Birds are able to fly in a flock moving so harmonically that it looks like one single large bird.

In 1986, Craig Reynolds described the behaviour of each animal as composed of four elements :

1. Follow the leader.
2. Keep distance to the other birds.
3. Head to the centre of the flock.
4. Adapt your speed to the one of the others.

In this applet you can influence the relative weight of each of these four components with the sliders.

See how the outer appearance of the swarm changes!


This simulation shall show, why self-regulation through feed-back can lead to a regular oscillation.

In a room, there is a radiator, symbolised by a blue rectangle at the botom left. The heating power can be adjusted higher or lower with the slider. In the middle of the room, there is a thermometer. In the graphics thereunder, the heating power is marked in blue, the measured temperature in red and the desired temperature in black.

If you try now to reach the desired temperature, it will take some time until the heat of the radiator has made an impact and reached the thermometer. One is inclined to regulate the heating too early into the desired direcion and so to oversteer the system. The graphical representation of the temperature distribution in the room is helpful for regulating it. You can just as well switch it off.

This oscillation effect is particularly striking when one leaves the regulation to a thermostat. This can be seleced on the bottom of the switching panel, with different heating powers to choose from. It works in the way that it increases the heat output as long as the actual temperature is beneath the target value, and turns down the heating as soon as the desired temperature is exceeded.

This form of regulation leads to an oscillating temperature, which is caused by oversteering. As it takes some time until heating up the radiator shows an effect, the reaction - cooling it down - is carried out too early and too hard. This induces an exaggerated counter-reaction and counter-counter-reaction.

For all periodic fluctuations in the simulations presented here, we can base on this mechanism of oversteering.


Let's just try to apply the methods of the Cellular Automata to economies.

Let's imagine the fields as houses. In each of them lives a family consuming goods, one per day. They can stock a maximum of ten goods in their cellar. The more it is filled, the more intensely blue appears the field.

Now, dealers walk from house to house, introducing new goods. Each one of them can carry three pieces. If a dealer can sell all of them, he and his boss will be satisfied. If it is less because the cellar is already full, it will be considered as a failure.

The size of the sales team depends on its success : If a dealer succeeds to sell all of his goods on five consecutive days, he will get some support. The new colleague simply appears on the field that he just has left. If, on the other hand, he comes back with remaining goods five days in a row, his career will end abruply - he just disappears without a trace.

The evolution on the macro level is also remarkable : At suitable conditions, the black population curve gets into oscillation, the number of traders - a measure of economic activity - going periodically up and down. The reason is : When there are (too) many traders, the market will be saturated more and more. This makes the demand go down. As a consequence, the supply will also decrease, which leads however to an under-supply. The increasing demand calls new dealers (supply) into action, and then ...

We have created an economic cycle !

In the real world, the ups and downs of supply and demand has an effect on the prices, playing a very important role in this process. We, however, have managed it here without any money to make the it work. This means that money is not really needed for the economical process. It's just an intermediate.

With the five sliders, the following parameters can be adjusted :
- How quickly the goods are consumed by the households.
- The storage capacity of the cellar.
- How many goods the dealer can carry.
- After how many successful days the dealer gets support.
- After how many unsuccessful days the dealer gives up.

If this reminds you of the story of the rabbit in the meadow (see "Predator-Prey Relationship") - this is neither accidental nor coincidental. I have just reworked the rabbit-program a bit.


Following the model of "Daisy World" by James Lovelock and Andrew Watson (1983), I have written a simplistic program about the potential impact of the flora on the climate.

It's about a distant planet, with daisies as the only life form. They thrive in heat - the hotter the better - but fold at frost. The sun shines with a uniform intensity warming up the planet's surface. Since the daisies are white, they reflect the sunlight back into space. Where they grow, it becomes therefore colder. There is however a compensating heat flow between differently warm areas of the ground.

The flowers are represented by white dots, the buds from which they blossom as small green points. Warm areas are colored red and cold ones purple.

On the sliders, it can be set - how sensitive to cold the flowers are - at which temperatures buds come out - how much time the buds need to blossom out.


Here, an influenza outbreak shall be simulated with a Cellular Automaton.

The playing board is initially white, which means full of healthy people. When you press the Start button, one or more fields are "infected" (colored red). Then, one of the four directly neighbouring fields is selected. If this one is red, the field gets also infected, if the other field is white (or blue), nothing happens.

After some time, he disease ends, and the convalescent is now immune for some time. The degree of immune protection is indicated by the intensity of blue staining. Unfortunately, the immunity will end after some time and the field will turn white again.

With the sliders, it can be adjusted - the probability of being infected when meeting an infected person.
- the duration of the disease in days (number of moves)
- The duration of the immunity.

The dynamics of an epidemic depends to a large extent on these three parameters.

It is particularly interesting that the same pattern can also be observed in completely different systems, for example in chemical reactions (Belussov-Zabotinsky reaction) or during signal transmission between slime molds (Dictyostelium discoideum).


This is the same program as above but it starts from different initial distributions. The upper and the lower side, such as the left and the right edge are linked again.

It turns out that there are certain stable waveforms.


In this game, the flu epidemic will be shown with the help of balls.

First, a number of healthy people move as a white ball across the playing field. When the button "Infection" is clicked, one ball is colored red. If it collides now with a white one, the white ball will also turn red - it has been infected.

After a while, the sick person gets well again and is immune for some time. This is marked with a blue color. Unfortunately, the immune protection disappears and the ball/person can be infected again.

The duration of illness and immunity can be adjusted with the sliders. A "day" takes here a quarter of a second.