Social biological organisms Aggregation patterns and localization Chad
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? Large-scale coordinated movement Parrish & Keshet, Nature, 1999 Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
What is an aggregation? Large-scale coordinated movement No centralized control Dorset Wildlife Trust 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 UNFAO 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 Sinclair, 1977 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) Newton’s 2 nd Law Selfpropulsion Friction Social interaction Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Levine et al. (PRE, 2001) Interorganism potential Discrete model example Interorganism distance Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Discrete model example Levine et al. (PRE, 2001) Levine’s simulation results: N = 200 organisms 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) Conservation Law Diffusion Advection Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Continuum model example Density dependent drift Nonlocal attraction Mogilner and Keshet (JMB, 1999) 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) Density Mogilner/Keshet’s simulation results: Space 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 Connect movement rules to macroscopic properties? Mathematical Description Stable groups with finite extent? Sharp edges? Constant population 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
Goals 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 Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
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 “Incompressible” or “Divergence-free” “Potential” (See, e. g. , A Mathematical Introduction to Fluid Mechanics by Chorin and Marsden) 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 “Incompressible” or “Divergence-free” “Potential” (See, e. g. , A Mathematical Introduction to Fluid Mechanics by Chorin and Marsden) Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Incompressible velocity Assume initial condition: r = r 0 r=0 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 Chad Topaz, UCLA Department of Mathematics (t) 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 Incompressible case “Swarm-like” for all time Rotational motion, spiral arms, complex boundary Vortex-like asymptotic states Fish, slime molds, zooplankton, bacteria, … Potential case Expansion or contraction of population Model not rich enough to describe nucleation 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 XX X X X 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 XX X Social repulsion Descend pop. gradients Short length scale (local) Strength ~ density Speed ratio r X X XX X X X XXX Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
1 -d steady states Set flux to 0 Choose Transform to local eqn. Integrate density integration constant slope speed ratio Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
1 -d steady states Ex. : velocity ratio r = 1, integration constant C = 0. 9 r slope f = 0 x density r = 0 Clump existence 2 param. family of clumps (for fixed r) Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Coarsening dynamics (example) Box length L = 8 p, 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 Slepcev, Topaz and Bertozzi (in progress) log 10(number of clumps) Previous work on split and amalgamation of herds: Stochastic models (e. g. Holgate, 1967) L = 2000, M = 750, avg. over 10 runs log 10(time) Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Energy selection Box length L = 2 p, velocity ratio r = 1, mass M = 2. 51 Steady-state density profiles Energy x max(r) Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Large aggregation limit Example: velocity ratio r = 1 Peak density Density profiles mass M 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. 5 r Preferred size is M/(1. 5 r) Independent of particular choice of K Generalizes to 2 d Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
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 Keshet, Watmough, Grunbaum (J. Math. Bio. , 1998) airborne locust density (x, t) ground locust density (x, t) Nonexistence of traveling band solutions (no swarms) 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 z [Uvarov (1977), Rainey (1989)] x (downwind) 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 z x (downwind) Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Locust swarms Topaz, Bernoff, Logan and Toolson (in progress) Gravity “Terminal velocity” G z x (downwind) 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) z x (downwind) 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) z x (downwind) Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Locust swarms Topaz, Bernoff, Logan and Toolson (in progress) H-stability Chad Topaz, UCLA Department of Mathematics Catastrophe IPAM, 3/2/2006
Locust swarms Topaz, Bernoff, Logan and Toolson (in progress) H-stability Catastrophe N = 1000 Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Locust swarms Topaz, Bernoff, Logan and Toolson (in progress) social interactions (catastrophic) wind vertical structure + boundary + gravity Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Locust swarms Topaz, Bernoff, Logan and Toolson (in progress) Are catastrophic interactions a reasonable model? Conventional wisdom: Species have a preferred interorganism spacing independent of group size (more or less) Nature says: Biological observations of migratory locust swarms vary over three orders of magnitude (Uvarov, 1977) Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
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