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

Chad Topaz, UCLA Department of Mathematics IPAM, 3/2/2006

Social biological organisms:Aggregation 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)


Goals

Give some details (thanks, Andrea!)

Highlight different modeling approaches

Focus on localized aspect of swarms(how can localized solutions arise in continuum

models?)


Background

Two swarming models

Future directions


What is an aggregation?

Parrish

& K

esh

et, N

atu

re,

19

99

Large-scale coordinated movement



Dorse

t Wild

life T

rust

Large-scale coordinated movementNo centralized control



UN

FAO


Interaction length scale (sight, smell, etc.) << group size



Sin

clair, 1

97

7



Sharp boundaries and constant population density





Sharp boundaries and constant population density

Observed in bacteria, insects, fish, birds, mammals…


Are all aggregations the same?

Length scalesTime scalesDimensionalityTopology


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)





Economic/environmental Defense/algorithms

See: Bonabeu et al., Swarm Intelligence, Oxford University Press, New York, 1999.




“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)




“Sociology”


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, …


Discrete model example Levine et al. (PRE, 2001)

Self-propulsio

n

Friction Socialinteracti

on

Newton’s 2nd Law


Discrete model example

Interorganism distance

Inte

rorg

anis

m

pote

nti

al

Levine et al. (PRE, 2001)


Discrete model example

Levine’s simulation results: N = 200 organisms

Levine et al. (PRE, 2001)


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


Continuum model example Mogilner and Keshet (JMB, 1999)

Diffusion Advection

Conservation Law



Densitydependen

tdrift

Nonlocalattractio

n

Nonlocalrepulsion



Mogilner/Keshet’s simulation results:

Space

Densi

ty


Bottom-up modeling?

Fish neurobiologyFish behaviorOcean current

profilesFluid dynamics

Resource distribution

MathematicalDescription


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)





Quantitative experimental data lackingGuide bottom-up modeling efforts



Deterministic motionConserved population

Attractive/repulsive social forces

MathematicalDescription

Stable groups with finite extent? Sharp edges?Constant population

density?

Connect movement rules to macroscopic properties?


Background

Two swarming models

Future directions


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


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


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


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


Incompressible velocity

Assume initial condition:

ρ ρ

ρ

Reduce dimension/Green’s Theorem:

decaying unit tangent


Incompressible velocity

decaying unit tangent

Lagrangian viewpoint self-deforming curve (t)


Example numerical simulation

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(N = Gaussian, …)


Overview of dynamics

Incompressible case

“Swarm-like” for all timeRotational motion, spiral arms, complex boundary

Vortex-like asymptotic statesFish, slime molds, zooplankton, bacteria,…


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,…


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


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


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)


1-d steady states

Set flux to 0ChooseTransform to local eqn.Integrate

density

slope

integration

constant

speedratio


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


Coarsening dynamics (example)

Box length L = 8, velocity ratio r = 1, mass M = 10


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)


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)


Energy selection

Box length L = 2, velocity ratio r = 1, mass M = 2.51

Steady-statedensity profiles

Energy

max(ρ)

x


Large aggregation limit

Peak density Density profiles

mass M

Example: velocity ratio r = 1


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


2-d simulation

Box length L = 40, velocity ratio r = 1, mass M = 600

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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


Background

Two swarming models

Future directions


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)


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



Social interactionsPairwise

Attractive/repulsiveMorse-type

x (downwind)

z



Gravity“Terminal velocity” G

x (downwind)

z



AdvectionAligned with wind

Speed UPassive or active (Kennedy, 1951)

x (downwind)

z



Boundary conditionImpenetrable ground

Locust motion on ground is minimalLocusts only move if vertical velocity is positive

(takeoff)

x (downwind)

z



H-stability Catastrophe



H-stability Catastrophe

N = 100

N = 1000


Locust swarms

social interactions (catastrophic)wind

vertical structure + boundary + gravity



Topaz, Bernoff, Logan and Toolson (in progress)


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)

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

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