Collaboration with Federico V

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The continuum description of individual-based
models


Cristóbal López
Cristóbal López
Cristóbal López
Collaboration with Federico Vázquez
http//ifisc.uib.es - Mallorca - Spain
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Outline of the talk
Hola tonto

• First part Generalities (Cristóbal)
• Introduction.
• Description of complex systems individual and
  continuum approaches.
• The need of proper continuum descriptions via an
  example.
• - Second part Applications to the voter model
  and language dynamics ( Federico)

totonto
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Introduction
Most real-life systems are complex many units
interacting in a non-linear way give rise to not
obvious (unexpected) collective behavior.

• Ant colonies.
• Plankton communities.
• Traffic.
• People.
• Cells populations, bacteria.
• Atoms.
• Stars.
• Etc..

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HOW TO STUDY COMPLEX SYSTEMS?
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(No Transcript)
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The discrete nature of organisms or chemical
molecules is missed in general when a continuum
approach (via density or concentration fields) is
used to model processes in Nature. This is
specially important in situations close to
extinctions, and other critical situations.
However, continuum descriptions have many
advantages stability analysis and pattern
formation.
Therefore, there is the need to formulate
Individual Based Models (IBMs), and then
deriving continuum equations of these
microscopic particle systems that still remain
discreteness effects.
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The individual level

• Some terminology
• Agent-based models (Sociology, Computer Science,
  Game Theory)
• Individual-based models (Ecology, Biology).
• Interacting particle systems (Physics).

• ABMs describe systems at a individual/microscopic
  level.
• An Agent-Based Model (ABM) simulates operations
  of multiple
• agents to recreate behavior of complex
  phenomena.

• Advantage ABMs can be used as computer
  experiments to
• explore the behavior of a system.

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Examples of interacting particle systems
Ecology
Species competition. Invasion processes.
Predator-prey systems (Lotka Volterra).
Biology
Epidemic spreading (ISI, IRSI). Allele
frequency (genetics). Bacteria dynamics.
Neural networks. Tumor growth.
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Social Science
Opinion spreading. Cultural propagation.
Language dynamics.
Surface Physics/ Chemistry
Catalytic reactions. Deposition/
reaction-diffusion/ aggregation.
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Description of ABMs

• Analytical treatment of systems provides insight
  of phenomena.
• But.., systems are composed by a huge number of
  agents!
• (many degrees of freedom).
• It is hard and unpractical to develop an
  analytical framework
• (equations) to describe the evolution of each
  single agent.
• Need to reduce number of degrees of freedom.

how?
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The continuum level

• Define a few collective variables that describe
  the system as a
• whole, or at least some of the variables.
• Still, a lot of information is obtained from
  this simplified viewpoint.
• Prediction of macroscopic behavior from a given
  microscopic
• dynamics (ABM) becomes relevant.
• Statistical Physics provides a suitable
  framework that relates
• micro with macro in systems with many
  particles/agents.

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• Equilibrium statistical physics thermodynamic
  relations
• between macroscopic/measurable variables
  (P,V,T) are
• derived from Hamiltonian-Equipartition
  functions.
• But.., most real-life systems are out of
  equilibrium.
• Analytical techniques to treat non-equilibrium
  problems that
• involve time dependence
• Master equations, Fokker-Planck equations,
• Langevin equations, etc.

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• EXAMPLES OF SYSTEMS OF INTERACTING PARTICLES
• Conserved number of particles
• Particles diffusing on a lattice (or in continuum
  space)
• Particles jump to a nearest neighbour with a
  given probability. Diffusion coefficients may
  depend on space, local ocuppancy of particles,
  etc...
• 2) Point particles interacting via some potential
  (typically pairwise). This is standard for
  studies of gases, liquids or crystals. It is used
  a Hamiltonian accounting for the interactions and
  the dynamics of the particles is given by

The continuum description of these systems is via
a continuity equation
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• B) Nonconserved number of particles
• Reaction-diffusion systems particles may appear,
  disappear or be changed into something else
  (chemical reactions, living organisms undergoing
  birth-death processes, biological populations
  dynamics)..
• If we denote by A and B two different species,
  seveeral reaction processes are possible
  (non-exhaustive list)
• i) Birth 0 ? A
• ii) death A? 0
• iii) Coalescence 2A ? A
• iv) contamination A B ? 2A
• v) transmutation A?B
• vi) death at contact AB ? A.

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Approaches to obtain macro evolution equations
The appropriate type of approach depends on the
topology of interactions between agents.

• MEAN-FIELD
• Rate equation for the time evolution of a global
  quantity.
• Ex density of particles, spin magnetization,
  population of species.
• Gives very good estimates on well mixed
  populations where every agent interacts
  with any other agent (complete graph or fully
  connected network).
• Simplest approach, but typically neglects
  spatial dependence, correlations and
  fluctuations.

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• PAIR APPROXIMATION
• Rate equation for the evolution of the global
  density of different
• types of pairs (neighboring sites).
• Account for nearest neighbor correlations, but
  neglects fluctuations.
• Used to obtain approximate solutions in square
  lattices.
• Gives some idea of spatial effects.
• Specially useful in complex networks, with very
  accurate results.

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• FIELD EQUATIONS
• Evolution equations for the spatiotemporal field
  (Langevin equation), or the distribution of
  probabilities (master or Fokker-Planck
  equations).
• Accounts for stochastic fluctuations associated
  to discreteness effects (agents).
• It contains the spatial dependence, so
  appropriate for spatially extended systems.
• Useful to study stability and and pattern
  formation.
• Natural framework to study critical phenomena,
  renormalization group, etc...

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WHAT IS A MASTER EQUATION?
It is a first order differential equation
describing the time evolution of the probability
of having a given configuration of discrete
states.
If P(N1, N2, ) probability of having N1
particles in the first node, etc
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MODELS WITH CONSERVED NUMBER OF PARTICLES
A system of N interacting Brownian dynamics
Gaussian White noise
Interaction potential
DENSITY
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• Is it really neccesary to take into account the
• individual character of the system in the
• continuum descriptions?

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EXAMPLE
One of the simplest ABM Brownian bug
model. Birth-death model with non-conserved total
number of particles
Young, Roberts and Stuhne, Reproductive pair
correlations and the clustering of organisms,
Nature 412, 328 (2001).
- N particles perform independent Brownian
(random) motions in the continuum 2d physical
space. - In addition, they undergo a branching
process They reproduce, giving rise to a new
bug close to the parent, with probability l (per
unit of time), or die with probability b .
The physical phenomenon that minute particles,
immersed in a fluid, move about randomly.
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LETS WRITE DOWN A MEAN-FIELD LIKE CONTINUUM
EQUATION
Modeling in terms of continuous concentration
field
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l gt b
If lgtb explosion If lltb extinction
If lb, simple diffusion
Total number of particles
l b
l lt b
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At the critical point (lb), fluctuations are
strong and lead to clustering
NOT SIMPLE DIFFUSION
Very simple mechanism Reproductive correlations
Newborns are close to parents. This is missed in
a continuous deterministic description in which
birth is homogeneous
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Making the continuum limit PROPERLY
Demographic noise
Fluctuations play a very important role and a
proper continuum limit must be performed.
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• Some bibliography
• Van Kampen,
• Gardiner,
• Birolil