Social biological organisms: Aggregation patterns and localization

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Social biological organismsAggregation patterns
and localization
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Swarm collaborators
Prof. Andrea Bertozzi (UCLA) Prof. Mark Lewis
(Alberta) Prof. Andrew Bernoff (Harvey
Mudd) Sheldon Logan (Harvey Mudd)Wyatt Toolson
(Harvey Mudd)
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Goals
Give some details (thanks, Andrea!) Highlight
different modeling approaches Focus on localized
aspect of swarms(how can localized solutions
arise in continuum models?)
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Background Two swarming models Future directions
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What is an aggregation?
Large-scale coordinated movement
Parrish Keshet, Nature, 1999
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What is an aggregation?
Large-scale coordinated movement No centralized
control
Dorset Wildlife Trust
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What is an aggregation?
Large-scale coordinated movement No centralized
control Interaction length scale (sight, smell,
etc.) ltlt group size
UNFAO
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What is an aggregation?
Large-scale coordinated movement No centralized
control Interaction length scale (sight, smell,
etc.) ltlt group size Sharp boundaries and constant
population density
Sinclair, 1977
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What is an aggregation?
Large-scale coordinated movement No centralized
control Interaction length scale (sight, smell,
etc.) ltlt group size Sharp boundaries and constant
population density Observed in bacteria, insects,
fish, birds, mammals
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Are all aggregations the same?
Length scales Time scales Dimensionality Topology
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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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Impacts/Applications
Economic/environmental Defense/algorithms
See Bonabeu et al., Swarm Intelligence, Oxford
University Press, New York, 1999.
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Impacts/Applications
Economic/environmental Defense/algorithms Sociol
ogy
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)
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Impacts/Applications
Economic/environmental Defense/algorithms Sociol
ogy
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Modeling approaches
Discrete(individual based, Lagrangian, )
Coupled ODEs Simulations Statistics Search of
parameter space Swarm-like states
Simple particle models (1970s) Suzuki, Sakai,
Okubo, Self-driven particles (1990s) Vicsek,
Czirok, Barabasi, Brownian particles (2000s)
Schweitzer, Ebeling, Erdman, Recently Chaté,
Couzin, DOrsogna, Eckhardt, Huepe, Levine,
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Discrete model example
Levine et al. (PRE, 2001)
Newtons 2nd Law
Socialinteraction
Self-propulsion
Friction
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Discrete model example
Levine et al. (PRE, 2001)
Interorganism potential
Interorganism distance
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Discrete model example
Levine et al. (PRE, 2001)
Levines simulation results N 200 organisms
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Modeling approaches
Continuum(Eulerian)
Continuum assumption Similar approach to
fluids PDEs Analysis Clump-like solutions Role of
parameters
Degenerate diffusion equations (1980s)Hosono,
Ikeda, Kawasaki, Mimura, Nagai, Yamaguti,
Variant nonlocal equations (1990s)Edelstein-K
eshet, Grunbaum, Mogilner,
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Continuum model example
Mogilner and Keshet (JMB, 1999)
Conservation Law
Diffusion
Advection
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Continuum model example
Mogilner and Keshet (JMB, 1999)
Densitydependentdrift
Nonlocalattraction
Nonlocalrepulsion
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Continuum model example
Mogilner and Keshet (JMB, 1999)
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Continuum model example
Mogilner and Keshet (JMB, 1999)
Mogilner/Keshets simulation results
Density
Space
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Bottom-up modeling?
Fish neurobiologyFish behaviorOcean current
profilesFluid dynamicsResource distribution
MathematicalDescription
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Pattern formation philosophy
Study high-level models Focus on essential
phenomena Explain cross-system similarities
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Pattern formation philosophy
Study high-level models Focus on essential
phenomena Explain cross-system similarities
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Pattern formation philosophy
Study high-level models Focus on essential
phenomena Explain cross-system similarities
Quantitative experimental data lacking Guide
bottom-up modeling efforts
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Pattern formation philosophy
Deterministic motionConserved populationAttracti
ve/repulsive social forces
Connect movement rules to macroscopic properties?
MathematicalDescription
Stable groups with finite extent? Sharp
edges?Constant population density?
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Background Two swarming models Future directions
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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
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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
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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)
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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)
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Incompressible velocity
Assume initial condition
r r0
W
r 0
Reduce dimension/Greens Theorem
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Incompressible velocity
Lagrangian viewpoint ? self-deforming curve
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Example numerical simulation
(N Gaussian, )
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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,
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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
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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
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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
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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
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1-d steady states
Set flux to 0ChooseTransform to local
eqn.Integrate
integrationconstant
speedratio
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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)
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Coarsening dynamics (example)
Box length L 8p, velocity ratio r 1, mass M
10
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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 groupingin Diffusion and Ecological
Problems, Springer (2001)
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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)
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Energy selection
Box length L 2p, velocity ratio r 1, mass M
2.51
Steady-statedensity profiles
Energy
x
max(r)
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Large aggregation limit
Example velocity ratio r 1
Peak density
Density profiles
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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
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2-d simulation
Box length L 40, velocity ratio r 1, mass M
600
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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
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Background Two swarming models Future directions
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Locust swarms
Keshet, Watmough, Grunbaum (J. Math. Bio., 1998)
Nonexistence of traveling band solutions (no
swarms)
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Locust swarms
Topaz, Bernoff, Logan and Toolson (in progress)
Model framework Discrete framework, N locusts 2-d
space,xxxxxxx Swarm motion aligned locally with
windUvarov (1977), Rainey (1989)
z
x (downwind)
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Locust swarms
Topaz, Bernoff, Logan and Toolson (in progress)
Social interactions Pairwise Attractive/repulsive
Morse-type
z
x (downwind)
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Locust swarms
Topaz, Bernoff, Logan and Toolson (in progress)
Gravity Terminal velocity G
z
x (downwind)
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Locust swarms
Topaz, Bernoff, Logan and Toolson (in progress)
Advection Aligned with windSpeed UPassive or
active (Kennedy, 1951)
z
x (downwind)
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Locust swarms
Topaz, Bernoff, Logan and Toolson (in progress)
Boundary condition Impenetrable groundLocust
motion on ground is minimalLocusts only move if
vertical velocity is positive (takeoff)
z
x (downwind)
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Locust swarms
Topaz, Bernoff, Logan and Toolson (in progress)
H-stability
Catastrophe
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Locust swarms
Topaz, Bernoff, Logan and Toolson (in progress)
H-stability
Catastrophe
N 100
N 1000
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Locust swarms
Topaz, Bernoff, Logan and Toolson (in progress)
social interactions (catastrophic)wind vertical
structure boundary gravity
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Locust swarms
Topaz, Bernoff, Logan and Toolson (in progress)
Are catastrophic interactions a reasonable
model? Conventional wisdom Species have a
preferred inter-organism spacing independent of
group size(more or less) Nature
saysBiological observations of migratory locust
swarms vary over three orders of magnitude
(Uvarov, 1977)