Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Social biological organisms: Aggregation patterns and localization

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Swarm collaborators Prof. Andrea Bertozzi (UCLA) Prof. Mark Lewis (Alberta) Prof. Andrew Bernoff (Harvey Mudd) Sheldon Logan (Harvey Mudd) Wyatt Toolson (Harvey Mudd)

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Goals Give some details (thanks, Andrea!) Highlight different modeling approaches Focus on localized aspect of swarms (how can localized solutions arise in continuum models?)

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Background Two swarming models Future directions

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 What is an aggregation? Parrish & Keshet, Nature, 1999 Large-scale coordinated movement

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 What is an aggregation? Dorset Wildlife Trust Large-scale coordinated movement No centralized control

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 What is an aggregation? UNFAO Large-scale coordinated movement No centralized control Interaction length scale (sight, smell, etc.) << group size

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 What is an aggregation? Sinclair, 1977 Large-scale coordinated movement No centralized control Interaction length scale (sight, smell, etc.) << group size Sharp boundaries and constant population density

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 What is an aggregation? Large-scale coordinated movement No centralized control Interaction length scale (sight, smell, etc.) << group size Sharp boundaries and constant population density Observed in bacteria, insects, fish, birds, mammals…

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Are all aggregations the same? Length scales Time scales Dimensionality Topology

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Impacts/Applications Economic/environmental $9 billion/yr for pesticides (all insects) $73 million/yr in crop loss (Africa) $70 million/yr for control (Africa) (EPA, UNFAO)

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Impacts/Applications Economic/environmental Defense/algorithms See: Bonabeu et al., Swarm Intelligence, Oxford University Press, New York, 1999.

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Impacts/Applications Economic/environmental Defense/algorithms “Sociology” Critical mass bicycle protest: "The people up front and the people in back are in constant communication, by cell phone and walkie-talkies and hand signals. Everything is played by ear. On the fly, we can change the direction of the swarm — 230 people, a giant bike mass. That's why the police have very little control. They have no idea where the group is going.” (Joel Garreau, "Cell Biology," The Washington Post, July 31, 2002)

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Impacts/Applications Economic/environmental Defense/algorithms “Sociology”

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Modeling approaches Discrete (individual based, Lagrangian, …) Coupled ODE’s Simulations Statistics Search of parameter space Swarm-like states Simple particle models (1970’s): Suzuki, Sakai, Okubo, … Self-driven particles (1990’s): Vicsek, Czirok, Barabasi, … Brownian particles (2000’s): Schweitzer, Ebeling, Erdman, … Recently: Chaté, Couzin, D’Orsogna, Eckhardt, Huepe, Levine, …

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Discrete model example Levine et al. (PRE, 2001) Self- propulsion Friction Social interaction Newton’s 2 nd Law

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Discrete model example Interorganism distance Interorganism potential Levine et al. (PRE, 2001)

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Discrete model example Levine’s simulation results: N = 200 organisms Levine et al. (PRE, 2001)

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Modeling approaches Continuum (Eulerian) Continuum assumption Similar approach to fluids PDEs Analysis Clump-like solutions Role of parameters Degenerate diffusion equations (1980’s): Hosono, Ikeda, Kawasaki, Mimura, Nagai, Yamaguti, … Variant nonlocal equations (1990’s): Edelstein-Keshet, Grunbaum, Mogilner, …

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Continuum model example Mogilner and Keshet (JMB, 1999) DiffusionAdvection Conservation Law

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Continuum model example Mogilner and Keshet (JMB, 1999) Density dependent drift Nonlocal attraction Nonlocal repulsion

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Continuum model example Mogilner and Keshet (JMB, 1999)

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Continuum model example Mogilner and Keshet (JMB, 1999) Mogilner/Keshet’s simulation results: Space Density

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Bottom-up modeling? Fish neurobiology Fish behavior Ocean current profiles Fluid dynamics Resource distribution Mathematical Description

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Pattern formation philosophy Study high-level models Focus on essential phenomena Explain cross-system similarities Faraday wave experiment (Kudrolli, Pier and Gollub, 1998) Numerical simulation of a chemical reaction-diffusion system (Courtesy of M. Silber)

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Pattern formation philosophy Study high-level models Focus on essential phenomena Explain cross-system similarities

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Pattern formation philosophy Study high-level models Focus on essential phenomena Explain cross-system similarities Quantitative experimental data lacking Guide bottom-up modeling efforts

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Pattern formation philosophy Deterministic motion Conserved population Attractive/repulsive social forces Mathematical Description Stable groups with finite extent? Sharp edges? Constant population density? Connect movement rules to macroscopic properties?

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Background Two swarming models Future directions

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Modeling goals: ≥ 2 spatial dimensions Nonlocal, spatially-decaying interactions Mathematical goals: Characterize 2-d dynamics Find biologically realistic aggregation solutions Connect macroscopic properties to movement rules Goals

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/ d continuum model Topaz and Bertozzi (SIAP, 2004) Assumptions: Conserved population Deterministic motion Velocity due to nonlocal social interactions Velocity is linear functional of population density Dependence weakens with distance

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Hodge decomposition theorem Organize 2-d dynamics via… Helmholtz-Hodge Decomposition Theorem. Let  be a region in the plane with smooth boundary . A vector field on  can be uniquely decomposed in the form (See, e.g., A Mathematical Introduction to Fluid Mechanics by Chorin and Marsden) “Incompressible” or “Divergence-free” “Potential”

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Hodge decomposition theorem Organize 2-d dynamics via… Helmholtz-Hodge Decomposition Theorem. Let  be a region in the plane with smooth boundary . A vector field on  can be uniquely decomposed in the form (See, e.g., A Mathematical Introduction to Fluid Mechanics by Chorin and Marsden) “Incompressible” or “Divergence-free” “Potential”

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Incompressible velocity Assume initial condition:     Reduce dimension/Green’s Theorem: decaying unit tangent

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Incompressible velocity decaying unit tangent Lagrangian viewpoint  self-deforming curve  (t)

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Example numerical simulation (N = Gaussian, …)

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Overview of dynamics Incompressible case “Swarm-like” for all time Rotational motion, spiral arms, complex boundary Vortex-like asymptotic states Fish, slime molds, zooplankton, bacteria,…

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Overview of dynamics Potential case Expansion or contraction of population Model not rich enough to describe nucleation Incompressible case “Swarm-like” for all time Rotational motion, spiral arms, complex boundary Vortex-like asymptotic states Fish, slime molds, zooplankton, bacteria,…

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Previous results on clumping nonlocal attractio n local dispersal Kawasaki (1978) Grunbaum & Okubo (1994) Mimura & Yamaguti (1982) Nagai & Mimura (1983) Ikeda (1985) Ikeda & Nagai (1987) Hosono & Mimura (1989) Mimura and Yamaguti (1982) Issues Unbiological attraction Restriction to 1-d

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Clumping model Topaz, Bertozzi and Lewis (Bull. Math. Bio., 2006) Social attraction Sense averaged nearby pop. Climb gradients K spatially decaying, isotropic Weight 1, length scale 1 X X X X X X XX X X X X X X X X

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Clumping model Social attraction Sense averaged nearby pop. Climb gradients K spatially decaying, isotropic Weight 1, length scale 1 X X X X X X XX X X X X X X X X Social repulsion Descend pop. gradients Short length scale (local) Strength ~ density Speed ratio r X X X X X X XX X X X X X X X X Topaz, Bertozzi and Lewis (Bull. Math. Bio., 2006)

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/ d steady states Set flux to 0 Choose Transform to local eqn. Integrate densityslope integratio n constant speed ratio

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/ d steady states Ex.: velocity ratio r = 1, integration constant C = 0.9 Clump existence 2 param. family of clumps (for fixed r) slope  = 0 density  = 0  x

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Coarsening dynamics (example) Box length L = 8 , velocity ratio r = 1, mass M = 10

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Coarsening “Social behaviors that on short time and space scales lead to the formation and maintenance of groups,and at intermediate scales lead to size and state distributions of groups, lead at larger time and space scales to differences in spatial distributions of populations and rates of encounter and interaction with populations of predators, prey, competitors and pathogens, and with the physical environment. At the largest time and space scales, aggregation has profound consequences for ecosystem dynamics and for evolution of behavioral, morphological, and life history traits.” -- Okubo, Keshet, Grunbaum, “The dynamics of animal grouping” in Diffusion and Ecological Problems, Springer (2001)

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Coarsening Previous work on split and amalgamation of herds: Stochastic models (e.g. Holgate, 1967) log 10 (time) log 10 (number of clumps) L = 2000, M = 750, avg.over 10 runs Slepcev, Topaz and Bertozzi (in progress)

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Energy selection Box length L = 2 , velocity ratio r = 1, mass M = 2.51 Steady-state density profiles Energy max(  ) x

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Large aggregation limit Peak densityDensity profiles mass M Example: velocity ratio r = 1

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Large aggregation limit How to understand? Minimize energy over all possible rectangular density profiles Results Energetically preferred swarm has density 1.5r Preferred size is M/(1.5r) Independent of particular choice of K Generalizes to 2d

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/ d simulation Box length L = 40, velocity ratio r = 1, mass M = 600

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Conclusions Goals Minimal, realistic models Compact support, steep edges, constant density Model #1 Incompressible dynamics preserve swarm-like solution Asymptotic vortex states Model #2 Long-range attraction, short range dispersal nucleate swarm Analytical results for group size and density

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Background Two swarming models Future directions

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Locust swarms airborne locust density (x,t) ground locust density (x,t) Nonexistence of traveling band solutions (no swarms) Keshet, Watmough, Grunbaum (J. Math. Bio., 1998)

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Locust swarms Topaz, Bernoff, Logan and Toolson (in progress) Model framework Discrete framework, N locusts 2-d space,xxxxxxx Swarm motion aligned locally with wind [Uvarov (1977), Rainey (1989)] x (downwind) z

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Locust swarms Topaz, Bernoff, Logan and Toolson (in progress) Social interactions Pairwise Attractive/repulsive Morse-type x (downwind) z

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Locust swarms Topaz, Bernoff, Logan and Toolson (in progress) Gravity “Terminal velocity” G x (downwind) z

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Locust swarms Topaz, Bernoff, Logan and Toolson (in progress) Advection Aligned with wind Speed U Passive or active (Kennedy, 1951) x (downwind) z

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Locust swarms Topaz, Bernoff, Logan and Toolson (in progress) Boundary condition Impenetrable ground Locust motion on ground is minimal Locusts only move if vertical velocity is positive (takeoff) x (downwind) z

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Locust swarms Topaz, Bernoff, Logan and Toolson (in progress) H-stability Catastrophe

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Locust swarms Topaz, Bernoff, Logan and Toolson (in progress) H-stability Catastrophe N = 100 N = 1000

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Locust swarms social interactions (catastrophic) wind vertical structure + boundary + gravity Topaz, Bernoff, Logan and Toolson (in progress)

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Locust swarms Are catastrophic interactions a reasonable model? Conventional wisdom: Species have a preferred inter- organism spacing independent of group size (more or less) Nature says: Biological observations of migratory locust swarms vary over three orders of magnitude (Uvarov, 1977) Topaz, Bernoff, Logan and Toolson (in progress)