Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006 Social biological organisms: Aggregation...

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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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(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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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)