FIGURE 4. Analytical and Monte Carlo simulation results.

a, b, Analytical results for the parameter values q = 1, k = 6, = 1/40 min^-1, and a = 0.1 mm min, which have been adjusted to experimental measurements. The curves show the proportion F_ij of the overall flow on each branch in the stationary state. a, In the absence of pushing ( = 0) the model predicts asymmetrical traffic above very low values of the overall flow of ants, independently of the traffic volume. b, When the proportion of pushed ants is high ( = 0.6), traffic is asymmetrical for moderate values of the overall flow of ants, but stable symmetrical traffic with F_ij = 0.5 is expected above a critical value of traffic flow, which is an increasing function of branch width. The symmetry-restoring transition from asymmetrical to symmetrical traffic flow corresponds to an inverse pitchfork bifurcation. c, d, Results of Monte Carlo simulations. At time t = 0, the pheromone concentration and the number of ants on each branch are set to zero. Ants arrive at choice point j at a rate _j(t) = . The probability of choosing the right or left branch at a choice point is governed by the choice function F_ij (see equation (3)). However, as soon as an ant has engaged on a branch, it may be pushed to the other branch with probability by an ant moving in the opposite direction coming from the second choice point. The pushed ant then continues its course on the alternative branch and lays a trail. For each value of , the simulations are averaged over 1,000 runs. The graphs show the relative frequency of simulations in which a certain proportion of traffic was supported by the right branch. c, Assuming a pushing probability equal to zero ( = 0), most simulations ended with asymmetrical traffic. However, when we used the experimental value = 0.6, most simulations for narrow branches resulted in symmetrical traffic. d, The proportion of simulations with = 0.6 in which asymmetrical traffic emerged is a function of the total number of ants crossing the bridge, just as in our experiments (compare Fig. 3b).