Every day there are flocks of birds, schools of fish, and colonies of ants, etc. that all exhibit the amazing power to move coherently together, to make a smart collective choice. How can they do it? This question has inspired generations of scientists to investigate this phenomenon- called collective animal behaviors.
In recent literature, the study of how animals move together is classified as collective animal motion. In 1995, a physicist Vicsek et al. proposed a “Self-propelled particle (SPP)” model to explain how a swarm of particles with constant speed can move in a common direction. It is not straightforward to see how, because each one particle just can process local information and lives in a noisy environment. In the model, they assumed a very simple interaction rule for each particle, that each one just averages the direction of its local neighbors at each time step as its new direction in the next time step.
They found that if the population density is higher than a critical value and the noise is lower than a critical value, the system exhibits some critical behaviors. At some critical points, it will switch from a disordered state in which many particles proceed their own direction to an ordered one in which all the particles moving in one common direction.
If a group of animals is given two or more options to choose, like two sheltering sites for an aggregation of cockroaches, two paths for a school of fish and two or more sheltering sites for a colony of ants, how do they make collective decisions? The study of this phenomenon is classified as animal collective decision-making. In this field, a well-recognized interaction rule is called quorum response, which is a nonlinear interaction rule and observed across many different species of animals. In quorum response rules, if an option is opted by more than a threshold number of individuals, this option becomes very likely chosen. On the other hand, if this option is opted by less than a threshold number of individuals, it will be neglected by this individual most of the time. This interaction is believed to make the animal group reach a consensus decision and at the same time, enhance the decision efficiency.
In our paper, we found that quorum response rules can also be applied in the field of collective animal motion. We proposed a model and assumed that each individual makes a directional decision from many (8 in the paper) choices based on its local neighbors’ moving directions. In our model, we found that this new type of interaction also results in the critical behaviors showed in SPP model. The group switches from a disordered state to an ordered state at some critical point. The analytic form of this interaction suggested an opportunity to apply a mean field theory in 1D with globally interacting individuals, so we can estimate the average time period between changes in the group direction. The results provide a limiting bound to simulation results.
Information entropy, proposed by Shannon in 1948, is a core concept in information theory. We applied it as a new order parameter to study a 2D model. Compared with the previous order parameters such as alignment, we find that besides the global order, information entropy can also capture the structural features of the local order of a system.
These findings are described in the article entitled Application of quorum response and information entropy to animal collective motion modeling, published in the journal Complexity. This work was led by Feng Hu from Chongqing Normal University.