To appear in Proc. 46th IEEE CDCSpatial Models of Bistability in Biological CollectivesDerek A. Paley, Naomi Ehrich Leonard, Rodolphe Sepulchre and Iain D. CouzinAbstract— We explore collective behavior in biological sys-tems using a cooperative control framework. In particular, westudy a hysteresis phenomenon in which a collective switchesfrom circular to parallel motion under slow variation of theneighborhood size in which individuals tend to align with oneanother. In the case that the neighborhood radius is less thanthe circular motion radius, both circular and parallel motioncan occur. We provide Lyapunov-based analysis of bistabilityof circular and parallel motion in a closed-loop system of self-propelled particles with coupled-oscillator dynamics.I. INTRODUCTIONIn this paper, we use a cooperative control approachto model collective motion in biology. Collective motionappears in natural systems ranging from molds [1] to lo-custs [2] to fish [3]. All of these systems exhibit collec-tive motion that ranges from relatively disordered (swarmmotion) to highly ordered (parallel motion). Another highlyordered type of motion observed in biological collectives iscircular motion [1]–[3], characterized by collective rotationabout a fixed point. Numerical investigations [4] of a re-pulsion/orientation/attraction (ROA) description of collectivebehavior [4], [5] indicates that changes in model parametersgenerate transitions between swarm, circular, and parallelmotion. In fact, slow variation of a single parameter relatedto interaction between individuals is observed to generatehysteresis in the transition between these motions.We infer from the presence of hysteresis in the ROAbehavior that there exists a parameter range in which bothcircular and parallel motion are stable, in a qualitative sense.We say that the ROA behavior is bistable with respect tocircular and parallel motion. We seek to prove the existenceof such a parameter range in a related model by showing thatboth circular and parallel motion are stable, in a Lyapunovsense. An earlier version of this work appeared in [6], wherewe studied a bistable model restricted to two individuals; inthis work, that restriction is lifted. Techniques that supportour results are drawn from our related work on collective mo-tion in engineered systems [7], [8] as well as from stabilitytheory, graph theory [9], and studies of consensus in time-varying systems [10]. This paper reveals that an alternateThis work was partially supported by an NSF GRF, ONR grants N00014–02–1–0826, N00014–02–1–0861 and N00014–04–1–0534, and the BelgianProgram on Interuniversity Attraction Poles, initiated by the Belgian FederalScience Policy Office. The scientific responsibility rests with its authors.D. Paley ([email protected]) is an assistant professor of aerospace en-gineering at the University of Maryland, College Park, MD 20742. N.Leonard ([email protected]) is a professor of mechanical and aerospaceengineering at Princeton University, Princeton, NJ 08544. R. Sepulchre([email protected]) is a professor of electrical engineering and computerscience at the University of Liege, B-4000 Liege Sart-Tilman, Belgium. I.Couzin ([email protected]) is an assistant professor of ecology andevolutionary biology at Princeton University, Princeton, NJ 08544.version of the circular formation control that appeared in [11]also stabilizes parallel formations.The paper is organized as follows. In Section II, wedescribe a collective behavior model including the ROA be-havior and, in Section III, we present an idealized version ofthe model that is mathematically tractable. Our approach todeveloping a provably bistable behavior depends on whetherthe interaction between individuals in the model is undirectedor directed; we describe the approach for undirected interac-tion in Section IV. The corresponding analysis for directedinteraction is omitted, and will be presented elsewhere.II. COLLECTIVE BEHAVIOR MODELEach individual in the collective behavior model is repre-sented by a particle (point mass). Let N denote the number ofparticles. Let I , (O, ˆex, ˆey, ˆez) denote an inertial referenceframe with origin O and orthonormal unit vectors ˆex, ˆey,and ˆez. Let rk(t) denote the position of particle k ∈ N ,{1, . . . , N} and ˙rk(t) denote the inertial velocity.Let Bkbe a moving reference frame with origin rk(t) andunit vectorsˆbk(t), ˆak(t) andˆbk(t) × ˆak(t) (for example, Bkcould be a Fermi-Walker frame [12], although this choice isnot critical to the development). Frame Bkis oriented suchthatˆbk(t) is always parallel to the direction of motion ofparticle k. Let skdenote the speed of particle k and assumethat sk> 0 is constant. The velocity ˙rk(t) expressed as avector component in frame Bkis ˙rk(t) = skˆbk(t).The trajectory of each particle is determined by a discrete-time update rule. Let ∆t denote the discrete time step. Wehave rk(t + ∆t) = rk(t) + skˆbk(t + ∆t), whereˆbk(t + ∆t)is a unit vector that represents the direction of motion ofparticle k at time t + ∆t. The direction of motion of particlek depends on the position and direction of motion of asubset of all of the particles. Let r(t) , (r1(t), . . . , rN(t))Tdenote the N × 1 matrix of particle positions and letˆb(t) ,(ˆb1(t), . . . ,ˆbN(t))Tdenote the N × 1 matrix of particledirection unit vectors. The direction of motionˆbk(t + ∆t)is a function—called a steering behavior—of the entries ofr(t) andˆb(t) and the time step ∆t.The subset of particles that interact with particle k and thenature of that interaction depends on the perceptual geometryof particle k. Consider M non-overlapping perceptual zonesassociated to each particle k that are fixed with respect toframe Bk. Let Γ(n)kdenote perceptual zone n ∈ {1, . . . , M}of particle k. If rj(t) ∈ Γ(n)k(t), then particle k perceivesparticle j and may respond to it; the response of particlek to the presence of particle j depends on n. Let N(n)k(t)denote the set of indices of all of the particles contained inzone Γ(n)kat time t. For each n, the collection of perceptualrelationships between all of the particles with respect to zoneΓ(n)kis called an interaction network, of which there are M.Let b(n)k(t + ∆t) denote the desired direction of motionof particle k with respect to the set N(n)k(t). We callb(n)k(t + ∆t), which is not necessarily a unit vector, abehavior rule. A steering behaviorˆbk(t + ∆t),