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Swarming of the Zooplankton Daphnia : Experiment and Theory

IPAM/UCLA Workshop on Biological and Artificial Swarms, 10/03. Swarming of the Zooplankton Daphnia : Experiment and Theory. Anke Ordemann and Frank Moss. Collaborators:.

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Swarming of the Zooplankton Daphnia : Experiment and Theory

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  1. IPAM/UCLA Workshop on Biological and Artificial Swarms, 10/03 Swarming of the Zooplankton Daphnia: Experiment and Theory Anke Ordemann and Frank Moss Collaborators: Ai Nihongi and Rudi Strickler, Great LAKES Water Institute, UWM Frank Schweitzer, FhI for Autonomous Intelligent Systems, Germany Lutz Schimansky-Geier and colleagues at HU Berlin Thanks to Wolfgang Alt, Beatrix Beisner, Winfried Lampert, David Russell, Allan Tessier, and Lon Wilkens Supported by the Alexander von Humboldt Foundation and the US Office of Naval Research

  2. Motivation • Vortex-swarming of ‘self-propelled’ animals Blue Planet: Seas of Life, Discovery Channel 2001

  3. Observation • Zooplankton have been observed to perform vortex-swarming in the field Oceanic zooplankton Anchylomera blossevilli vortex-swarming in the field[Lobel and Randall, J. Plankt. Res. 8, 253 (1986)] • Zooplankton Daphnia performs vortex-swarming in our lab

  4. How does this multi-agent vortex form? Transition to vortex(Top-view, very high Daphnia density) CCW vortex with spiral arms  Water inside vortex turns in same direction as Daphnia -Hydrodynamic coupling/alignment of agents

  5. Daphnia Facts and Experimental Setup: • Daphnia are photosensitive • Attracted to visible light • Repelled by ultraviolet • Blind to infrared • Predator avoidance strategies: 2D r • Diel Vertical Migration • Swarm Formations 3D ‘Strickler Lab’

  6. Observation ofmanyDaphnia: Five successive positions of Daphnia taken in intervals of 0.3s (black/white inverted). Bottom-view of many Daphnia moving around light shaft (radius r=5mm). Movie is speeded up 4 times.  Many Daphnia circle in both directions around light shaft, frequently change their circling direction 3D view of many Daphnia moving around light shaft.

  7. Observation ofsingleDaphnia: • y [mm] x [mm] Bottom view of individual Daphnia moving around light shaft. Movie is speeded up 3 times. Track of one Daphnia individually circling horizontally around light shaft (146s). • Single Daphnia circle individually in both directions around light shaft, frequently change their circling direction - Circular motion is not an emergent property of large Daphnia concentrations

  8. How to characterize observed circular motion? - M=624 moves from 4 different animals - Average speed vavg = 5.71 ± 1.35 mm/s Distribution P(θ) of angle θ between Daphnia heading and direction to light for every move. Distribution P(Lang) of angular momentum Lang for Daphnia moves. Distribution P(MC) of number of successive moves MC heading clockwise/counterclockwise before changing the circling direction. P(Lang) • <MC> = 11.8 (more than one turn on average)

  9. Motion of single Daphnia indarkness y[mm] - M=1599 moves from 8 different animals - Cumulative recorded time: trec = 455s - Average speed vavg = 6.18 ± 0.57 mm/s x[mm] Track of one Daphnia, M =200, trec =60s. Distribution of absolute turning angle (DTA) of Daphnia between successive moves. DTA also measured for zooplankton copepod: Schmitt and Seuront, Phys. A 301, 375 (2001).

  10. Avoidance maneuver of Daphnia indarkness non-living obstacle bumping

  11. 2D particle models for self-propelled agents • Boids [C. Reynolds, Comp. Graph. 21, 25 (1987)] • Biological Models [Review: Parrish et al., Biol. Bull. 202, 296 (2002)]  Self-Propelled Interacting Particles(SPIP) [Vicsek et al., PRL 75, 1226 (1995); Vicsek: Fluctuations and Scaling in Biology, Oxford, 2001]  SPIP with short range attraction and long range repulsion[Levine et al., PRE 63, 017101 (2001)] • Interacting Active Brownian Particlesw/ internal energy depot Single Active Brownian Particle[Schweitzer et al., PRL 80, 5044 (1998)] Global coupling via mean angular momentum of swarm [Schweitzer et al., PRE 64, 021110 (2001)] Particle-Particle Interaction [Erdmann et al., PRE 65, 061106 (2002)] Hydrodynamic Interaction [Erdmann and Ebeling, FNL 3, 2145 (2003)] Avoidance Maneuvers [Mach et al., preprint]

  12. VicsekModel •  Self-Propelled Interacting Particles [Vicsek et al., PRL 75, 1226 (1995)] • N locally aligning particles with noise and constant velocity v • Periodic boundary conditions • Parameters: density of particles and amplitude of noise (a) initial random setting (b) low density, low noise (c) high density, high noise (d) high density, low noise Order parameter: average momentum

  13. LevineModel Short range repulsion and long range attraction added to the model of Self-Propelled Interacting Particles [Levine et al., PRE 63, 017101 (2001)] • N particles with mass mi, position ri, velocity vi, experience a self-propelling force fi with fixed magnitude α; friction coefficient γ • Direction of fieither parallel to vi or by averaging over neighboring vj • Attractive force between particles with interaction range la • Short-range repulsive force between particles with interaction range lr without alignment: or with alignment (range lc): (a) initial random setting (b), (c), (d) after 20, 50, 300 iterations without alignment

  14. Active Brownian ParticleModel  Single Active Brownian Particle (ABP)with internal energy depot [Schweitzer et al., PRL 80, 5044 (1998)] • Particle with mass m, position r, velocity v, self-propelling forceconnected to energy storage depot e(t); velocity dependent friction γ(v) • External parabolic potential U(r) and noise F(t) -Energy depot: space-dependent take-up q(r), internal dissipation ce(t), conversion of internal energy into kinetic energy d2e(t)v2 - Energy depot analysis (for q(r) = q0 ): γ(v)

  15. Hopf bifurcation: bifurcation parameter  Noisy Fixed Point Noisy limit cycle  Single Daphnia show the limit cycle (circular motion in a central attractive potential) of the ABP Model – but how and why do Daphnia do it?

  16. Interacting Active Brownian Particles [Schweitzer et al., PRE 64, 021110 (2001); Ebeling and Schweitzer, Theory Biosci. 120, 20 (2001); Erdmann et al., PRE 65, 061106 (2002); Erdmann and Ebeling, FNL 3, 2145 (2003); Mach et al., preprint] • Additional global coupling of the agents to the center of mass of the swarm unbroken symmetry/ circling in both directions CCW and CW motions equally probable.  Daphnia at low density show the non-symmetry broken motion (CCW and CW equally probable), at high density spon-taneous symmetry breaking (vortex state) -What can cause symmetry to break/ the vortex to form?

  17. Symmetry breaking could be induced by: • Hydrodynamic interactions (as modeled via Oseen-contributions added to ABP model [Erdmann and Ebeling, FNL (2003)]) • Avoidance maneuvers (as simulated by ‘privat spheres’ modeled by repulsive potentials added to ABP model [Helbing, Molnár, and Schweitzer (1994); Mach et al., preprint])

  18. Model for Daphnia: Random Walk with short-range temporal correlation and attraction to light • Basic model for Daphnia in darkness:Random Walk (RW)where direction of next time step ti+1 is randomly chosen from observed DTA • Daphniain light field: After RW attime ti chooses angle from DTA, it gets an additional kick of strength r*L/(L-1) towards the light(parabolic potential) and the final heading is rescaled to unit length (0  L < 1) -strength L of light attraction - distance r of RW to light - angle θ between RW heading and direction to light - turning angle α

  19. Simulation results for attraction strength L=0.4: 5 0 Light at (0,0) -5 -5 0 5 Simulation of one RW with DTA and attraction to light. Start at (2,10), t = 100 time steps. Typical track of one RW with DTA and attraction to light. Start at (2,10),t= 100.  RW with DTA in parabolic potential shows circular motion in both directions for intermediate attraction strength L = 0.4

  20. Simulation results for various L: P(θ) <MC> L θ L=0.4 L=0.4 P(Lang) P(MC) MC

  21. Model: Interacting RW with DTA • Simple model for (vortex-) swarming Daphnia • Indirect inter-agent interactions via water drag are incorporated by adding ‘alignment’ or ‘water drag’ kick proportional to The direction of the kick is CW for V > 0, CCW for V < 0. Parameters: strength of kick, range a of interaction O t Typical example for time evolution of O = (NCW  NCCW )/Ntot with Ntot = 30.

  22. Discussion: • Simple RW model with DTA and attraction to light successfully models observed circular motion of individual Daphnia • Sufficient ingredients/local rules for circular motion are • Self-propelled particles (finite non-zero velocity) • Assigned preference to move in ‘forward’ direction • - Confinement of incompressible particles/Attraction • For vortex swarming to take place symmetry breaking is necessary. This is caused by alignment added to the above rules: • Alignment of neighboring agents [Levine at al. (2001)] • Hydrodynamic interactions(observed in experiment and modeled byOseen-contributions [Erdmann & Ebeling (2003)], RW Model) • - Avoidance maneuvers(observed in experiment and by ‘privat spheres’ modeled by repulsive potentials [Mach et al., preprint])

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