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Spatial Models of Bistability in Biological Collectives



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To appear in Proc 46th IEEE CDC Spatial Models of Bistability in Biological Collectives Derek A Paley Naomi Ehrich Leonard Rodolphe Sepulchre and Iain D Couzin Abstract We explore collective behavior in biological systems using a cooperative control framework In particular we study a hysteresis phenomenon in which a collective switches from circular to parallel motion under slow variation of the neighborhood size in which individuals tend to align with one another In the case that the neighborhood radius is less than the circular motion radius both circular and parallel motion can occur We provide Lyapunov based analysis of bistability of circular and parallel motion in a closed loop system of selfpropelled particles with coupled oscillator dynamics I INTRODUCTION In this paper we use a cooperative control approach to model collective motion in biology Collective motion appears in natural systems ranging from molds 1 to locusts 2 to fish 3 All of these systems exhibit collective motion that ranges from relatively disordered swarm motion to highly ordered parallel motion Another highly ordered type of motion observed in biological collectives is circular motion 1 3 characterized by collective rotation about a fixed point Numerical investigations 4 of a repulsion orientation attraction ROA description of collective behavior 4 5 indicates that changes in model parameters generate transitions between swarm circular and parallel motion In fact slow variation of a single parameter related to interaction between individuals is observed to generate hysteresis in the transition between these motions We infer from the presence of hysteresis in the ROA behavior that there exists a parameter range in which both circular and parallel motion are stable in a qualitative sense We say that the ROA behavior is bistable with respect to circular and parallel motion We seek to prove the existence of such a parameter range in a related model by showing that both circular and parallel



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