We're going to do uh, predator-prey
cycles. We're going to do rabbits eating foxes, and we're going to try and use maths to- - (Brady: Rabbits are gonna eat foxes?)
Okay, we're not going to do-
Well, I mean, maybe? We are doing foxes eating rabbits.
Can we mathematically model and explain and predict the changes in the population numbers of that interaction between a predator,
in this case a fox, eating the prey, a rabbit. u are going to represent our prey.
(Brady: Why u?) We tend to use u and v
a lot in maths. We can't use p because it's prey and predator.
So u is going to be the the rabbit, the prey. Here we go - attempt to draw a rabbit. Some ears, a leg.
Do they have whiskers? All right, there's a rabbit. I think that's a pretty good rabbit. - (Brady: That's not bad)
Okay, and then v is going to represent our fox.
They've got like a big tail, a body, pointy ears and like a pointy no- oh god!
Is that a fox? We have a rabbit and a fox and we want to try and write down
equations that are going to explain the interaction between them. You know,
the rabbits will keep growing and populating but as they do there's more food for the foxes
so the foxes might start to grow and then there's too many foxes.
So there's like a complicated interaction between the two.
And that's sort of ultimately what leads to this really nice predator-prey dynamics in in the model.
We want to try and figure out the change in the population
of the prey
over time. So that means writing down a derivative. So it's the change in the population, du,
with respect to time. If we have more rabbits
we expect they're you know, they're going to repopulate. This is, you know,
it's, it's a actual metaphor or idiom or whatever, isn't it about like rabbits?
(Brady: They're gonna breed like rabbits?) - Breed like rabbits, there's the word! I was thinking of something else.
So we're gonna breed like rabbits. If we have more rabbit this is gonna grow. Population gets bigger,
we get more rabbits. - (Brady: The more rabbits you have the more rabbits you get)
Exactly. The change in the population will also be affected by the foxes.
And that's going to be a decrease.
If there's more foxes to eat the rabbits, they will go down.
So that's going to be a minus. Because it depends on both fox number and rabbit number
it's going to be a u times a v. So that sort of shows that this is the rabbits growing by themselves,
and then this is being eaten by the foxes. So we can put here reproduction for this term,
and then this is what we would call the predation term. So our rabbit's being eaten by foxes.
Now if we want to write down the same for the foxes,
so we're going to start with the rate of change of the fox - the predator - population
over time. Now, the foxes will grow when there's more food. - (Brady: Because they're not as prolific at breeding?)
We're going to put in a constant,
let's just call it alpha. And this is going to be uv, this is a plus, because it's going to increase the fox population,
so more food available. And then now what you're going to see
is the negative bits that causes the fox population to decrease is when there are too many foxes. Because if there are loads of foxes
there's a lot of competition
for catching and eating the rabbits. Whereas the rabbits don't face that problem, because we're assuming there's always enough lettuce.
(Brady: Grass, or...?)
Whatever rabbits - I used to feed my rabbit lettuce. - (Brady: Yeah, but not in the wild!)
All right, grass. So whatever it is the rabbits eat
there's going to be, we're going to assume there's lots of it
so they're just going to keep growing. But the foxes are going to decrease when there's more foxes because there's more competition
to eat those rabbits. So this is going to be minus
alpha times v. Now the alpha here is just a constant- - (Brady: The same alpha or?) - It's the same,
it's the same alpha.
So I've kind of skipped over a step because what we've actually done, is if you wrote out the full model
you'd have different constants on each of the four terms and then you do
the classic applied maths trick of
non-dimensionalise everything to get rid of all of these constants.
So I've just skipped over that and all, all that you end up with when you do that, and you can work through this,
is you just get the one constant, which is the same thing. This is now going to be our model,
so this bit here is competition.
So we've got the rabbits, they grow because they reproduce, and they go down because there's more foxes. The foxes, there's more food available
so they increase with the rabbit population, and they decrease with their own population because of competition.
So now we have a set of equations. du by dt depends on u and v,
and dv here depends on u and v.
So this is what makes them what we call a coupled set of equations. Because you can't just solve this one.
Because you need to know v. But to get v you have to solve this one and you need to know u.
So this is like the the sort of complex dynamics between the predator-prey because they are so
interdependent on one another. So normally to solve a differential equation you do an integral, you use some, you know tricks of the trade.
But in the predator-prey interaction in math biology, you just draw lots of really nice pictures and it's super cool.
So that's what we're going to do. We're gonna try and draw out
what does this look like in what we call the phase plane. Our two axes: so now
there are a few things we can draw straight away here.
So the only information we have to construct our phase plane is going to be these two equations.
So a good starting point is think about what happens on the two axes.
So if we plug in
v being zero, is gonna be our starting point. So v being 0 is the
u axis here. So if we plug that in, that
disappears so that tells us that the change in u
is equal to the value of u. So if I'm, let's say here, at some positive number of u,
v is zero so that term disappears. And now I've got the change in the population is equal to a positive number, because I'm
at a positive number.
So the change is positive - means it's increasing.
So if I'm here, I have to be going that way. There's nothing else I can do, I'm just gonna keep going along here.
Which makes complete sense. There are zero predators.
So if you're a rabbit, you're just breeding all day long and growing and growing and growing your population. Similarly
along this one we have that u is zero,
so we have no prey. Now we'll come at this from the biology angle.
We've got no prey and we have some number of predators, some number of foxes. They're a bit screwed, let's be honest.
They've got nothing to eat. They can't go and eat that lettuce
so they're going to decrease and you're going to go down like that. And you get the same in your equations because you take this,
plug in u equals naught, you get rid of that term and you've got the change in v equal to something negative.
So you're going down. So now we actually want to go back to our equations and think about things that we call
steady states. So this is a particular population number
where
both the prey and the predator, the rabbit and the fox, are happy. It sticks at a constant number,
there's no change. Once you get to this point, this particular number of rabbits and corresponding number of foxes, once you get there
everyone's just happy and there's no change. And we call this a steady state.
And so what that means mathematically in this equation
is that u will be equal to some constant number. And so if u is constant it doesn't change with time, because it's constant,
that's the sort of definition of constant. So that term goes. So to find which values of u
our population is happy at we just solve this equal to zero.
So we've got to solve u minus uv
equals naught. We need to find values of u and v which will satisfy this equation.
So the first one would be u can be zero, then that works. And then the second one you could also have,
if u is not zero,
then if v was equal to 1 you'd have u minus u.
Now for a proper steady state, remember, we needed both populations to be happy.
So they sort of, that's why it's called a steady state,
they reach this point and they just sit there and just bob along without changing.
So if this one doesn't change we've got the same idea here. This is alpha times v
times u minus 1 must be zero. So if u is naught, that was option one,
then that's naught,
then the only way for that to be zero is for v
to be zero. Because the u term has vanished. So if u is naught, v is naught.
And that is one steady state. And if v is 1,
that's a 1. So u has to be 1 to make this zero now. So u is 1. So these are our only two possible
steady states. These are the two values where our population is happy. So now we can go back to our phase plane.
So we have one of them here. And now this makes complete sense
for no predators and no prey to be a happy place for the population.
(Brady: There's nothing there?)
There's nothing there. So it ain't gonna change. There's nothing that can be - you know, they can't reproduce, they can't kill each other,
it just ain't happening.
It's just so nice when you do the maths and you get this answer like oh, yeah
the population is happy when there's nothing. And then you think about it, as you said, oh, yeah: there's nothing there, great.
That's exactly what we expect. It's a really good sanity check where you're doing the right thing
and your maths is correct. So we can either have that and they'll happily sit there,
but then what we can also have is one and one. What you can now do with this
particular steady state is something called a stability analysis.
This is a bit more advanced, um sort of a dynamics or mechanics type thing.
And what we can work out is we can work out the behaviour around this particular point. It turns out that in this situation
you get something which is called a centre. And near this point of the population the population
pathways look like this.
So you do this stability analysis, and it tells you that these things go round in circles.
Now we can actually complete the rest of our phase plane, because the way the phase plane works is these
arrows tell you how the population behaves when you start at a certain number.
So we said here if I start with this number of predators the phase plane says I'm just going to decrease and die out.
Which makes complete
biological sense because there is zero prey here. So they all just die. And similarly if I start here, where there are no predators,
I'm just going to increase forever. Now we've done our fancy stability analysis to say that near this point
they sort of go around in circles.
But we don't yet know which way around they go. And what we have to do to fill that in is look at the rest
of the phase plane. Because there's a really important rule about phase planes that your trajectories,
these pathways you're drawing, they can never cross. And all of the directions sort of follow.
It's like a flow. So you have water going this way and water going this way,
it wouldn't make sense to be sort of going like that.
It would be going against the flow. Each pathway influences the pathway next to it. So what's actually happening here is you've got really big
cycles, looking something like this, going all the way around
this particular point. This (1,1) steady state situation. And so this is all consistent, everything fits, all of these directions work.
(Brady: Yeah, I see what you mean. It feels intuitively right, but up the top there)
(the, the rabbits are going the wrong way-) - If you think about what's happening here, so they're going opposite to down here
but they've crossed through this sort of threshold- - (Brady: That magical pole)
That magical stability point. That, that bit where the population's happy.
So, so what's actually happening up here is if you sort of approximately look at the numbers, your prey is on around 1
and your predator is like way above that. So there's, there's a lot of predators per rabbit.
So that's why it kind of makes sense for the rabbits to start coming down again,
because you've got more pred- you've got a surplus of predators.
So as a rabbit, you know,
you've not just got one person after you've got like several. So we can now, now we've got this phase plane,
what we do is we can pick a point on here and then follow the arrows and that will tell us how the
population of the rabbits and the foxes will change in relation to one another.
Now that we've got the phase plane. So even though we haven't solved any equations, we've just turned our equations
into a picture and the picture then gives us the answer.
So we now think about uh starting at a certain point and what will happen to the rabbits and the foxes. Let's say we start
here. So this is going to be population number and this is going to be time. Because this is ultimately what the real question was
at the start. We want to know how do their populations change over time. - (Brady: Population of rabbits or foxes?)
We're going to do rabbits first. As you move anywhere on this diagram both of them change because you're in your two coordinate system.
We start here. We're at about 1, right, so it doesn't really matter. Let's just call that our starting point at the beginning. Now
what's happening?
We are increasing, increasing, increasing, increasing just in thinking about what's happening to you our prey. So we increase
and then we get to some maximum value of u and then we start to come back down.
And we go all the way down past where we were, just looking at the rabbits. And then we go below where we were,
hit a minimum, and then we come back to this point. Okay, probably looks quite familiar.
So it's some kind of oscillation,
it's like a cos or a sin or something, some kind of wave-like behaviour in the rabbit population.
And then we're gonna just continue to go round and round this
because we can't cross, we can't leave, it sort of
goes round in circles. So we just continue
to do this. So this
is the prey - the rabbits.
There you go, rabbits.
Now, the foxes. So at our starting point the foxes are below 1. It doesn't really matter where, but they're below this value.
Let's say they start down here. As we move around they slowly
increase, very slowly increase,
then they increase a bit faster. They slowly increase,
then they come around and we get to a maximum all the way up here
and then they come back down to, it turns out approximately this minimum point. So we've got a similar behaviour that we saw with the
rabbits, except when the rabbits started here,
which is sort of halfway between their max and min, the foxes are starting approximately at their minimum value. So they have the same
oscillation. So this is the predator, spikey ears and little teeth.
I don't know if that's a fox,
it's like a vampire. Fox, rabbit. As we go around this particular loop the rabbits
start to increase and increase and increase. And as the rabbits are increasing there's quite a lot of food.
So the foxes are like oh, this is great. There's going to be more and more foxes because there's lots to eat,
so there's plentiful food and supply, they're going to reproduce lots,
they're gonna have lots of kids, or whatever little baby foxes are called. And then they're going to start to increase, but then as there's
more foxes
the rabbits have a lot of predators after them.
So now the rabbits are scared for their lives and the rabbits are getting killed a lot
and so the rabbits start to go down.
But the foxes are still increasing. But now the rabbits have gone down, there's loads of foxes, there's not many rabbits;
the foxes are gonna die out. They've got a food shortage now. So they had a food surplus down here, now
they've got a food shortage. So down here there's more rabbits than foxes,
and then at this critical point there's now more foxes than there are rabbits.
(Brady: Even when the rabbits started falling, for a while it was still good times for them.) - It was, it's like a lag.
It's lagging a little bit behind. So they're both doing this oscillation of up and down
but it's just, the fox just lags a little bit behind the rabbit because until, you know,
if you imagine it's one rabbit per one fox, until you cross through that it's always good for the foxes,
there's always a surplus of food. There's more food than they need. So they're just going to keep reproducing.
And then, you know,
perhaps it's due to uh the sort of gestation time or something, for the for the baby to be born,
but then it sort of, it lags behind a bit and then all of a sudden the rabbits have gone down
'cause there's too many foxes and then the foxes see their decline. - (Brady: And you've drawn these - on, on this graph)
(they're, they're at similar sizes but presumably the absolute numbers could be different. It could be a billion rabbits to a thousand foxes-)
Absolutely, absolutely. So that's the, the reason we've done the non-dimensionalisation
right at the very beginning.
So this sort of happy place is (1,1).
That doesn't mean one rabbit and one fox. That sort of means the one here is like
the perfect number of rabbits. And this is the perfect number of foxes. We've sort of scaled it out in that non-dimensionalisation.
So I've got rid of the units. I would imagine there would need quite a lot of rabbits to sustain a single fox. The
non-dimensionalisation - it maintains the really important stuff. So it tells you that you have this oscillation, this
predator-prey cycle. It tells you that's going to happen, but it doesn't tell you the exact numbers, but like
from a sort of
population dynamics idea you don't really care too much about the exact numbers.
You just want to know that there's going to be peaks and troughs in each of the populations. And you can look at actual data
for any, pretty much any predator-prey
situation in nature, you see exactly this. You have this, the boom
in the prey followed by a boom in the predator,
and then the decrease. Exactly
in this setup. And it all just follows from sort of those simple equations
we wrote down at the very beginning and then drawing the pretty picture of the phase plane.
[Preview] Just to emphasise the the level of favouritism of these equations,
the full Navier-Stokes equations as written down on our piece of paper. - (Brady: Hang on which way around are we?) - If we're talking this-
if - yeah. So those should be exactly the ones I've written down. - (Brady: There we go.) - The little guy and the big guy.
That's right, yeah, and just zigzagging back and forth, um sort of living around zero but in a really complicated way.
And so the thought was that, okay if-