Numerics

Equations 1, 2 and describe the system, but for the purposes of simulation it is necessary to introduce some discretization of space and time, and to translate the noisy behavioral function observed experimentally into transition rules on this discrete space [9]. These discrete rules are merely tools for the approximation of a continuous model, and other discretizations are possible.

The agent (referred to as an ``ant") will be allowed to move from site to site on a square lattice. We allow each ant to take one step on the lattice of points (cells) at each time step. As a result of discretizing the space, an individual ant at each time step finds itself in one of these cells, and its sensory input is the concentration of pheromone in its own cell and each of the eight neighboring cells. In addition, each ant leaves a constant amount of pheromone at the node in which it is located at every time step. This pheromone decays (is reduced by a certain percentage) at each time step. Toroidial boundary conditions are imposed on the lattice to remove, as much as possible, any boundary effects. The transition rates from cell i to cell j are proportional to . The normalized transition probabilities on the lattice to go from cell k to cell i are then given by

where the notation indicates the sum over all the cells j which are in the local neighborhood of k. measures the magnitude of the difference in orientation (direction) to the previous direction the last time the ant moved. Since we are using a neighborhood composed of the cell and its eight neighbors on a square lattice, can take only the discrete values 0-4, and it is sufficient to assign a number for each of these changes of direction. Here we used weights of (same direction) , and , , and (u-turn). Once the parameters , and are set a large number of ants can be placed on the lattice at random positions, the movement of each ant can be determined randomly taken from the distribution given by . We usually take the initial condition of the pheromone to be zero at every point on the lattice. Every time step each ant leaves a quantity (here ) of pheromone in each cell, and the total amount of pheromone in each cell is decreased at a rate (here ) at the end of each time step. The evolution of the system is simulated numerically and the pattern of lattice sites that contain ants is displayed in the figures.