Fig. 4: Collective response to external forces.
From: Maximal response to a mechanical leader at critical group size in ant collectives
In (A–D), the heatmap illustrates the switching probability per time bin derived from empirical data, plotted as a function of both the duration \(\left(\Delta t\right)\) and magnitude of the externally applied force \(\left({F}_{{{{\rm{ext}}}}}\right)\). The data has been binned using k-nearest neighbors and averaged through a simple moving window across all group sizes (the raw empirical data is presented as scatter points in Fig. S6 of the Supplementary Text). The empirical data consists of multiple trials for each group size: (A) n = 31 for small, (B) n = 85 for intermediate, (C) n = 201 for large, and (D) n = 260 for very large groups. G–J Present the results from simulations of the microscopic model conducted over a 100 × 100 grid of Fext and Δt, averaged across 100 runs for each parameter pair as heatmaps. The color scale in both heatmap panels ranges from dark red, indicating a high probability of group switching, to dark blue, indicating resistance to switching. The phase boundary corresponds to the yellowish-green area in the heatmaps, where the probability of switching is 50%. It is fitted with exponential functions from Eqn. (13), shown as dashed lines. E, K Show the mean switching probability per time bin as a function of the magnitude of the externally applied force (Fext), independent of the duration of the force from empirical data and numerical simulations, respectively. F, L Show the mean switching probability per time bin as a function of the forcing duration (Δt), independent of the magnitude of the external force from empirical data and numerical simulations, respectively. The error bars represent the standard deviations from the mean values. Source data are provided in the Source data file.
