(494c) Coarse Collective Dynamics of Animal Groups

Many animal groups such as fish schools and bird flocks display remarkable collective behavior such as coherent group motion and transitions between different group configurations. A small number of "informed" individuals in such animal groups, namely those with a preferred direction of motion, are known to influence the foraging behavior of the entire group and direct its approach to a target. Understanding the mechanisms of information transfer in these and other biological systems is a problem of fundamental interest.

Individual-based simulations that consider detailed local interaction rules between group members have achieved considerable success in this arena, demonstrating the emergence of collective dynamics from local behavior. These detailed simulation approaches are, however, computationally expensive thus limiting the length and time scales they can explore. Coarse-grained approaches using macroscopic PDEs that effectively average the local interaction rules are unable to predict the full spectrum of collective behaviors observed in animal groups. This suggests that much of the interesting collective group behavior/dynamics can be robustly simulated only by describing the very large number of local interactions among group members. We develop an "equation-free" approach to collective dynamics that sets simulation protocols indicating when/where to perform short bursts of the detailed "local" model. Identification of meaningful simulation observables is a critical component of our approach that enables macroscopic simulation to be performed with sparse usage of the local model. We use a recently-developed dimensionality-reduction approach to "detect" good observables by performing eigen-computations on data generated by local model simulation ?bursts?.

We consider a 1D model of group motion where each individual is characterized by its position, velocity, and direction of travel. These quantities are updated for each individual at each time step using a set of local rules based on "zones" of information surrounding each individual. Different rules are operative in the inner zone of deflection, where collisions with near neighbors are avoided, and in the outer zone of attraction, where alignment with distant individuals is imposed. One individual in the swarm is "informed" having a preferred direction of travel. Our simulations with this local model indicate stick-slip-type behavior, with instances where the swarm is stuck in place with individuals vibrating in place and also periods where the swarm flows in the preferred direction of the informed individual.

Short bursts of the local model simulation are performed and the results processed by a dimensionality reduction technique (diffusion map approach) that automates detection of the reaction coordinate. The diffusion map for this system orders high-dimensional points according to the extent of their stick-slip. We show that the ?automated" reaction coordinate found by the eigencomputations is closely related to the distance of the informed individual from the centroid of the group. Identification of the slow variables for the system allows us to construct an effective potential that quantitatively describes the stable and metastable states of the system. This capability to "detect" the reaction coordinates on-the-fly suggests the power of this approach for the acceleration of more complicated simulations (in 2D/3D) where the reaction coordinates are poorly understood. This work is in collaboration with the group of Prof. Coifman at Yale, and Prof. Moehlis at UCSB.