean's speed times the time. That will give me the y coordinate of the
Dean because that's how far the team has traveled at any given time T. Let's just compute
some of the differences between these coordinates. Down here you'll see the difference in the
x coordinates. The zombies’ x coordinate was x. The Dean's x coordinate was x0 so that's the
difference in the two x coordinates. The difference in the two y coordinates will just be the
Dean's y coordinate minus the zombies’ y coordinate, S sub D of T minus y. This particular line
that you see right here you'll remember from calculus. This is what we call the tangent line.
The tangent line has a slope that is the derivative. In fact, the derivative, dy dx of this particular
curve that we see right here is the slope of this particular line right here. But it's something
that you learn very early is that slope is always given by taking rise over run. That says dy dx
should be the rise over the run in this particular case which means dy dx is equal to s sub d t-y.
That's the rise divided by x0 minus x. That's the run. This is a differential equation. And it's a
differential equation that if we can solve we will then know what this curve is. This has to be
satisfied by the x and y that give me the coordinates of the zombie. Let's take a look at that
equation. We're looking at that equation. We would like to know if I have an equation in this
form, what does y look like? I want you all to be thinking in your own head I bet I have a guess
as to what that's going to be. Everybody sort of try to think in your head, okay, what's that
going to be. And we're going to call the q the ratio of speed, the speed of the Dean to the
speed of the zombie, so the relative speed, so we get some kind of parameter there that we can
use. Has everybody figured out what they think that looks like? Well, this is it. Anybody come
up with that one? It's bad. It's nasty. This is actually kind of a nasty equation, but it turns out
this is the solution for that relatively simple differential equation assuming that q is the speed
of the Dean divided by the speed of the zombie. This is kind of ugly, however, we can still use it
very effectively to determine when the zombie is going to catch the Dean giving a particular
value of the relative speed q. Let's look at that equation. I'll put it at the top there, and you'll
notice that capture is going to occur when they have the same x-coordinate. The zombie is
following the Dean along, but once the zombie has the same x-coordinate as the Dean, the
zombie will have caught the Dean. That means capture is going to occur when the x-coordinate
of the zombie equals the x0 which is the x-coordinate of the Dean. Notice what happens when
x equals x0. In that case, this term right up here disappears. That's all 0, so all this big mess up
here is gone and all and all we're left with is this. The y-coordinate of capture occurs at q over
1-q squared times x0. Let's take a look at that in various situations and see what that is telling
us. Capture is going to occur in this particular situation here and here we have this situation
where we had the Dean starting at x0 and look at the case where q equals a half. That means
the Dean is moving at half the speed of the zombie. When the Dean moves at half the speed of
the zombie plugging q equals a half in here, we get y equals two thirds of x0. That means the
Dean gets about two thirds of x0 up this line before the zombie catches him. That is not a good
situation. That's bad. On the other hand, what if q is equal to three quarters, so the Dean is
moving at three quarters of the speed of the zombie? How far does the Dean get? This time
the Dean gets to 12/5 of x0. Okay? So the Dean gets quite a bit of a ways up. That's not so
bad. And, in fact, that's far enough to make it to the Bronfman Science Center, so the Dean
makes it to safety in the case that the ratio of their speeds is three quarters. That's good news.
That's great. What would happen if q equaled 1? That means the Dean and the zombie are
actually going at exactly the same speed. In that case you'll notice the zombie is always going
to be following the Dean and, in fact, in that case capture never occurs. Y equals infinity. The
Dean the demon never gets caught. The bad news is, unfortunately, as I told you, that, in fact,
it was q equals three quarters and yes, the Dean did make it into the Bronfman Science Center.
Unfortunately, the Bronfman Science Center was full of zombies and it really didn't turn out so
well. That's too bad for the Dean. Now I want to look at a different situation. This is the
situation where I had a zombie chasing someone who was moving in a straight line. But a
particular situation that occurred when this all started on the campus at Williams College was a
circle pursuit where I was riding a bike and I was trying to attract zombies away from some
other people. The situation was this one. I'm riding a bike around in a circle. The zombie is
going to chase me but still the zombie is always going to be headed straight towards where I
am. The zombie’s direction vector, also called the tangent vector to its curve will always be
pointed straight at me. We want to figure out what that's going to look like. My coordinate
first of all we have to figure out. I'm on a circle of radius capital R. I'm going to call my angular
speed omega. So my ankle is going to be omega times T, any given time. My x-coordinate I'll
call x sub A for Adams, y sub A for Adams. And you'll notice that x sub A is always going to be
equal to R cosign omega T, basic trigonometry, this divided by this. And y sub A is going to be R
sine omega T so I can figure out very easily coordinates of my point on a circle as I'm moving
around that circle at a given speed. The zombie, so this is now the zombies’ coordinates again.
The zombies’ coordinates, the zombie is moving along a curve and I'm going to assume that the
curve, the zombies’ coordinates right now x of Ty of T, so I have a curve given by x Ty of T. And
if I want to find a direction vector that always has to point towards me, that direction vector is
simply going to be the derivative of this particular parametric equations of this curve. The
velocity vector which is a derivative x prime y prime is tangent to the curve and it has a length
equal to the speed of the zombie. That's the thing that I want to have pointed me. It's actually
the zombies’ velocity vector and also keep in mind that it has length equal to the speed of the
zombie, because we're going to use that to our advantage in just a second. The zombies’
velocity vector is supposed to be pointing at me at position A, just as I said, so this is position,
so this is the zombies’ velocity vector right here. That says this thing x prime for the zombie, y
prime for the zombie points at x of A y of A, so I now have one vector that's pointing straight at
me and that's the zombies’ velocity vector. Now there's this second factor that is pointed
straight at me. If I just take the coordinates that I have, xA yA and I subtract the coordinates of
the zombie has, that will give me the vector zA and that is a vector that point straight at me
because it is, in fact, the fact that starts at this zombies’ position and ends at my position. This
is now a second vector that points straight at me at any given time from the zombie to me. I
want to try to get rid of the length of vector. I don't like the length of the vector so I'm going to
divide the vector by its length and that gives me what's called a unit vector pointed straight at
me. A unit vector just means that its length is 1. Whenever I divide a vector by its length,
which is this here, I get a vector that has unit length 1, so now I have created a vector that
points straight at me with unit length. Again, we are going to let Sc be the speed of the zombie
as we did before and I'm going to multiply this vector by Sc and by doing that I have created a
vector that points at me. That was the unit vector that pointed me, but I've manipulated its
length to be the length of the velocity vector because I multiplied it by the speed. That makes it
the velocity vector. There is only one vector that point straight at me but also has length equal
to the speed and that is the velocity vector. We have now created a vector that is the velocity
vector right here. If I just write out what that means, the velocity vector was x prime of zy
prime of z and over here I've got all of this. This can be unpacked. I can look at the xcoordinate and the y-coordinate of this and when I do that I get the xdt, that's the derivative up
here is equal to this expression and dy dt is equal to this expression. Just by looking at the two
components of this vector and breaking it up into these two equations. And this is a pair of
what are called coupled differential equations. If we can solve this system of two differential
equations, we can then figure out what the path of the zombie will be as it chases me along this
circle. The last thing I'm going to do here is, remember, we did figure out what x sub A and y
sub A are. Those are my coordinates as I ran around the circle, so I'm going to add those into
the top. I'm going to place x sub A by R cosign omega T and y sub A by R sine omega T. They
are messy. They are ugly, but these are the two equations the dictate the motion of the zombie
as I'm riding in a circle on my bike this way. I'm riding in a circle on a bike. Great. So all we
have to do is solve these equations. I would be wonderful. It can be done. Nobody has ever
been able to solve these equations. These are very difficult nasty equations to solve, so you
can't solve them, so what do you do instead? You just run a computer using this and see what
the paths look like. Let's take a look and see what happens if you use these equations to
govern the path of the zombie. Here's the first situation. The zombie is in red and it's chasing
me as I ride around on the bike and you'll watch what happens. It seems to be sort of coming in
on little dotted circle there, that red dotted circle. Let's try another one. This time I started the
zombie up here. I'm going to start the zombie down here this time. Still chasing me, still
always trying to head straight towards me using its velocity vector it's headed towards me.
Once again, it seems to be circling in on that red circle. It's kind of interesting that it's doing
that. In fact, that will happen all the time no matter where you start the zombie. The zombie
will always have a path that slowly works its way inland towards that red circle. If you start it
over here it looks like this. If you started over here it looks like this. That happens again and
again and again. That circle is called the limit cycle. That's the limit cycle you're finding the
zombies are always ending up nearer and nearer to the limit cycle. If you start a zombie right
on the limit cycle as we have right here and you know that it's tangent vector is starting out
pointing at me right at the start, and then you let it go what happens is this. It actually follows
me around the whole way around, always stays exactly the same distance from me all the way
around that circle and ends up in the same place that it started at if I end up where I started. It
just stays right there the whole time. What's happening is its tangent vector is always pointed,
it's velocity vector is always pointed straight at me and as we go around it just stays pointed
straight at me all the way around. It never varies. It's always pointed in the same direction as
we go around there. What happens if we start with a bunch of zombies all following me?
Notice what they do. The guy almost caught me there. You'll notice that they bunch, they
bunch on that limit cycle. They all bunch up and they all do this and it doesn't matter where
they start, the only assumption I'm making that I didn't mention before, the only assumption
I'm making is that all of the zombies are traveling at the same speed. The zombies are traveling
at the same speed, but if they do that they will bunch just like this. You can use that to your
advantage as which occurred in this particular situation, that, again, occurred on the Williams
campus. There are two students trying to escape. I am going to ride my bike and see if I can
help them. Let's see what we can do here. Of course, safety first. Put the helmet on. All right.
I am going to try to attract the zombies to me. Here we go. Wish me luck. Let's see what
happens. All right, you zombies, hey, follow me. Come on, you zombies, follow me over here.
Come on. You're not so smart anymore. You used to be, but you are not anymore. Are you?
That's good. Good zombies. Come on. It's working like a charm. Nothing like an afternoon
bike ride with a bunch of zombies behind you in a pack. I'm writing in a big circle around the
quad right now. You'll notice they are starting to follow me. They are starting to group
together and the idea is the zombies are always going to head towards where I am and what
that means is that their tangent vectors are always pointed straight at me. Because of that it
turns out that no matter where they start they end up actually grouping in a clump and they
end up following me on a circle that has a smaller radius than my circle. My circle is bigger than
theirs because I am writing faster than they are moving, but I end up getting all of them to
follow me. Eventually, they will be altogether in which these people can escape. The zombies
are following me. Okay. You folks, I think you are clear. You can get out of here and get to
safety as quickly as possible. It looks like they're good. Now I just have to get to safety, so hang
on just a second. I'm going to bring it around. Come on you zombies. So there you get the
idea. You can actually use this to your advantage if you know the fact that these zombies are
going to follow you in a circle and they are going to clump together that's a huge advantage.
Now I want to talk a little bit about the long-term prognosis. It has been a few years now since
this whole thing started. We have survived this far, but how long will this go on? First, let's let
H of t be the number of humans at any given time. Let Z of t be the number of zombies at any
given time. Let's let the dHdt be the rate of change in the number of humans and let DZdt be
the rate of change in the number of zombies. Let's first think about what happened initially.
Initially, for the zombies there was unlimited resources and those resources were us. They had
plenty to eat. They had plenty of people to convert into zombies and so there was this
immense growth in the number of zombies and that was exponential growth that you see right
here. That's exponential growth. Initially, when the initial moments of infection occurred, we
saw exponential growth in the number of zombies. But that could not be sustained forever
because eventually, the number of people out there starts to drop. The number of zombies
grows and the number of people drops, and so eventually you find yourself in a case of logistic
growth and this is the case of logistic growth where you can think of this as the rate of change
in the number of zombies will be proportional to the number of zombies times the number of
people left who have not been killed or turned into zombies. We're losing a lot of people here.
This is still proportional to this because this is really keeping track of the number of interactions
that there are between zombies and people. If there are very few people left then there's not
going to be a lot of interactions. If there are very few zombies left there won't be a lot of
interactions, so if either of these two terms gets small there won't be a lot of interactions. If
both of these terms are reasonably big you'll get more interactions and you'll get more people
turning into zombies and the growth in the number of zombies will continue. If I multiply that
out and then think about what's going on, here is the curve that comes from that so-called
logistic growth curve. And you'll notice this first term. When Z is small this term is a lot bigger
than this one because these squares are going to be smaller than Z so this is a big term and this
is actually exponential growth. That's the growth that you saw initially is right here; that's the
exponential growth you see in the beginning, but then it tails off because of this term right
here. In fact, you can see it better on this one. As soon as P0 minus Z gets small the Zdt is
going to go to 0 which means this thing has to flatten out. It has to go to have a derivative of 0
which means it is getting closer and closer to a derivative of 0 which means it is going to tail off
with time. That's the medium long term. Now I want to talk about the really long term. Sorry,
this is the equation that you get from that logarithmic growth and so that's logistic growth.
Now I want to talk about the greater long-term situation. This is very long-term. It turns out
that these two differential equations that give me the rate of change in the number of humans
and the rate of change in the number of zombies govern what's going to happen. This first
term I want you to think about if there were no zombies, whatsoever, and humans have lots of
resources, then in fact the rate of growth of the number of humans is just going to be
proportional to the number of humans. Humans have a tendency to breed and as they breed
they create more humans. That's what that term represents. But, of course, there is
interaction between humans and zombies once zombies are around and every time there is a
human zombie interaction, not in every case, but often in those cases we lose some humans
each time that happens which is why we have this minus beta times H times Z. That's the rate
of change in the number of humans. For the rate of change in the number of zombies, if there
are no humans around the number of zombies will drop and the reason the number of zombies
will drop is because as we said before they have to have sustenance in order to survive. They
do starve to death. Zombies can starve to death unlike what a lot of people assume. You will
lose zombies if there are no humans, but every time there is a human zombie interaction that
can increase the number of zombies and so that's why we have this term right here. These two
equations are supposed to govern what happens to the number of humans and the number of
zombies. Let's look at those two equations. I'm going to factor a capital H out of the top one.
I'm going to factor a capital Z out of the bottom one and then let's look at the critical points.
Critical points is where the rate of change in the number of humans and the rate of change in
the number of zombies is 0. Those numbers are not changing. Those critical points are going to
correspond to solutions of this and there are two possibilities. You could have both H and Z be
zeros. That would certainly make this true. And you can have H equal gamma over delta which
would make this 0 and Z equals out for over beta which would make this 0. Those are two
possibilities for so-called critical points where we would be in stable populations. Let's talk
about the first one. When H and Z equals 0 it is definitely true that the rate of change in the
number of humans and the rate of change in a number of zombies are both 0. The good news
is the zombies are all dead. The bad news is we're dead too. That is not a great situation. We
don't really want to be in that situation. We would like to analyze what happens in the other
situations, so I'm going to use this fact that by the chain rule I can write this dZdt in this way
and then I'll divide through and I'm going to look at what is the rate of change of the number of
zombies depending on the number of humans. I'll get that by taking the equation I had for dZdt
and dividing it by the equation I had for dHdt, so I get an expression that looks like this. I am
now going to look at that in the ZH plane, so each point in the ZH plane will correspond to a
particular number of humans and a particular number of zombies and this will correspond to a
slope. I'm going to draw those slopes, what's called the slope field in the HZ plane. H is on this
axis down here. Z is on this axis over here. You remember that critical point which was Z
equals alpha over beta. H equals gamma over delta. That's this point right at the center. But
at these other points I can have horizontal slopes here. That's when dZdt equals 0, all the way
along there. Over here I have vertical slopes. That's when dHdt was 0 and so dZ over dH will be
infinity, and then I can also draw some other slopes. This slope field exists and from that slope
field I can create solution curves that just touch. These are the tangent lines to the solution
curves and they look something like this. We want to think about what happens if we start on
one of these solution curves and we're just going to travel around it as the number of humans
and the number of zombies varies. We're going to start at point over here and so here's the
situation. This is the situation we started in. Initially when the whole zombie infection started
there were very few zombies and there were a lot of humans. We were right here. Then what
happened was the number of zombies started to increase very fast, so the zombies went up
very fast and as the number of zombies went up the number of humans started to go down. As
the zombies increased in number the humans started to drop. We found ourselves in the
position like this, lots of zombies and very few humans. Then because the zombies had nothing
to eat the number of zombies started to drop. They started to starve to death and they didn't
have anyone to convert to become zombies and so this thing started dropping down here and
we found ourselves in a situation where the number of zombies and the number of humans
was low. Now in that situation because there were so few zombies the human race had a
chance to recover, had a chance to procreate and the numbers started to grow back again. And
this is where we are today. We're back here at the point where we started at where there are
very few zombies but very many humans. Now, of course, the zombies are poised to make a
comeback because there's humans to eat and humans to convert and were going to find
ourselves going around this path again and again and again, so we find ourselves in a situation,
in a cyclic situation. It's a cyclic situation where the number of humans is large and then it
drops down low. The number of zombies is large and then it drops down and we just keep
going again and again and again like that. Of course, this is the case that we care about the
most, but there are other situations where this occurs. A very famous case on Isle Royale
National Park which is in Lake Superior, this is a national park in Lake Superior and that Eileen
had been colonized by moose and the moose had lived there for a long time and they were on
the order of 2500 moose on that island when one day two wolves swam across to the island
and established their own little colony on the island and they started eating the moose. The
moose population suddenly dropped and as the moose population dropped the wolves were
thriving until the moose population got too low at which point the wolves started starving to
death. Then the moose had a chance to recover, which they did, and then the wolves had a
chance to recover and this cyclic property, these so-called predator prey models occurred
actually in this particular situation and they have been keeping track of the number of wolves
and the number of moose now for close to 100 years, I believe it is and they see this cyclic
behavior that occurs right there. It's not just for zombies that this occurs. Anyway, we're going
to be trapped in this situation of this cyclic relationship between the number of zombies and
the number of humans for who knows how long unless we can be smart. That's the trick. I'm
just going to end with a couple of credits here. Keep in mind, try to be clever and then use the
zombies’ lack of intelligence to your advantage.
[music]. Thank you very much for listening. I really appreciate you coming and listening. Thank
you very much. [applause]. Thank you. And I would be happy to take any questions or
comments or suggestions as to how to survive the zombie apocalypse, anything you find useful.
Yeah, please.
>>: You were talking about the zombies as opposed to their relationship with the wolves and
the moose at the national park. And you get to the point where two wolves actually swim
across and I assume there was a similar thing when the zombies started destroying us all it was
a research monkey in a lab that got out and 28 days later or whatever, is there a way to
integrate some of that into mathematics? Like here's the unknown that may come in and
here's the unknown that may come in and actually destroy that, the loop that you had
[indiscernible]?
>> Colin Adams: Once the loop gets established as in the case of the wolves, as in the case of
zombies, in the case of the zombies it was the lab at Harvard where they were experimenting
and maybe did some things they shouldn't have done. But once it's established than that loop
can continue and as you can come up with something else. Maybe you're suggesting doing
something else to disrupt the loop? Is that what you mean?
>>: [indiscernible] zombies, you know, they would eat the zombies then and then you would
have predator prey predator prey interacting and I wonder if that would create a more complex
loop.
>> Colin Adams: You know, sometimes when you try things like that you regret it later.
[laughter].
>>: Superduper zombies [laughter].
>> Colin Adams: That's a better idea. Maybe, I mean maybe it would work, I don't know.
There is some risk in creating super zombies I think. And the other thing is that zombies have
an evolutionary avoidance of eating each other. That's one of the things and so if you are going
to come up with a super zombie it might be difficult to do because it's to their evolutionary
advantage not to eat each other which is, in fact, the reason that these viruses that infect them
don't allow them to do so. That's my understanding of it.
>>: Thank you.
>> Colin Adams: Other questions or comments? Yeah?
>>: What happens to people that don't pass calculus?
>> Colin Adams: They're all gone now, actually. It's very sad, but they're mostly gone. Yeah, no
it's…
>>: I try to tell my kids that, so thank you.
>> Colin Adams: That's the hope is that people will take this to heart and learn calculus so that
they'll be safe in a situation like this. There are some interesting questions, actually, that come
up. Here's one that somebody asked me when I gave this talk a couple of weeks ago actually
and I think it's an interesting question. We know what happens when you take a circle and you
ride in a circle and the zombie’s heading straight for you. We know what the path of the
zombie does. It moves in on that perfect circle limit cycle. What if you are on an ellipse? Just
change it a little bit. Instead of being on a circle, you are on an ellipse. You sort of squashed it
down a little bit. You're going around the ellipse. The hard part is it's even hard to figure out
what the parameterization of an ellipse is if you are going a constant speed. That's not so easy
to even figure out, but let's say you are doing that. You are going on a constant speed. You're
on your bike but you are riding in an ellipse. What path does the zombie take? Is there even a
limit cycle? I don't even know the answer to that, quite honestly. It's not even clear that there
is a limit cycle, and if there is, what is that limit cycle? What does it look like? And it's kind of
interesting because when you play around with it you can see that it does some very weird
things and so there are a lot of questions like that that nobody has ever thought about, I don't
think as far as I know. The mathematics of the zombie pursuit, it's a whole new field. Yeah?
>>: Kind of, not a higher level question, but some of the references you made to the Math
Gene by Devlin. I think he wrote a book about the idea that in the case of the dog that would
go and catch the ball, and so I know there has been a lot of research about this and there is talk
about whether we are actually doing those equations in our head natively or something that we
have actually evolved that actually does it. In the case of your example of the zombies, why
they go in straight lines, which makes all this possible, what part of that brain do you think
would be the thing that is actually destroyed and prevents them? I mean, you go from the
mammalian to the reptile brain. The dog has a mammal brain so it's able to actually do it.
>> Colin Adams: It's really interesting because although it's true that Elvis is smart enough.
Elvis is a really smart dog. I should say Elvis was a really smart dog. Elvis passed away last
summer. It was very sad. Elvis was a really smart dog and dogs when they chase a rabbit do
not try to cut off the rabbit. They always head for straight where the rabbit is, in fact. So the
vast majority of dogs don't know that they should try to cut off. It's tricky with a rabbit, I guess,
because if you had to where the rabbit is going to be, the rabbit often goes the other direction.
The rabbit is smart enough to then head in a separate direction, so the rabbit is not going in a
straight path. But dogs tend to head for where the animal is and most animals when they are
hunting will do that. They won't cut off the animal. They will head for where the animal is at
that particular instant.
>>: Have you seen Jurassic Park and the raptors [indiscernible]
>> Colin Adams: Yeah?
>>: Can math provide hope for a professor that is slower than the professor that actually made
it to the center? Because if zombies are just generally faster…
>> Colin Adams: Yes. Certainly, one of the things that we talk about in the book is the fact that
if you think of the various speeds that people have as a bell curve, as a normal distribution, that
there are certain ones and one of the characters argues that we shouldn't try to help these
people who are down at this end of the bell curve because they don't have a chance, that if you
are moving slower than the zombie then you are in trouble. And I think that's true. I don't
know, there's no curve that you can create that will protect you if you are slower than the
zombie. Eventually, the zombie will catch you, so you are stuck especially if the zombie is
headed straight for you. If the zombie was doing something else you might be able to figure
out something. I suppose if the zombie was always heading for where you were you could
maybe turn around and go the opposite direction before the zombie realized it you could make
it longer for it to actually, yeah, but in general, no, no solution for that. I'm afraid that group is
mostly gone by now, unfortunately. That's true.
>> Amy Draves: We have questions from folks online. One of them is just the you know the
history of the word zombie?
>> Colin Adams: I don't know the history of the word zombie. It's very interesting because I've
gotten a little bit of flack because I picked a particular model for zombies which is not
necessarily the standard model. The standard model is they are the walking dead and they
don't need sustenance and they essentially live forever until their bodies decay away. I needed
this model because I wanted to be able to do predator prey models and I wanted to be able to
have them starve to death and so I needed, for my equations to be able to work, I needed this
particular model. But some zombie aficionados were very unhappy about that, so I obviously
am not as knowledgeable about zombies as some of the other people who are out there who
take it very seriously as it turns out.
>> Amy Draves: The other question was around, we're having a little bit of an argument
around, when I'm under attack by a zombie how will I have time to stop and make these
calculations? Is there a simple tool that will help me figure it out on the run? So I suggested he
study ahead, study up now.
>> Colin Adams: I think somebody has to create an app. Somebody's got to create an app to
protect yourself.
>> Amy Draves: I'm not going to remember unless I'm constantly doing it. It's just not practical
to practice it every day. Do you have any suggestions as to what to do when the apocalypse
hits other than studying calculus books?
>> Colin Adams: Obviously, my recommendation is studying calculus and if you can keep
yourself alive long enough while you are studying calculus it will be to your advantage,
definitely. Beyond that, there are various things that people say. For instance, they say move
into the second floor of your house and then destroy the staircase. I think that's true because
zombies can't climb. In the book I have the characters actually move up to Canada because it
turns out that the cold temperature causes the zombies to hibernate because they didn't
devolve into this earlier evolutionary state and then that allows me to do Newton's law of
cooling, so of course I manipulate things to do the things I want to do. And what they do is they
move up to Canada and they build tree houses and they live in tree houses up in Canada where
the zombies can't get them and where they can pull up the rope ladder at night and they are
safe. So little things like that I think that you have to use to your advantage. Yeah?
>>: Was a coincidence that you mentioned reptile brain? That sort of scientifically is not
accurate but it's a widely read book among trial lawyers.
>> Colin Adams: Is that right?
>>: Yeah, it's called the reptile and it's that you want to pick a jury and then make your trial
arguments based on the fact that you are appealing to the alleged reptile portion of the brain.
And the way you stated it is very similar language. But it's not true at all.
>> Colin Adams: It's not true at all but there is an underlying earlier… I still believe, you can tell
me I'm wrong, but I still believe that we did evolve from animals of a much different type and
there's probably some portions of our brain that came from that, right? [laughter]. You know,
I've always loved that idea, but you don't like it. I'm shocked by how similar my thought
processes are to my dog’s. I'm shocked by that I'm shocked by how smart my dog is and how
the relationship that we have and the fact that we can have all that and I am convinced that
that is because we evolved from similar animals. My guess is that there really is some stuff
that's left over from some point that formed the basis of what became human, although, it
might not be a reptile necessarily.
>>: [indiscernible] trial lawyers [indiscernible]
>> Colin Adams: I can consult. I'll make a fortune. It would be great.
>> Amy Draves: We have time for one more if anyone has one? Otherwise, Colin, thank you so
much.
>> Colin Adams: Thank you. Thanks for coming everybody. Thank you. [applause]