So let me walk you through an example here. There's one agent based model, which is based on the Lotka-Volterra equation, its predator-prey dynamics. We start with one model where we have a predator and a prey, and that will get us to the homogeneous equilibrium state. So we have wolves and sheep both walk randomly around. I have a 100 sheep and 50 wolves. They walk randomly around they're blind, the wolves lose some energy with every step they take and they can run out of energy and then they die, or they randomly find a sheep, they will eat the sheep and will that replenish their energy as well. The idea is now, well, how does that evolve over time? They also have a certain reproductive rate if they survived them. You can see here first of all we start out with more sheep twice as many sheep as well as the red line. The red line down here the sheep the black line is the wolves and the sheep keep on increasing. So the sheep flourish there more sheep than wolves, so they have an advantage, they can flourish, they can reproduce better. You see as the sheep increase also the wolves will increase because now the wolves has more food. Now the wolves have more food, that's great, oh, the wolves when so big that the wolves population is so big actually the sheep population started to decline. The sheep population started to decline because there were too many wolves, well that's bad for the wolves. Now, you can see here in the graph that tracks the number of sheep and wolf. There wolves also start to decline because they're not enough sheep around anymore. Now the wolves started to die out, but the sheep bounce back. The wolves also going to bounce back then right, there's some wolves around over them, now than other enough sheep for them to find. Sheep are increasing and increasing, but the wolves now, the few wolves that survived that also now have way more than enough food, so they are really in a luxurious situation right now running into sheep every step they take and they really start to populate. They start to populate so much that they, well, they crushed the sheep population. Actually, that's something humankind always struggles with, right? We're often prone to overdo it. Now what will happen if you overdo it? Well, now the wolves have nothing to eat left, they really overdid it, they have nothing to eat left and they're going to go extinct. They killed their necessary resource by being way too greedy. There's still some wolves around but eventually, if there's nothing to eat, they will go see, they will have one lonely wolf, that's all we have left and at the end, but that's likely a equilibrium state. It's homogeneous state nobody survived. That's an outcome and that's an attractor they can be attracted to. Let's run it again to see if you get the same result. Again, our sheep start to increase, we have twice as many sheep as wolves, that's just how we set it up you can play around with it, you will find that NetLogo library. The wolves increase they almost killed the sheep, but you see some sheep survived. Two, three, fourth sheep, they just didn't run into these wolves. There were some two, three, four lucky sheep, that didn't run into the wolves. The wolves didn't find them. The wolves died out. What happened now the wolves died out. Well, feel free to go back and replay that one, the wolves died and what happens now with the sheep, but they have nobody to prey on them. They basically, well, that's a lot of sheep, the sheep inherit the Earth, very funny NetLogo, very funny. But the sheep inherited the Earth, that's what actually happened. Let's try to do that again, same first to cycles at the beginning we see the same. The sheep go and wolves go and what we find here again, sheep's died out, we find one equilibrium and nobody survives because the sheep didn't make it and the other equilibrium that we can find the other homogeneous state that we can find, is that, the sheep inherit the Earth, because you can see here a few sheep are still around the wolves don't find them these three sheep are still around make it in wolves, it's too sparse the wolves don't find them. Now there are four sheep, they bounce back but it's too late. It's too late for the wolves, the last wolves will die out not close enough to the sheep and now, well, nobody can stop the sheep anymore. The sheep will inherit the earth. So but that's another equilibrium homogeneous. So we have these, these are called attractors. These attractors that the system gets attracted to either the end stage is all sheep or nobody. In this model, these are the only two solutions. Now you can have an invariant distribution and I have these terms here because some people call it differently depends if you're from physics or from some other discipline. So it's an invariant distribution, a stationary distribution, an invariant measure or a stationary measure. So a mix of this, but that's the idea and this end we don't have this distribution that I showed you before we just have two and it can be 40 percent of the time. This happens 60 percent of the time that happens. These two and that's one possible outcome, and that's quite frequent as we saw for example in Schelling's models that you have this. Is very good because then you can make pretty good predictions if you have these attractors. Let's look at another model, cycles. So the cycle the ideas that something goes around in cycle, in a periodic orbit, in a season for example, a season as a cycle, an economic cycle they're actually called cycles. Here what we introduce now is we get rid of the wolves, but we have the sheep dependent on grass. So the sheep eat grass, the grass doesn't move around when it gets eaten up it replenishes as well. But the grass doesn't move around which gives us a little bit a different dynamic. You can see we get cycles here, because cycles here the grass goes down as the sheep increase and they balanced each other off. We have less grass that affects the sheep, the sheep go down. If we have less sheep the grass can grow back. If you have more grass, the sheep say, "Let's go for the grass." So this sheep go up, and how this system is established here how I basically set it up if we get these very smooth cycle is in the monitor down there, began very smooth cycles the sheep fluctuate a little bit more than the grass because they're dependent on the grass one follows also the other the grass goes there, but first you can say the sheep follow it because the sheep prey on the grass, they are the predators. But you get a very smooth periodic orbit, a very smooth cycle in this, and that also gives you predictability. You can say, well, this happens and this happens same as you can predict after day comes night, and after summer comes fall and then comes winter. So you can make predictions with that, but it's not like you get pulled to one attractor and just stay there and nothing else changes anymore you are in equilibrium. Equilibrium means that nothing else changes anymore, right? So that's another outcome that you can have in complex adaptive systems which are very well modeled with agent-based computer simulations. Then again, the random as I said, well, honestly if something is truly random you don't have information you can not do science, rarely. So what we're going to skip that one for now and go to the last one which is where we need all of you help us future researchers which is the system is complex and social systems are complex systems. So, now we bring all of that together. We have our grass and our sheep. Our sheep eat the grass, but the grass grows back and our wolves prey on the sheep. As we have a three variable problem interdependent among them and we start our simulation. We can see here at the bottom, first we have a lot of grass on green line, that's the amount of grass, the sheep increased. Yes, the sheep and the grass they balanced, we saw that the grass goes down the sheep go up, but if there's not enough grass the sheep could okay. So these two fluctuate we've seen that cycle. Well, now but the sheep also depend on the wolves, and the wolves now prey on the sheep which means the sheep go down. Now this sheep go down as right, now the sheep go down the grass can go up. Now the grass goes up the sheep should bounce backwards, if the sheep could bounce back then the wolves also increase which affects the sheep, which then if the wolves increase the sheep will go down which allows the grass to go up, right? So that like a smooth cycle, the wolves really mess this smooth cycle, but you can see there is something predictable, there is certainly a trade off between grass and sheep, we can see that clearly and wolves. There's also a dependency between the wolves and the sheep, we can also see that clearly. I mean, that the sheep go up the wolves can thrive, but like the best you could describe that to say it's complex. Now it's not complicated in a way, that's really complicated we only have three variables here. It's not tremendously complicated, but it's complex. You can make some predictions but not really perfect predictions are not smooth predictions. There's randomness and evolve even so there are some local structures that you can identify. Often, that's what you most often find in social systems, complex adaptive systems and ecosystems, of course, as well. Now imagine in society on ecosystems there has not only three variables. That's a quite simple model and you already get patterns like that. Now try to make predictions in reality with society, that the best thing you can say it is complex, so that brings us to our last one. These are the four outcomes, I just wanted to show you to them. So you have is not like anything can happen even so the last categories I said we run it or you have to dig a little bit deeper into the different kinds of complexity, but homogeneous and equilibrium where you then every time you run it you might get a slightly different attractor. In the Lotka-Volterra model, we've got two very clean ones either they are only sheep or there's nobody. In the Shelling we also got equilibrium, but we're always a little bit different every time you run it, so you get your invariant distribution, you're stationary measure. Then there can be a cycle, up and down very nice and smoothly a lot of systems work with that and that's very useful economic growth and recession. We can actually measure them as well, and we've seen them over hundreds of years now. Randomness is truly chaotic. That's that's a tough nut to do science, is a tough nut to crack to do science with and then complex that's actually the biggest group where you can see most of the patterns in especially in social science.
