Scale-free movement patterns in termites emerge from social interactions and preferential attachments

As the number or density of interacting individuals in a social group increases, a transition can develop from uncorrelated and disordered behavior of the individuals to a collective coherent pattern. We expand this observation by exploring the fine details of termite movement patterns to demonstrate that the value of the scaling exponent μ of a power law describing the Lévy walk of an individual is modified collectively as the density of animals in the group changes. This effect is absent when termites interact with inert obstacles. We also show that the network of encounters and interactions among specific individuals is selective, resembling a preferential attachment mechanism that is important for social networking. Our data strongly suggest that preferential attachments, a phenomenon not reported previously, and favorite interactions with a limited number of acquaintances are responsible for the generation of Lévy movement patterns in these social insects.

Global behavioral traits in social insects represent a trade-off between individual and collective actions. In termites, where neuter individuals (workers and soldiers) are blind, short-range local interactions among conspecifics are known to generate large-scale spatial and temporal patterns of organization including sophisticated nest mounds, tunneling systems, soil patterns, and worker survival and foraging strategies (1–7). At the heart of collective social patterns are individual behaviors that are amplified or modified in a process known as social facilitation. In recent years, it has become important to study the details of the individual basis of termite behavior to better understand socially facilitated patterns arising at a large scale (5, 8).Regarding foraging and spatial exploration, it is well known that individual termite workers forage inside underground or wood-carved tunnels with a few examples of species foraging in the open (9). Laboratory observations have established that individual termite spatial exploration is highly directional with distances traveled following self-similar scale-free patterns (10) in a way that resembles passive floaters in near-chaos turbulent fluids, prompting the idea that generic physical phenomena may be at play. In ants, another social group, it was observed that density-dependent interactions among workers are responsible for a transition from chaos to periodic patterns of activity (11, 12), while in the gregarious locust a critical transition was observed in the coherence of the collective movement patterns when the size of the group was increased (13).Lévy walks (LW) are random walks composed of clusters of multiple short steps with longer steps between them. This pattern is repeated across all scales with the resulting clusters creating fractal patterns that have no characteristic scale. Because there is no characteristic scale, the overall length of LW is dominated by the longest step taken and, while the step-length variance grows over time, it nonetheless remains finite even when unbounded by biological and ecological considerations. The hallmark of Lévy walks is a distribution of step lengths, l, with a heavy power-law tail as described by the formula f(l)∼l−μ, where ∼ means “distributed as” and μ is the scaling exponent with 1 < μ < 3 as a condition which ensures that the distribution can be normalized with probabilities that sum to unity and is characterized by a divergent variance. When μ is close to 1, movements are close to being ballistic and when μ < 3, they are effectively Brownian (scale finite rather than scale-free). It has been hypothesized that LW may be an efficient way of exploring space when searching (14–17). It is now well established that many social insects including bumblebees (18), honeybees (19), ants (20), and termites (10) perform LW when engaged in foraging activities. LW have also been identified in swarming bacteria (21) and in spider monkeys (22) which live in social groups. Similarly, theoretical studies have shown how LW might arise in systems composed of interacting individuals (23). However, most of the experimental studies in these insects—and in other animals in general—have focused on individuals acting in the absence of interactions with conspecifics. Here we report on an experimental study—with strong theoretical support—of collective patterns where the aim is to explore in detail how social interactions influence the motion mode of individuals in a social context. For this we discuss three complementary experimental designs, each aimed at exploring different aspects of interacting termite motion. The experiments detailed below are 1) social interactions and collective motion, 2) motion with passive obstacles, and 3) annular constrained motion. We also develop computer simulations to uncover the possible mechanism involved in the generation of LW from collective behaviors.