Coarse-grained analysis of stochasticity-induced switching between collective motion statesAllison Kolpas, Jeff Moehlis, and Ioannis G. Kevrekidis doi:10.1073/pnas.0608270104 published online Mar 27, 2007; PNAS This information is current as of March 2007. Supplementary Material www.pnas.org/cgi/content/full/0608270104/DC1Supplementary material can be found at: www.pnas.org#otherarticlesThis article has been cited by other articles: E-mail Alerts. click hereat the top right corner of the article orReceive free email alerts when new articles cite this article - sign up in the box Rights & Permissions www.pnas.org/misc/rightperm.shtmlTo reproduce this article in part (figures, tables) or in entirety, see: Reprints www.pnas.org/misc/reprints.shtmlTo order reprints, see: Notes:Coarse-grained analysis of stochasticity-inducedswitching between collective motion statesAllison Kolpas*, Jeff Moehlis†‡, and Ioannis G. Kevrekidis§Departments of *Mathematics and†Mechanical Engineering, University of California, Santa Barbara, CA 93106; and§Department of ChemicalEngineering and Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544Edited by Simon A. Levin, Princeton University, Princeton, NJ, and approved February 15, 2007 (received for review September 20, 2006)A single animal group can display different types of collectivemotion at different times. For a one-dimensional individual-basedmodel of self-organizing group formation, we show that repeatedswitching between distinct ordered collective states can occurentirely because of stochastic effects. We introduce a frameworkfor the coarse-grained, computer-assisted analysis of such stochas-ticity-induced switching in animal groups. This involves the char-acterization of the behavior of the system with a single dynami-cally meaningful ‘‘coarse observable’’ whose dynamics aredescribed by an effective Fokker–Planck equation. A ‘‘lifting’’procedure is presented, which enables efficient estimation of thenecessary macroscopic quantities for this description through shortbursts of appropriately initialized computations. This leads to theconstruction of an effective potential, which is used to locatemetastable collective states, and their parametric dependence, aswell as estimate mean switching times.coarse-graining 兩 equation-free 兩 individual-based model 兩self-organization 兩 schoolingFish, birds, and honey bees, as well as many other animalgroups, display collective types of motion such as schooling,flock ing, and swarming (1, 2). A single animal group can displaydif ferent types of c ollective motion at different times, with ⬍ 1day of residence time in each state (3). Although such transitionsc ould be due to changing behavioral rules or environmentalfactors, they also can occur entirely due to stochastic effects, aswill be demonstrated for the model considered in this paper.One class of biologically motivated, individual-based modelsfor group formation, frequently used for schooling fish, abstractsan imal behavior by placing zones around individuals in whichthey respond to others through repulsion, alignment, and/orattraction (4–11). In the three-dimensional model of Couzin etal . (10), long-time steady-state computations revealed fourdif ferent types of st able collective motion in different parameterregions: swarm, torus, dynamic parallel, and highly parallel. Itwas also shown that by changing the quantitative features of thebehavioral rules (increasing or decreasing the radius of align-ment), the collective state of the school could be changed.In ref. 10, stochasticity is incorporated by adding a smalldeviation to the heading of each individual obtained from thedeter ministic evolution algorithm. Our simulations show that ifone instead considers relatively rare but substantial variations,namely that there is a small probabilit y of each individualchanging its direction substantially from that obtained from thedeter ministic algorithm, then for certain parameter regionsmultiple suc cessive transitions bet ween the torus and the dy-namic parallel state can occur. See supporting information (SI)Fig. 7.In this paper, we study a one-dimensional individual-basedmodel for group formation with stochasticity included along thelines of the variation described above. This system exhibitsrepeated stochasticity-induced sw itching bet ween distinct or-dered c ollective motion states. This sw itching appears similar innature but is, as we will show, dif ferent in detail, f rom resultsobt ained by using other models. In those cases, collective motiontransitions between ‘‘symmetry-related’’ states [e.g., bet weenclockwise and c ounterclockwise motions for marching locustsc onstrained to a ring (12), or the ‘‘alternating flock’’ in refs. 13and 14], stochastically driven transitions between ordered anddisordered st ates mediated by clustering (15), mixed-phase statesat phase transition boundaries (16), or transitions that do notoc cur repeatedly (17) were observed.Individual-based models are often used in the study of an imalgroups because they can incorporate biologically realistic be-havioral responses and social interactions that might be discon-tinuous (e.g., characterized by thresholds or if/then r ules) orstochastic in nature; they also can support c omplex networktopologies, allow for individual variability, and enable the studyof the relationship between adaptive individual behavior andemergent properties (18, 19). Most analysis of individual-basedmodels, however, relies on long-time simulations, which can beextremely costly and difficult to interpret and analyze (2, 19). Inthis paper, we introduce a framework for the coarse analysis ofstochasticit y-induced switching between collective motion statesfor individual-based models. We characterize the behavior of themodel with a single ‘‘c oarse observable,’’ A(t), a scalar variablethat quantifies the global structure of the school. We showc omputational ev idence to support that A(t) parameterizes aone-dimensional, attracting, invariant ‘‘slow manifold,’’ whichcharacterizes the long-term dynamics of the system. This sug-gests that we can use an effective Fokker–Planck (FP) equationto describe the dynamics of the probability distribution P(A),whose drift and diffusion c oefficients are determined by theshort-time evolution of the first two moments of A. We locallyestimate these coefficients by