Social biological organisms: Aggregation patterns and localization

1. Social biological organisms:Aggregation patterns and localization

2. Swarm collaborators Prof. Andrea Bertozzi (UCLA) Prof. Mark Lewis (Alberta) Prof. Andrew Bernoff (Harvey Mudd) Sheldon Logan (Harvey Mudd)Wyatt Toolson (Harvey Mudd)

3. Goals Give some details (thanks, Andrea!) Highlight different modeling approaches Focus on localized aspect of swarms(how can localized solutions arise in continuum models?)

4. Background Two swarming models Future directions

5. What is an aggregation? Large-scale coordinated movement Parrish & Keshet, Nature, 1999

6. What is an aggregation? Large-scale coordinated movement No centralized control Dorset Wildlife Trust

7. What is an aggregation? Large-scale coordinated movement No centralized control Interaction length scale (sight, smell, etc.) << group size UNFAO

8. 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

9. 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…

10. Are all aggregations the same? Length scales Time scales Dimensionality Topology

11. 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)

12. Impacts/Applications Economic/environmental Defense/algorithms See: Bonabeu et al., Swarm Intelligence, Oxford University Press, New York, 1999.

13. Impacts/Applications Economic/environmental Defense/algorithms “Sociology” Critical mass bicycle protest: "The people up front and the people in back are in constantcommunication, 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)

14. Impacts/Applications Economic/environmental Defense/algorithms “Sociology”

15. 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, …

16. Discrete model example Levine et al. (PRE, 2001) Newton’s 2nd Law Socialinteraction Self-propulsion Friction

17. Discrete model example Levine et al. (PRE, 2001) Interorganism potential Interorganism distance

18. Discrete model example Levine et al. (PRE, 2001) Levine’s simulation results: N = 200 organisms

19. 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, …

20. Continuum model example Mogilner and Keshet (JMB, 1999) Conservation Law Diffusion Advection

21. Continuum model example Mogilner and Keshet (JMB, 1999) Densitydependentdrift Nonlocalattraction Nonlocalrepulsion

22. Continuum model example Mogilner and Keshet (JMB, 1999)

23. Continuum model example Mogilner and Keshet (JMB, 1999) Mogilner/Keshet’s simulation results: Density Space

24. Bottom-up modeling? Fish neurobiologyFish behaviorOcean current profilesFluid dynamicsResource distribution MathematicalDescription

25. Faraday wave experiment(Kudrolli, Pier and Gollub, 1998) Numerical simulation of achemical reaction-diffusion system(Courtesy of M. Silber) Pattern formation philosophy Study high-level models Focus on essential phenomena Explain cross-system similarities

26. Pattern formation philosophy Study high-level models Focus on essential phenomena Explain cross-system similarities

27. Pattern formation philosophy Study high-level models Focus on essential phenomena Explain cross-system similarities Quantitative experimental data lacking Guide bottom-up modeling efforts

28. Pattern formation philosophy Deterministic motionConserved populationAttractive/repulsive social forces Connect movement rules to macroscopic properties? MathematicalDescription Stable groups with finite extent? Sharp edges?Constant population density?

29. Background Two swarming models Future directions

30. 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

31. 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

32. “Incompressible”or“Divergence-free” “Potential” 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)

33. “Incompressible”or“Divergence-free” “Potential” 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)

34. unit tangent decaying Incompressible velocity Assume initial condition: r = r0 W r = 0 Reduce dimension/Green’s Theorem:

35. unit tangent decaying W(t) Incompressible velocity Lagrangian viewpoint  self-deforming curve

36. Example numerical simulation (N = Gaussian, …)

37. 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,…

38. 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

39. Previous results on clumping nonlocal attraction local dispersal Mimura and Yamaguti (1982) Kawasaki (1978)Grunbaum & Okubo (1994) Mimura & Yamaguti (1982)Nagai & Mimura (1983) Ikeda (1985)Ikeda & Nagai (1987)Hosono & Mimura (1989) Issues Unbiological attraction Restriction to 1-d

40. X X X X X X X X X X X X X X X X 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

41. X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Clumping model Topaz, Bertozzi and Lewis (Bull. Math. Bio., 2006) Social repulsion Descend pop. gradients Short length scale (local) Strength ~ density Speed ratio r Social attraction Sense averaged nearby pop. Climb gradients K spatially decaying, isotropic Weight 1, length scale 1

42. density slope 1-d steady states Set flux to 0ChooseTransform to local eqn.Integrate integrationconstant speedratio

43. x 1-d steady states Ex.: velocity ratio r = 1, integration constant C = 0.9 r slope f = 0 density r = 0 Clump existence2 param. family of clumps (for fixed r)

44. Coarsening dynamics (example) Box length L = 8p, velocity ratio r = 1, mass M = 10

45. 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)

46. Coarsening Slepcev, Topaz and Bertozzi (in progress) Previous work on split and amalgamation of herds: Stochastic models (e.g. Holgate, 1967) L = 2000, M = 750, avg.over 10 runs log10(number of clumps) log10(time)

47. Energy selection Box length L = 2p, velocity ratio r = 1, mass M = 2.51 Steady-statedensity profiles Energy x max(r)

48. mass M Large aggregation limit Example: velocity ratio r = 1 Peak density Density profiles

49. 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

50. 2-d simulation Box length L = 40, velocity ratio r = 1, mass M = 600