The Form of Swarms: A Primer of Swarm Equilibria

The Form of Swarms: A Primer of Swarm Equilibria Andrew J. Bernoff Harvey Mudd College Chad M. Topaz Macalester College Supported by: National Science Foundation DMS-0740484

Andrew J. Bernoff Harvey Mudd College ajb@hmc.edu Chad M. Topaz Macalester College ctopaz@macalester.edu Abstract: Biological aggregations (swarms) ehibit morphologies governed by social interactions and responses to environment. Starting from a particle model we derive a nonlocal PDE, known as the aggregation equation, which describes evolving population density. The solutions to the aggregation equation can ehibit a variety of behaviors including spreading without bound, concentrating into δ- functions and formation of compactly supported equilibria. We describe some tools for investigating the asymptotic behavior of solutions. We also study equilibria and their stability via the calculus of variations which yields analytical solutions. Finally we present a case study about how these methods can be used to construct a model of locust swarms. Useful References: A. J. Bernoff & C. M. Topaz, Nonlocal aggregation models: A primer of swarm equilibria. SIAM Review 55 (2013) 709-747. Note the introduction and epilogue to this paper contain a potentially useful literature review. A. J. Leverentz, C. M. Topaz & A. J. Bernoff. Asymptotic Dynamics of Attractive-Repulsive Swarms, SIAM J. Appl. Dyn. Sys. 8 (2009) 880-908. C. M. Topaz, A. J. Bernoff, S. Logan & W. Toolson, Aggregations, Interactions, and Boundaries: A Minimal Model for Rolling Swarms of Locusts, Eur. Phys. J. - Spec. Top. 157 (2008) 93-109.

Many aggregations have sharp boundaries. Sinclair, 1977

Biological question How are individual behavior and group behavior connected? Mathematical question How do different social interaction kernels affect the (asymptotic) macroscopic behavior of a solution?

A Continuum Model of Swarming Three Key Ideas: Continuum model of swarm density,. Individuals interact pairwise and social forces are additive. Repel at short distances (collision avoidance) Attract at longer distances (swarming behavior) Kinematic Model (velocity social forces) Previous Work: Bodnar & Velazquez (2006) considered well-posedness and blow-up in 1D. Bertozzi & Laurent (2007) proved aisymmetric blow-up in N-dimensions. Bertozzi & collaborators (2008...) working on general blow-up in N-dimensions.

Computing the Velocity (y) Velocity for a test mass at v() = Z 1 y= 1 dy from the density (y) f s ( y) (y) dy f s Where f s () is the social force induced by the mass distribution with density (y). f s ()

Conservation of Mass J J The flu, J, is given by: J = v So the equation of conservation of mass is: @ @t + @J @ =0 ) t +(v ) =0 And the velocity is given by v = Z 1 1 f s ( y) (y) dy f s f s ()

Conservation of Mass (y) The flu, J, is given by: J = v So the equation of conservation of mass is: @ @t + @J @ =0 ) t +(v ) =0 And the velocity is given by v = Z 1 1 f s ( y) (y) dy f s f s ()

Continuum vs. Discrete Continuum Model: Difficult to simulate, particularly, in higher dimensions Simulation difficulty independent of number of particles. Relationship to biology not always clear. Much better tools for analysis Discrete Model: Easy to simulate a small number of particles. Easy to generalize to higher dimensions. Relationship to biology well understood. Difficult to analyze or find eact solutions. z 1.2 0.8 0.4 (b) 0 0.5 1 1.5 2

Meet Andy. SIAM J. APPLIED DYNAMICAL SYSTEMS Vol. 8, No. 3 2009 Society for Industrial and Applied Mathematics Asymptotic Dynamics of Attractive-Repulsive Swarms Andrew J. Leverentz, Chad M. Topaz, and Andrew J. Bernoff Andy Leverentz HMC 2008

The Continuum Model (1D) Conservation of Mass Odd Finite first moment Cont. & piecewise diff. ( 0) Jump discontinuity ( = 0) {Crosses zero for eactly one Morse Social Force f s () (0, 1 F ) Piecewise Linear Social Force f s () (0, F ) ( L, 1) fs() fs() ( 1, 0) (1, 0) (0, 1 +F ) ( L, 1) (0, F )

A Menagerie of Behaviors 0.2 Spreading 0.4 Steady state 0.15 0.3!() 0.1!() 0.2 0.05 0.1 0!20!10 0 10 20 0!4!2 0 2 4 2 (c) 1.5 (d) 1.5 Blow-up 1 Blow-up!() 1!() 0.5 0.5 0!2!1 0 1 2 0!4!2 0 2 4

Morse Social Force: A Phase Diagram F L

Cumulative Mass Function Define the cumulative mass function: Where satisfies a wave equation And the velocity can be approimated as Where. Note that as the density is positive,, is increasing.

Burger s Equation The dynamics are dominated by - If social forces are repulsive at small distances and density is bounded for all t. - If social forces are attractive at small distances and will form a shock. But as, Shocks = Density δ-functions!! See Bodnar & Velazquez (2006)

Spreading Swarms Epand velocity at long lengthscales: Where is the first moment Substitute into C of M,, to yield the porous media equation, Barenblatt Solution which has spreading solutions for κ< 0.

Barenblatt Solution Rescaled Density Approaches Barenblatt Solution RMS width has proper time dependence

Finding Steady States!() 0.4 0.3 0.2 (b) Steady state When β > 0, κ< 0 we epect steady solutions. 0.1 0!4!2 0 2 4 β α Energy first variation Fredholm Integral Equation Details Fredholm integral equation V ( z z )ρ(z ) dz = λ f(z), Coming Later Where α < z < β is the support of the solution Mass Constraint β α ρ(z ) dz = M

Phase Diagram Revisited Morse Social Force: F L

Some conclusions Repulsion (attraction) at the origin suppresses (causes) blow-up and density concentrations. First moment governs spreading at long lengthscales. Spreading governed by porous media equation. Steady states can be found via integral equations. Open Questions Necessary and sufficient conditions for spreading and convergence to Barenblatt solution.

A fly in the ointment... This potential has β > 0, κ> 0 which suggests spreading Social Force Density Plot RMS Width Final Steady State

Let s focus on locusts...

Locusts form giant, destructive swarms. National Geographic, A Perfect Swarm $10 billion/yr for pesticides (all insects) $100 million/yr for control (Africa) $100 million/yr in crop loss (Africa) (EPA, UNFAO) 10 10 locusts 10-100 km/day days or weeks 100 km 2

With gravity, swarm forms a bubble. Catastrophic Gravity G = 1 Sticky boundary

With wind, swarms roll Catastrophic Wind U = 1 Takeoff Resting Landing Uvarov, Grasshoppers & Locusts (1977)

Can we eplain the swarm morphology? Steady Catastrophic Locust Swarm 1.2 (b) z 0.8 0.4 Empty Gap 0 0.5 1 1.5 2 Mass concentration on ground

A Continuum Swarming Model Velocity Conservation of Mass q() Eternal Force (0, 1 F ) (0, 1 +F ) Morse Social Force Finite Mass Compact Support Pairwise, additive Antisymmetric Finite first moment Jump discontinuity ( = 0) Bodnar & Velasquez (2005,2006)

Energy for the Continuum Model Energy Eternal Potential Energy Dissipation Morse Potential Energy dissipated unless equilibrium Connection to optimal transport Pairwise, additive [ Q ()= - q(), F () = - f() ] Repulsive at small and attractive at large Symmetric Pointy at ( = 0)

First Variation and Steady Solutions Let where is a steady solution and is a zero mass perturbation. Then Where the first variation is given by: For the first variation to vanish, in the support of the solution: Z Q( y) (y) dy + F () = for 2. which is a Fredholm Integral Equation. Constant energy per unit mass

Second Variation and Stability Suppose Z is an equilibrium solution satisfying: Q( y) (y) dy + F () = for 2. Then for stability two conditions are sufficient: I) The energy per unit mass minimizes the induced potential, (): Z Q( y) (y) dy + F () = () II) Second Variation is positive definite: W 2 [, ] = Z for zero mass perturbations,. Z Q( y) () (y) d dy > 0 () () =

Finding Equilibrium States Equilibrium solutions satisfy an integral equation 0.4 Equilibrium (b) state 0.3 Energy first variation Fredholm integral equation!() 0.2 0.1 0!4!2 0 2 4 Fredholm Integral Equation Mass Constraint Where α < z < β is the support of the solution

Finding Equilibrium States!() 0.4 0.3 0.2 Equilibrium (b) state 0.1 0!4!2 0 2 4 Energy first variation Fredholm integral equation invert integral operator ODE If then So applying to yields

Finding Equilibrium States!() 0.4 0.3 0.2 Equilibrium (b) state 0.1 0!4!2 0 2 4 Energy first variation Fredholm integral equation invert integral operator ODE Linear ODE Solving Plug in and satisfy mass constraint Solution Yields ρ(z) = A + B cos (μz) in the support of the solution.

Continuum solutions agree with discrete (numerical) ones. 30 ρ vs. z Steady Solutions (N = 100) F = 0.5, L = 2 20 10 0.5 4.5 ρ(z) = A + B cos (μz)

A Primer of Swarm Equilibria

Repulsive Forces on a Finite Interval Laplace Repulsive Potential: Finite Interval: -d < < d Equipartition of mass between concentrations at the endpoints and classical solution in interior. Mass Concentrations can t be found by classical integral equation methods Numerics (a) 0.3 ρ 0.15 0 1 0.5 0 0.5 1

Repulsive Forces in a Quadratic Potential Laplace Repulsive Potential: Quadratic eternal potential: Solution is compactly supported on. Jump discontinuities at edge of support Global energy minimizer. 1.5 Numerics (c) 1.4 ρ 1.3 0 = = = 0.4 0.2 0 0.2 0.4

Repulsive forces in a gravitational potential Laplace Repulsive Potential: Half line: If M < g all mass concentrates at origin. If M > g some mass concentrates at origin and the remainder forms a compact swarm. Global minimizer and probably global attractor. 0.6 Numerics (b) 0.4 ρ 0.2 0 0 0.25 0.5 0.75 1 As a swarm model: - Concentration on ground (good) but no gap (bad).

Catastrophic Morse Potential in Free Space Morse Potential in catastrophic regime. Solution is compactly supported on. Width of support independent of mass!! Jump discontinuities at edge of support. Local energy minimizer. Probably a global energy minimizer. Numerics (a) Morse Potential Analytic Solution 0.3 ρ 0.15 0 3 2 1 0 1 2 3

H-Stable Morse Potential on bounded interval Morse Potential in H-stable regime. No solution in free space (swarm spreads forever). Mass concentrations at endpoints of interval. Global energy minimizer. Probably global attractor. Morse Potential 0.265 Numerics (b) ρ 0.26 = = = 0 1 0.5 0 0.5 1

Quasi 2D-Potential Suppose that we look at a quasi-2d mass distribution This has an associated potential, : Q 2D (z) = Z 1 = 1 Q( p 2 + z 2 ) d = Z 1 e p 2 +z 2 = 1 This allows steady swarms with a gap at zero: d Airborne swarm z Repulsion from mass concentrated on ground ~ dz 2 ln(z) Gravity + gz = Stable gap (for some parameters) Empty Gap Mass concentration on ground

The quasi 2D potential yields dynamically stable bubbles. Organism column Density profile ρ 1D z ρ Quasi 2D 35 insects z

Quasi 2D Locust Swarms Quasi 2D Laplace Repulsion Potential. Gravity & Ground: - Linear potential on a half space. Mass concentration on ground Gap above ground. Continuous family of solutions Solution is only a local swarm minimizer. Quasi-2D Numerics (d) 8 6 ρ() 4 2 0 0 4 8 12 1.2 (b) (c) 12.8 z 0.8 Airborne swarm Λ() 12.4 0.4 Empty Gap 0 0.5 1 1.5 2 Mass concentration on ground 12 0 4 8 12

2D Morse Equilibria 2D Catastrophic Swarm (d) 0.6 Why are there mass concentrations at the edges? 8 6 2D Morse Ribbons (d) 0 ρ() 4 2!0.6!0.6 0 0.6 0 0 4 8 12 Mass concentration on ground z Airborne swarm Louis Ryan HMC 2012 Empty Gap

2D Morse - Ribbon Morse Potential Quasi-2D Morse Potential Q 2D (z) = z 1 Quasi-2D Morse Potential ( ) Q 2D (z) Z L L Z 1 = 1 z 2 ln z Q( p 2 + z 2 ) d Integral Equation (Carleman) y 2 ln y (z y) dz = z <L Density 600 500 400 300 200 z L (z) -L Numerical Quasi 2D Swarm Carleman (z) = C L 2 z 2 z <L 100 0 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 Position

2D Morse - Circular Swarm Density [ ρ(r) ] density 180 160 140 120 100 80 60 40 20 2D Catastrophic Swarm (Continuum Model) Plot of Swarm Density for C = 1.500000e+000, L = 2, n = 1600 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 radius Radius [r] Found as an Energy Minimizer Singular mass concentration at the edges (r) C p R r 2D Catastrophic Swarm (Particle Simulation) 0.6 0!0.6 (d)!0.6 0 0.6 Dimensionality of local minimizers of the interaction energy D Balagué, JA Carrillo, T Laurent, G Raoul Archive for Rational Mechanics and Analysis (2013)

Conclusions Energy methods allow us to analyze equlibria and their stability. Equilibria satisfy a Fredholm Integral Equation. This yields some analytical solutions. Equlibria often contain concentrations on boundaries. For locust model, quasi-2d repulsive potential and gravity yield a ground concentration together with a gap. 2D Morse yields inverse square-root mass singularities on boundaries. Inverse square-root singularities appear generic in 2D for pointy repulsive potentials. z 1D Quasi 2D Empty Gap Mass concentration on ground Bernoff & Topaz Nonlinear Aggregation Equations: A Primer of Swarm Equilibria. SIREV (2013).