Wheat Fields and Emergence

My daily cycling takes me through wheat fields, ponds, and pastures, and I take great delight riding through nature. At first I was not familiar with the terrain at all, and I had to carefully inspect where the grass was high, where there were logs blocking my way, and which canals were easiest to go through. It took me a couple of weeks to establish which path offered the smoothest ride, as opposed to ones filled with menacing thorns and other plants. This got me thinking: how are these paths created? Starting with raw nature, untouched fields, how does it come about that some routes are easier to ride through than others?

Some tracks and hiking routes were obviously designed and crafted with thoughtful intent – they are marked with rocks or signs, and may be gravelled, filled with man-made steps, or even paved. But other nature trails seem more spontaneous – we can easily discern that they are trails alright, but it doesn’t seem like any serious national nature organization created them.

I suspect that these paths appear spontaneously by random people and animals who walk the fields. The whole process is known as “emergence”: the development of a complex system or pattern out of a simple set of rules or behaviors. Usually, these rules do not at first seem correlated to the grand phenomenon.
A popular example of emergence is that of schools of fish, flocks of birds, or herds of wildebeest. These animals group together and move collectively, like a single living organism, as portrayed in this video. It turns out that this behavior can be explained by just three simple rules: animals try not to collide with each other; if not close to collision, animals move towards each other; animals align their speed and direction with that of their neighbors.
A possible emergence explanation for trails in wheat fields might go like this: suppose that there is a field of wheat, standing untouched by man. A person wanting to cross it from one side to the other would have to force his way through the tall stalks. In doing so, he bends and breaks some of the plants along the way. The next time he walks down that path, it will be easier for him – there is less upright and vigilant wheat to block his way. It thus makes sense for him to walk the same path twice. But now we have a positive feedback mechanism – when he takes the same route again, he will trample more wheat, making it even easier the third time, and so on. Eventually, if he takes the same course again and again, the wheat will be completely trampled, and an easy to travel, spontaneous trail will have emerged.

I devised a simple model for trail emergence through a field of wheat, based on the above explanation. It goes as follows: we start with an empty field of wheat, portrayed as a yellow square:

Every once in a while, people (also called agents) spawn in the center left, and start walking towards the right. Every iteration, an agent goes one step to the right, and randomly chooses between one step up, one step down, or going straight. When an agent walks through the wheat field, he tramples it. However, the wheat, if left alone, rejuvenates, and eventually grows back to full height after a while.
A single agent walking through the field looks like this (click the animated gif to see the animation. If the gif doesn’t play more than once, you might have to tell your browser to force-reload the image page):

The more yellow a square is, the healthier and stronger the wheat there. Notice how the trail disappears behind the agent as the wheat heals and grows (this is just a simulation, in real life it would take quite a while for the wheat to recover; however, it would also take more time between crosses).
In this animation, we have several agents going forth, one after the other (click the animated gif to see the animation):

Essentially, in this manner, the agents are simply performing a group of random walks – randomly choosing the direction to go in. Notice how some squares turn redder, because they were stepped on multiple times, meaning the crops there were quite heavily flattened.

Until now the model had no interactions between the agents. Let us put in the feedback mechanism. It is a simple one: when choosing a new direction to go in, the agents don’t pick uniformly at random. Rather, they tend to do what’s easiest for them, and walk through bent and broken wheat – through already existing trails. The more damaged the wheat is in a certain direction, the higher the chances they will go there. More precisely, the chance to pick a certain direction is inversely proportional to the height of the wheat in that direction.

So here is how a scenario might go like: the first agent to cross the wheat field takes a random zigzagging path, according to the normal rules of random walk. He leaves behind him a trail of slightly trampled stalks. The second agent to cross is more inclined to take the same path, although he sometimes deviates. The wheat in the original path will be even more trampled than before. When enough agents have crossed, the wheat in the original trail will be totally crushed, and almost everyone will pick that trail. Because almost no one goes through other paths, the rest of the wheat field will heal, and we are left with a single, powerful route that was created spontaneously. Not one person planned it. In fact, it was chosen entirely at random, with a random walk. It is a completely arbitrary trail, no better than any other trail you might have carved through the field. However, feedback effects have strengthened it, and now this is the only walkable trail there is; you are stuck with it. You have very little hope of convincing anyone to try and create a new one.
Here are two movies showing the creation of such a trail (the simulations ran for 700 steps. Click the animated gifs to see the animation):

We can clearly see how one path develops, getting stronger and stronger as more and more agents travel through it. It is interesting to note that trails developed this way will tend to follow the central line cutting straight across the field. This is because in essence you combine several random walks one on top of the other, and their average value is zero.
In contrast, this is the same simulation with the same parameters for 700 steps, but with the people taking completely random paths (click the animated gifs to see the animation):

We see that the starting zone is very red, which means that the grass there is very trampled, because everyone crosses through there; however, they soon start to diverge, and no real trail is created.

Of course, the exact creation of the path depends strongly on the simulation parameters. For example, if the wheat grows back too fast, then it’s impossible for any trails to be created. There is also a dependence on the spawning parameters – how many people cross the wheat field, and where. In this example, I have more than doubled the number of agents, but have allowed them to start anywhere on the east edge – not just in the center.

As can be seen, no trail persisted for a very long time. Trails are created, but because less agents reach them, they eventually fade out. However, red streaks are easily noticeable throughout the entire field.

We tend to think that most of the things around us are there for a reason. We constantly try to look for explanations to all the phenomena we see, often seeking ones which assume design or purpose. This is a good thing, and it helps us manage in the world. But it is a hard fact: much of what we see is arbitrary. Actions which cannot be anything but random are the seeds to much of the complexity we see around us. In these simulations, the totally random choices of a minute number of people eventually influenced everyone who wished to cross the wheat field. No one in the simulations “intended” to create a trail. No one gave any consideration at all to how such a path should look like. Nevertheless, a trail was born.
I personally think emergence is beautiful. Letting go of man-made constructs such as “design” and “purpose” is a crucial step towards understanding nature.

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