We studied a wide range of scent laying and decay rates, as well as different densities of ants. This is the simplest local, memoryless, homogeneous and isotropic model which leads to trail forming that we know of, and the formation of trails and networks of ant traffic is not imposed by any special boundary conditions, lattice topology, or additional behavioral rules. The required behavioral elements are stochastic, nonlinear response of an ant to the scent, and a directional bias. Furthermore the parameters of the system need to be tuned somewhat to the appropriate region, and not any nonlinear rule of the above type will do. If the nonlinear response or the directional bias are removed no lines form, and lines that are already formed do not persist.
Well defined trails form in the region above the phase transition line (ordered phase), but near the transition from disorder. Further away from the order-disorder line the clumping tendency overcomes the directional bias, and no lines form.
In every run observed within the part of the parameter space indicated in Figure 2, a final system of trails emerged and survived as long as the simulation was run (Fig. 1). The self-organization of regular patterns of movement of the ants occurs in two stages: an initial condensation phase, which occurs during approximately the first 500 time steps, and a simplification phase. In the first stage a network of trails forms within a few hundred time steps. The trails at first typically form a network of many branches. All of the branches initially appear stable, but after several hundred time steps, some abruptly disappear, decreasing the length of the network.
In this second stage the network of ant traffic resembles qualitatively some network type experiments done with real ants[7] and the network theory of [9], (and the basic discussion of the properties of these networks given there holds). The ants can neglect a branch by chance, and if the fluctuation in the density of the ants over the branch happens to be great enough, the neglect builds on itself and the trail disappears. As a result, the complexity of the network tends to decrease with time until a final, stable network is reached.
The final line in all runs has been a loop (sometimes with sub-loops). Most often, the trail exploits the periodic boundary conditions by forming a more-or-less straight line that wraps around the torus. This of course reflects the situation that a line ``free end" is clearly unstable due to the directional bias. That lines which feed back on themselves invariably form is thus not surprising. It is interesting to note here the qualitative similarity of these loops to autocatalytic sets of chemical reactions.[19,20] The instability of the free ends just illustrates the sensitivity to boundary conditions when the information flows in an active way. The separate issues of how these basic structures are incorporated into a larger colony context will be taken up elsewhere, and we believe they deserve further mathematical analysis.
The main conclusion to be drawn here is that simple osmotropotaxic scent following of the very simple kind described above is not only sufficient to allow for trail following behavior as shown in [3,4], but sufficient to produce evolution of complex pattern of organized flow of social insect traffic all by itself.
Erik Rauch