Squirrels. Squirrels. And depending on how you feel about them you may be thinking, yay, these lovely
cute little critters or you may be thinking of rats with fluffy tails. But however you feel
about them I'm gonna be talking about the maths of species conflict. There are two species of squirrel
in Britain: there's the red and the grey. And the grey have supplanted the red and pushed them out,
primarily up to Scotland, and some there's parts in in North Wales. Dominantly if you see a squirrel
in the UK it's going to be a grey squirrel. So how do we understand that? Well we try and
write down interaction equations between the two squirrel populations. We're going to be looking at
the grey squirrel and the red squirrel population, and we're going to write down some equations that
tell us how they interact. Grey squirrel population changes with time, so t is going to be time, the red
population is going to be changing over time. Now we can include space in this and understand how
they were pushed further north and further north but generally we don't really need to we- if you're
considering an island we can just consider how that conflict occurs. So we'll we'll throw away
space for now and just consider an island of grey and red squirrels. And what we say is that the grey
squirrels undergo binary competition, as do the red squirrels. In fact, the equations I'm going
to write are going to be pretty damn similar. So all this says is that the population of squirrels
grows proportional to how many squirrels there are, but if there are too many squirrels they compete.
And that could be for resources, partners, homes,
nuts - can be here for anything. But there's some competition keeping their population finite. Same
with the red, exactly the same. But we're going to add in a new term and we're going to add in
alpha GR here and beta GR here. Minus alpha GR is the competition of the grey and the red for the
grey squirrels; beta GR is the competition for the grey and red squirrels for the red. It's how they
interact; so it's always going to be a negative effect, the competition is never going to help
the species, but the size of alpha and beta are going to be how good they are at competing. Now
that's a lot of maths, I fully understand that, but one thing I love about mathematical biology
is that you get so far and so much information from drawing pictures. I- if you can- you know a
picture is worth a thousand words, and so if you can draw a picture that immediately shows
the answer to these equations you can convince parliament - policy makers - how to save the
red squirrel. So what we're going to do is we need to understand a little idea called a nullcline.
The nullclines in this are where the populations aren't changing, okay? So when we set these to zero
that's when the nullcline is. So we set this left hand side to zero, we have some conditions linking
the red and green- red and green? The red and grey which will give us some curves, and they are the
nullclines. So in this first case here we have G equals zero when G is equal to zero, that will be
satisfied, or we could have 1 minus G minus alpha R equals zero. When Gt equals zero, that's the grey
squirrel nullcline. This one: if R is zero that's also satisfied, we could have 1 minus R minus B
G equals - and that's the red squirrel nullcline. Now what we can do is rearrange these and what we find
is - if we put grey as the dominant population - we can rearrange this to 1 minus alpha R and this
one would be then G is equal to 1 minus R over beta. Okay so they're just two lines. So I have four
nullclines, two for each population. One G equals zero and g equals 1 minus alpha R for the grey
squirrels; R equals zero and G equals 1 minus R over beta for the red squirrels. Lines: so G equals
zero is a fantastic line to draw, as is R equals zero, and these are just two lines connecting G
and R, G and R. Well there are different ways that these lines can cross. Understanding how they cross
predicts what will happen with these populations. You can do all this analytically, you
can work it out with numbers and show that all of this is concrete but when you can draw
pictures why would you do that? So I'm going to be looking at my population uh in terms of two
axes; we're going to have the red population along the x-axis and the grey squirrels along the y-axis.
You'll notice that these lines are negative so as I increase R g decreases. So for the grey squirrels
we have G equals zero, so this is one nullcline along there, GT equals zero; and the other line is
somewhere - don't really care, it's just a line. Now for the red squirrels: we have R equals zero, that's
up here, and then again another line. But does my line cross like that? Does it cross like that? Does
it cross there? Or does it cross like that? There are four different cases we have to consider and
they each tell us something different about what could happen. This one is the red squirrels along
there is zero, and so let's put the red nullcline underneath the grey squirrels and along that line
R dot is equal to zero. I'll quickly explain this: a dot can stand for Rt, just a simple way so I don't
have to keep writing t. And so along here we have G dot equals zero. Now we could we could draw lots of
more detail on here but from this setup of the nullclines the G is dominating the dynamics. And
the other thing we can say is, where is the system going to go? I will pick a point in here and that
will be a setup of the system; the system under these equations will drive that trajectory to
a place where it will stop moving. Where could it go? Well if it stops moving then both the reds stop
moving in time and the greys stop moving in time. So wherever our red and our black line cross that's
a potential place where this trajectory could end up. Okay so there are three places it could end up,
it could end up at (0,0), that's the case when my red and grey populations are extinct. It could
end up here, that's where my great population is extinct and my red survive. It could end up
here, my red die out but my grey has survived. And in this setup of the nullclines it's actually
the grey that's going to dominate. So all of my trajectories are going to end up being pushed
towards this one. And the reason comes from this line being the furthest one out, it's dominating the
system, the grey squirrels are out-competing the red okay. And that's why this line is furthest out.
Right so let's- that's one setup we could have of these nullclines. The the grey line does not have
to be the furthest one out, because the red line is furthest out. So then guess what? The red squirrels
will dominate and they will head to the steady state. Why don't we ever stop at (0,0)? Because
there's a positive population force, so no matter how small your population starts they will grow. So
what we say is this is an unstable equilibrium. If we have no squirrels to start with we never will,
but any small amount of squirrels will push this away. So it's really these two that we have to
worry about. So what will happen here is the population will tend to this one. It's because
the red nullcline is furthest out so the reds are dominating the system; they're
out- competing the greys. Now, okay it tells us that can happen, but that's obviously not the case that's
happening in the UK unfortunately. Again that's not the only two ways that this could happen.
What's another way that they could cross? Well we could have grey squirrel nullcline there and
the red squirrel nullcline there. This is one of the interesting cases, because again you'll
have the cases of: everything extinct, red extinct, grey extinct but now they're crossing at a new
point we didn't have previously. So we have a fourth solution. And now in this setup we
actually find that this state here is stable, it's attractive. Because there's no dominant pairing, we
don't kill off either of them, they can actually survive in harmony, so the trajectories will push
the populations here. Again, unfortunately not how the UK is working out. So what we can say, what's
happening in this case is alpha and beta in our equations initially are small, we could we can say
that. Because what's really happening is they're not really competing. We could take this to a limit
and say alpha and beta are zero, they're really not competing at all. They're living together
but they're each doing their own thing. And in that case what you find is they quite happily
live together, you have your nuts I'll have mine, and we'll all live happily families, small
competition. What happens when competition is big? Well this gives us our fourth case; so there's the
grey squirrel, and this is the red squirrel. Again we have that fourth one but we've swapped the
nullclines so we have there, there there and our fourth one. Because we've swapped the nullclines we are now
in the case when the competition is very big. Both are trying their best and in this case the steady
state isn't stable anymore, it's unstable. So in this case we have the four points again that
we're going to head to but this time it's pushing us away from the state. So our population out here
will push us to the case where the red has died out; and if we start out here it will push to the
case where the grey is died out. And again if we start with a small batch of both we'll be sucked
in. Now which one will we choose all depends on where we start. So it's a very hard thing. You
need to know what your setup is to know will the red win out or will the grey win out.
Now of course in Britain unfortunately we are in this case here where the greys are easily
dominating the reds and the greys are winning out; but from that piece of mathematics we get to
see the whole idea of what biology can produce. We can produce cases where you're interacting
peacefully; you're interacting aggressively - you're out trying to out-compete each other. You're trying to
out-compete each other separately; that one of the cases that you're not even trying, you know your
the greys are really winning over the reds, here the reds are winning out over the greys. But through
that one equation we can see the entire plane of all mathematical competitive.
(Brady: You said we're now in this situation here. Is it possible it started in that situation there though?)
Ok yeah, I mean the greys have always dominated the reds; they're they're better hunters, they're bigger,
they're more aggressive. So- but the question is could we ever move it to say this? Or perhaps,
you know, if we want to eradicate the grey - I'm going to sit on the fence, I I love all animals -
but do we want to eradicate the grey? Or do we want to allow them to live in harmony?
Well in that case that tells you exactly what you need to do, you need to change the dynamics,
you need to change their competitive abilities. Either you swap the nullclines around, you make
the reds more aggressive in some way; or you move it such that they don't compete. You give them two
different food sources okay? Greys love sausages and the reds love peanut butter. If you could find
something like that then they could both survive and be happy. But unfortunately just because we can
say how it can work doesn't mean it's possible to push us in this direction. - (Tom some of this kind of)
(feels like just kind of stuff that's quite obvious and intuitive and you've really complicated it and)
(formalised it with graphs and equations where I could have told you: the greys are better hunters)
(and therefore- Why do you need this mathematical formalism for things that seem intuitively obvious)
(when you think about them?) - Yep no: a) it gives you a way of framing that argument. Because there are a
lot of arguments, and you could say greys are better hunters. Okay, well is that the reason? I haven't put
the fact that they're hunt- the better hunters in here; I've said that they compete better than
the reds. That could be for mating reasons, it could be for roosts where they live. All I'm saying is
they're competing better. So you've specified one point, I've taken out a strategy. Another thing it
gives us is being able to see, well how can we- what is the strategy space that we're working in? You
again have said: okay the greys have won out, that's the case we're dealing in. That doesn't tell us how
we get to the greys and reds living together or the reds surviving. This comes down to a question about
the pandemic that's happening a lot: why do we need mathematical modellers? We can see how many people
are ill; we can see how many people are dying; we've got the data. That's true but it doesn't
tell us what the best route of getting out of this mess is. And that's what maths gives you, it gives a
rigorous framework of language to understand: I've been in one situation, how do I get to another?
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..okay and from this what you get is a relation between time of first interaction and that's
proportional to how far away you are from them - space is being x and the rate of diffusion