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Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Social biological organisms:Aggregation patterns and localization
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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?
Parrish
& K
esh
et, N
atu
re,
19
99
Large-scale coordinated movement
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
What is an aggregation?
Dorse
t Wild
life T
rust
Large-scale coordinated movementNo centralized control
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
What is an aggregation?
UN
FAO
Large-scale coordinated movementNo centralized control
Interaction length scale (sight, smell, etc.) << group size
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
What is an aggregation?
Sin
clair, 1
97
7
Large-scale coordinated movementNo centralized control
Interaction length scale (sight, smell, etc.) << group size
Sharp boundaries and constant population density
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
What is an aggregation?
Large-scale coordinated movementNo 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 scalesTime scalesDimensionalityTopology
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)
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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
€
r vi = velocity of ith organism
€
r xi = position of ith organism
Discrete(individual based, Lagrangian,
…)
Coupled ODE’sSimulations
StatisticsSearch 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)
Self-propulsio
n
Friction Socialinteracti
on
Newton’s 2nd Law
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Discrete model example
Interorganism distance
Inte
rorg
anis
m
pote
nti
al
Levine et al. (PRE, 2001)
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Discrete model example
Levine’s simulation results: N = 200 organisms
Levine et al. (PRE, 2001)
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Modeling approaches
Continuum(Eulerian)
Continuum assumptionSimilar approach to fluids
PDEsAnalysis
Clump-like solutionsRole of parameters
Degenerate diffusion equations (1980’s):Hosono, Ikeda, Kawasaki, Mimura, Nagai, Yamaguti, …
Variant nonlocal equations (1990’s):Edelstein-Keshet, Grunbaum, Mogilner, …
€
v( r x, t)= velocity field
€
ρ( r x, t)= population density field
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Continuum model example Mogilner and Keshet (JMB, 1999)
Diffusion Advection
Conservation Law
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Continuum model example Mogilner and Keshet (JMB, 1999)
Densitydependen
tdrift
Nonlocalattractio
n
Nonlocalrepulsion
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)
Mogilner/Keshet’s simulation results:
Space
Densi
ty
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Bottom-up modeling?
Fish neurobiologyFish behaviorOcean current
profilesFluid dynamics
Resource distribution
MathematicalDescription
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Pattern formation philosophy
Study high-level modelsFocus on essential phenomena
Explain cross-system similarities
Faraday wave experiment(Kudrolli, Pier and Gollub, 1998)
Numerical simulation of achemical reaction-diffusion
system(Courtesy of M. Silber)
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Pattern formation philosophy
Study high-level modelsFocus on essential phenomena
Explain cross-system similarities
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Pattern formation philosophy
Study high-level modelsFocus on essential phenomena
Explain cross-system similarities
Quantitative experimental data lackingGuide bottom-up modeling efforts
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Pattern formation philosophy
Deterministic motionConserved population
Attractive/repulsive social forces
MathematicalDescription
Stable groups with finite extent? Sharp edges?Constant population
density?
Connect movement rules to macroscopic properties?
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
Modeling goals:≥ 2 spatial dimensions
Nonlocal, spatially-decaying interactions
Mathematical goals:Characterize 2-d dynamics
Find biologically realistic aggregation solutionsConnect macroscopic properties to movement rules
Goals
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
2-d continuum model Topaz and Bertozzi (SIAP, 2004)
Assumptions:Conserved populationDeterministic motion
Velocity due to nonlocal social interactionsVelocity 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
(See, e.g., A Mathematical Introduction to Fluid Mechanics by Chorin and Marsden)
€
r v
“Incompressible”or
“Divergence-free”
“Potential”
€
r v =∇⊥Φ +∇Ψ
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
(See, e.g., A Mathematical Introduction to Fluid Mechanics by Chorin and Marsden)
€
r v
“Incompressible”or
“Divergence-free”
“Potential”
€
K =∇⊥N +∇P
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Incompressible velocity
Assume initial condition:
ρ ρ
ρ
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 (t)
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Example numerical simulation
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are needed to see this picture.
(N = Gaussian, …)
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Overview of dynamics
Incompressible case
“Swarm-like” for all timeRotational motion, spiral arms, complex boundary
Vortex-like asymptotic statesFish, slime molds, zooplankton, bacteria,…
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Overview of dynamics
Potential case
Expansion or contraction of populationModel not rich enough to describe nucleation
Incompressible case
“Swarm-like” for all timeRotational motion, spiral arms, complex boundary
Vortex-like asymptotic statesFish, slime molds, zooplankton, bacteria,…
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Previous results on clumping
nonlocal
attraction
local dispers
al
Kawasaki (1978)Grunbaum & Okubo
(1994)Mimura & Yamaguti
(1982)Nagai & Mimura (1983)
Ikeda (1985)Ikeda & Nagai (1987)
Hosono & Mimura (1989)
Mimura and Yamaguti (1982)
IssuesUnbiological 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 attractionSense averaged nearby
pop.Climb gradients
K spatially decaying, isotropic
Weight 1, length scale 1
XXXX
X
XX X
XX
XX
X
X
X
X
€
l
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Clumping model
Social attractionSense averaged nearby
pop.Climb gradients
K spatially decaying, isotropic
Weight 1, length scale 1
XXXX
X
XX X
XX
XX
X
X
X
X
€
l
Social repulsionDescend pop. gradientsShort length scale (local)
Strength ~ densitySpeed ratio r
X
XX XX
XX X
XX
XX
X
XX
X
Topaz, Bertozzi and Lewis (Bull. Math. Bio., 2006)
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
1-d steady states
Set flux to 0ChooseTransform to local eqn.Integrate
density
slope
integration
constant
speedratio
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
1-d steady states
Ex.: velocity ratio r = 1, integration constant C = 0.9
Clump existence2 param. family of clumps (for
fixed r)
slope = 0
density ρ = 0
ρ
x
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Coarsening dynamics (example)
Box length L = 8, 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
Previous work on split and amalgamation of herds:Stochastic models (e.g. Holgate, 1967)
log10(time)
log
10(n
um
ber
of
clum
ps) L = 2000, M = 750, avg.over 10
runs
Slepcev, Topaz and Bertozzi (in progress)
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Energy selection
Box length L = 2, velocity ratio r = 1, mass M = 2.51
Steady-statedensity profiles
Energy
max(ρ)
x
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Large aggregation limit
Peak density Density profiles
mass M
Example: velocity ratio r = 1
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Large aggregation limit
How to understand?Minimize energy
over all possible rectangular density profiles
ResultsEnergetically 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
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Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Conclusions
GoalsMinimal, realistic models
Compact support, steep edges, constant density
Model #1Incompressible dynamics preserve swarm-like
solutionAsymptotic vortex states
Model #2Long-range attraction, short range dispersal
nucleate swarmAnalytical 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
airborne locust density
(x,t)
ground locust density
(x,t)
Nonexistence of traveling band solutions (no
swarms)
Keshet, Watmough, Grunbaum (J. Math. Bio., 1998)
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Locust swarms Topaz, Bernoff, Logan and Toolson (in progress)
Model frameworkDiscrete framework, N locusts
2-d space,xxxxxxx Swarm motion aligned locally with wind
[Uvarov (1977), Rainey (1989)]
x (downwind)
z
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Locust swarms Topaz, Bernoff, Logan and Toolson (in progress)
Social interactionsPairwise
Attractive/repulsiveMorse-type
x (downwind)
z
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Locust swarms Topaz, Bernoff, Logan and Toolson (in progress)
Gravity“Terminal velocity” G
x (downwind)
z
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Locust swarms Topaz, Bernoff, Logan and Toolson (in progress)
AdvectionAligned with wind
Speed UPassive or active (Kennedy, 1951)
x (downwind)
z
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Locust swarms Topaz, Bernoff, Logan and Toolson (in progress)
Boundary conditionImpenetrable ground
Locust motion on ground is minimalLocusts only move if vertical velocity is positive
(takeoff)
x (downwind)
z
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Locust swarms Topaz, Bernoff, Logan and Toolson (in progress)
H-stability Catastrophe
Chad Topaz, UCLA Department of Mathematics 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
social interactions (catastrophic)wind
vertical structure + boundary + gravity
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are needed to see this picture.
Topaz, Bernoff, Logan and Toolson (in progress)
Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006
Locust swarms
Are catastrophic interactions a reasonable model?
Conventional wisdom:Species have a preferred inter-organism spacing independent
of group size(more or less)
Nature says:Biological observations of
migratory locust swarms vary over three orders of magnitude
(Uvarov, 1977)
Topaz, Bernoff, Logan and Toolson (in progress)