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
Motivation
• Vortex-swarming of ‘self-propelled’ animals
Blue Planet: Seas of Life, Discovery Channel 2001
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
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
Daphnia Facts and Experimental Setup:
•Daphnia are photosensitive
 Attracted to visible light
2D
 Repelled by ultraviolet
r
 Blind to infrared
•Predator avoidance strategies:
 Diel Vertical Migration
 Swarm Formations
3D ‘Strickler Lab’
Observation of many Daphnia:
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.
Observation of single Daphnia:
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
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.
P(Lang)
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.
- <MC> = 11.8
(more than one turn on average)
Motion of single Daphnia
in darkness
- M=1599 moves from 8 different animals
- Cumulative recorded time: trec = 455s
- Average speed vavg = 6.18 ± 0.57 mm/s
y
[mm]
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).
Avoidance
maneuver of
Daphnia in
darkness
non-living
obstacle
bumping
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 Particles w/ 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]
Vicsek Model
 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

1
N
v
i
i
Levine Model
 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 selfpropelling force fi with fixed magnitude α; friction coefficient γ
- Direction of fi either 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
mi  t vi   fi   vi  U
t ri  vi
without alignment: fi  vi
or with alignment (range lc):

fi   v j exp  ri  rj / lc
j i

(a) initial random setting
(b), (c), (d) after 20, 50, 300 iterations without alignment
Active Brownian Particle Model
 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 force
connected to energy storage depot e(t); velocity dependent friction γ(v)
- External parabolic potential U(r) and noise F(t)
m t v  d2 e(t ) v   (v ) v  U (r )  F (t )
t r  v
- Energy depot: space-dependent take-up q(r), internal dissipation c e(t),
conversion of internal energy into kinetic energy d2 e(t)v2
 t e ( t )  q ( r )  c e ( t )  d 2 e( t ) v 2
- Energy depot analysis (for q(r) = q0 ):
d 2 q0
 (v )   0 
c  d2 v 2
γ(v)
Hopf bifurcation:
bifurcation parameter 
  0;
e.g. q0  0
Noisy Fixed Point
q0
c
   0
 0 d2
  0; q0  q
crit
0

0 c
d2
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?
 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
CCW and CW motions
equally probable.
 unbroken
symmetry/ circling in
both directions
 Daphnia at low density show the nonsymmetry broken motion (CCW and CW
equally probable), at high density spontaneous symmetry breaking (vortex state)
- What can cause symmetry to break/ the vortex to form?
Symmetry breaking could be induced by:
Hydrodynamic interactions (as modeled via Oseen-contributions
added to ABP model [Erdmann and Ebeling, FNL (2003)])
mt vi   F vF   (vi )vi  U (ri )  F (t )
rij  rij 
R
vF (rij )    
 vj
2
rij 
j rij 

Avoidance maneuvers (as simulated by ‘privat spheres’ modeled
by repulsive potentials added to ABP model [Helbing, Molnár, and
Schweitzer (1994); Mach et al., preprint])
m  t vi  d 2 e(t ) vi   (vi ) vi  U (ri )  F (t )   f ij
p
 Ri 
f ij  V ( Ri ) 
exp    (r i  r j )
 Ri
  
j i
Model for Daphnia: Random Walk with shortrange 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
• Daphnia in light field:
After RW at time 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
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
Simulation results for various L:
P(θ)
<MC>
θ
L
L=0.4
P(Lang)
L=0.4
P(MC)
MC
Model: Interacting RW with DTA
• Simple model for (vortex-) swarming Daphnia
Indirect inter-agent interactions via water drag are incorporated by adding
ACW  ACCW
‘alignment’ or ‘water drag’ kick proportional to
V
Na
,
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.
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
by Oseen-contributions [Erdmann & Ebeling (2003)], RW Model)
- Avoidance maneuvers (observed in experiment and by ‘privat
spheres’ modeled by repulsive potentials [Mach et al., preprint])