A Primer of Swarm Equilibria

We study equilibrium configurations of swarming biological organisms subject to exogenous and pairwise endogenous forces. Beginning with a discrete dynamical model, we derive a variational description of the corresponding continuum population density. Equilibrium solutions are extrema of an energy functional and satisfy a Fredholm integral equation. We find conditions for the extrema to be local minimizers, global minimizers, and minimizers with respect to infinitesimal Lagrangian displacements of mass. In one spatial dimension, for a variety of exogenous forces, endogenous forces, and domain configurations, we find exact analytical expressions for the equilibria. These agree closely with numerical simulations of the underlying discrete model. The exact solutions provide a sampling of the wide variety of equilibrium configurations possible within our general swarm modeling framework. The equilibria typically are compactly supported and may contain δ-concentrations or jump discontinuities at the edge of the support. We apply our methods to a model of locust swarms, which in nature are observed to consist of a concentrated population on the ground separated from an airborne group. Our model can reproduce this configuration; quasi-two-dimensionality of the model plays a critical role.

[1]  Gaël Raoul,et al.  Nonlocal interaction equations: Stationary states and stability analysis , 2012, Differential and Integral Equations.

[2]  Klemens Fellner,et al.  Stability of stationary states of non-local equations with singular interaction potentials , 2011, Math. Comput. Model..

[3]  J. Carrillo,et al.  Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations , 2011 .

[4]  T. Laurent,et al.  Lp theory for the multidimensional aggregation equation , 2011 .

[5]  G. Raoul,et al.  STABLE STATIONARY STATES OF NON-LOCAL INTERACTION EQUATIONS , 2010 .

[6]  Leah Edelstein-Keshet,et al.  Inferring individual rules from collective behavior , 2010, Proceedings of the National Academy of Sciences.

[7]  Andrea L. Bertozzi,et al.  Finite-time blow-up of L∞-weak solutions of an aggregation equation , 2010 .

[8]  T. Laurent,et al.  The behavior of solutions of multidimensional aggregation equations with mildly singular interaction kernels , 2009 .

[9]  Andrea L. Bertozzi,et al.  Blow-up in multidimensional aggregation equations with mildly singular interaction kernels , 2009 .

[10]  Andrew J. Bernoff,et al.  Asymptotic Dynamics of Attractive-Repulsive Swarms , 2008, SIAM J. Appl. Dyn. Syst..

[11]  J. Carrillo,et al.  GLOBAL-IN-TIME WEAK MEASURE SOLUTIONS, FINITE-TIME AGGREGATION AND CONFINEMENT FOR NONLOCAL INTERACTION EQUATIONS , 2009 .

[12]  Paul A. Milewski,et al.  A simple model for biological aggregation with asymmetric sensing , 2008 .

[13]  Andrew J. Bernoff,et al.  A model for rolling swarms of locusts , 2007, q-bio/0703016.

[14]  Andrea L. Bertozzi,et al.  Finite-Time Blow-up of Solutions of an Aggregation Equation in Rn , 2007 .

[15]  A. Bertozzi,et al.  State Transitions and the Continuum Limit for a 2D Interacting, Self-Propelled Particle System , 2006, nlin/0606031.

[16]  D. K. Weaver,et al.  Grasshoppers and locusts , 2007 .

[17]  A. Bertozzi,et al.  Self-propelled particles with soft-core interactions: patterns, stability, and collapse. , 2006, Physical review letters.

[18]  M. Bodnar,et al.  An integro-differential equation arising as a limit of individual cell-based models , 2006 .

[19]  Darryl D. Holm,et al.  Formation of clumps and patches in self-aggregation of finite-size particles , 2005, nlin/0506020.

[20]  A. Bertozzi,et al.  A Nonlocal Continuum Model for Biological Aggregation , 2005, Bulletin of mathematical biology.

[21]  Marek Bodnar,et al.  Derivation of macroscopic equations for individual cell‐based models: a formal approach , 2005 .

[22]  Darryl D. Holm,et al.  Aggregation of finite-size particles with variable mobility. , 2005, Physical review letters.

[23]  H. Chaté,et al.  Onset of collective and cohesive motion. , 2004, Physical review letters.

[24]  Andrea L. Bertozzi,et al.  Swarming Patterns in a Two-Dimensional Kinematic Model for Biological Groups , 2004, SIAM J. Appl. Math..

[25]  Y. Tu,et al.  Moving and staying together without a leader , 2003, cond-mat/0401257.

[26]  A. Mogilner,et al.  Mathematical Biology Mutual Interactions, Potentials, and Individual Distance in a Social Aggregation , 2003 .

[27]  Cédric Villani,et al.  Optimal transportation, dissipative PDE’s and functional inequalities , 2003 .

[28]  I. Couzin,et al.  Collective memory and spatial sorting in animal groups. , 2002, Journal of theoretical biology.

[29]  H. Chaté,et al.  Active and passive particles: modeling beads in a bacterial bath. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  W. Rappel,et al.  Self-organization in systems of self-propelled particles. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  Herbert Levine,et al.  Cooperative self-organization of microorganisms , 2000 .

[32]  A. Mogilner,et al.  A non-local model for a swarm , 1999 .

[33]  Leah Edelstein-Keshet,et al.  Do travelling band solutions describe cohesive swarms? An investigation for migratory locusts , 1998 .

[34]  Julia K. Parrish,et al.  Animal Groups in Three Dimensions: Analysis , 1997 .

[35]  Vicsek,et al.  Novel type of phase transition in a system of self-driven particles. , 1995, Physical review letters.

[36]  R. C. Rainey,et al.  Migration and meteorology. Flight behaviour and the atmospheric environment of locusts and other migrant pests. , 1989 .

[37]  W. Loher.  Polymorphisme Phasaire et Biologie des Acridiens Migrateurs. Les Grands Problemes de la Biologie, publiee sous la Direction du P.-P. Grasse.Frederic O. Albrecht , 1968 .

[38]  G. Varley,et al.  Grasshoppers and Locusts , 1967 .

[39]  Frédéric O. Albrecht,et al.  Polymorphisme phasaire et biologie des acridiens migrateurs , 1967 .

[40]  C. Breder.  Equations Descriptive of Fish Schools and Other Animal Aggregations , 1954 .