Cellular Automata

Cellular Automata are dynamic systems which feature a lot of individual processing units and the status of each unit usually depends on some kind of (simple) function that depends on temporal or spatial neighbors. 1D CAs can be thought of a line of nodes that have some status depending on the previous nodes. 2D CAs start with such a line and add, for each step, another line. Each node then has a predecessor (which is "above" it) and a successor (whis is "below" it). Each nodes status is influenced by a specified amount of predecessors (and its neighbors) and itself influences its successors. 3D CAs, then, can be thought of flat surfaces consisting of individual processing units. For every step, another surface is added, until the whole system resembles a cube, or a huge stack of paper.

They have a lot of practical and theoretical implications. One can easily model dynamic systems, for example how stock markets evolve when each investors considers only his "neighbors" actions from previous points in time. Similarly, one can model spatial dimensions. For example, a traffic model can be constructed, where all actions of individual cars (braking, accelerating), depend on the respective cars in front and behind them.

I played around quite a bit with them and tried to build some models of my own. Maybe someone starting to deal with Cellular Automatas can use some of them. Or maybe someone reads this and becomes interested in constructing such models him/herself. Really any spreadsheet software can be used to produce such models within models and obviously, various programming languages lend themselves to be used as well.

1) ONE SPECIES SURVIVAL
For this CA, I tried to model the survival or (non-survival) of a single species, which needs more than one individuum to reproduce. Each node represents a spatial unit where one individuum could be. Each line is intended to stand for one generation. The "5-neighborhood" is observed. So, if there is more than one individual in the last five spots, an offspring is produced and takes the spot (1/red) with probability 1-p. Otherwise, the spot remains empty (0/white). The initial distribution is randomly distributed with 50/50 probalitity. One can see how clusters of individuums "survive". On the edges of such groups, however, spots can become empty and lone individuals cannot reproduce. Depending on p, the species can die off. If p is reasonably low, the species will flourish and occupy nearly all spots.

1-species survival with p=0.4

1-species survial with p=0.31

1-species survival with p=0.2

2) FOREST FIRE
I tried to model how forest fire spread. This is a little complexer CA, as the nodes have more than 2 statuses and there are various parameters. All nodes represent a spatial unit where there is either nothing (0/white), a tree (1/green) or an ongoing fire (2/yellow). Each line represents one time step. Each node's status only depends on its own status before. p1 is a parameter that models the likelihood that no tree grows on an empty space (fertility factor). p2 indicates the likelihood that an ongoing fire burns out during one period. The last parameter, p3, simply represents the likelihood that a tree is not starting to burn in the next period. Depending on all factors, spots can remain empty for long, trees start to burn at alarming rates or fires rage on for several periods. These parameters can be used to model, for example, dry forests with high danger of ignition. Similarly, it is easy to simulate forests where trees are spaced out wide, meaning that fires are likely to die out.

forest fire with p1=0.9, p2=0.2 and p3=0.995