Social biological organisms:
Aggregation patterns and localization
QuickTime™ and a
H.263 decompressor
are needed to see this picture.
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
QuickTime™ and a
H.263 decompressor
are needed to see this picture.
$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, …)
r
xi = position of ith organism
r
v i = velocity of ith organism
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 2nd 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)
r
(x , t)  population density field
r
v(x , t)  velocity field
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
v field
on  can be uniquely decomposed in the form
r
v    
“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
v field
on  can be uniquely decomposed in the form
K    N  P
“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:
0

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, …)
QuickTime™ and a
YUV420 codec decompressor
are needed to see this picture.
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
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
X
X
XX
X X
X
Social repulsion
Descend pop. gradients
Short length scale (local)
Strength ~ density
Speed ratio r
X
X
X
X
X
XX
X
X X
X XX
X
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
integratio
n
constant
density
speed
ratio
slope
Chad Topaz, UCLA Department of Mathematics
IPAM, 3/2/2006
1-d steady states
Ex.: velocity ratio r = 1, integration constant C = 0.9

slope f = 0
x
density  = 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 = 8p, 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)
log10(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
log10(time)
Chad Topaz, UCLA Department of Mathematics
IPAM, 3/2/2006
Energy selection
Box length L = 2p, velocity ratio r = 1, mass M = 2.51
Steady-state
density profiles
Energy
x
max()
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 1.5r
Preferred size is M/(1.5r)
Independent of particular choice of K
Generalizes to 2d
Chad Topaz, UCLA Department of Mathematics
IPAM, 3/2/2006
2-d simulation
Box length L = 40, velocity ratio r = 1, mass M = 600
QuickTime™ and a
Video decompressor
are needed to see this picture.
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 = 100
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
QuickTime™ and a
H.263 decompressor
are needed to see this picture.
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