1. Collaboration with Federico Vázquez http://ifisc.uib.es - Mallorca - Spain The continuum description of individual-based models Cristóbal López

2. http://ifisc.uib.es Outline of the talk: totonto 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)

3. http://ifisc.uib.es Most real-life systems are complex: many units interacting in a non-linear way give rise to not obvious (unexpected) collective behavior. Introduction Ant colonies. Plankton communities. Traffic. People. Cells populations, bacteria. Atoms. Stars. Etc..

4. as CONTINUOUS FIELDS HOW TO STUDY COMPLEX SYSTEMS?

5. or as DISCRETE INDIVIDUALS

6. 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.

7. http://ifisc.uib.es The individual level 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. Some terminology: Agent-based models (Sociology, Computer Science, Game Theory) Individual-based models (Ecology, Biology). Interacting particle systems (Physics).

8. http://ifisc.uib.es Epidemic spreading (ISI, IRSI). Allele frequency (genetics). Bacteria dynamics. Neural networks. Tumor growth. Biology: Species competition. Invasion processes. Predator-prey systems (Lotka Volterra). Ecology: Examples of interacting particle systems

9. http://ifisc.uib.es Opinion spreading. Cultural propagation. Language dynamics. Social Science: Catalytic reactions. Deposition/ reaction-diffusion/ aggregation. Surface Physics/ Chemistry:

10. http://ifisc.uib.es 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. Description of ABMs how?

11. http://ifisc.uib.es 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. The continuum level

12. http://ifisc.uib.es 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.

13. http://ifisc.uib.es EXAMPLES OF SYSTEMS OF INTERACTING PARTICLES: A)Conserved number of particles: 1)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

14. http://ifisc.uib.es B) Nonconserved number of particles: 1)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: A+B  A.

15. http://ifisc.uib.es 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. The appropriate type of approach depends on the topology of interactions between agents. Approaches to obtain macro evolution equations

16. http://ifisc.uib.es 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.

17. http://ifisc.uib.es 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...

18. 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(N 1, N 2, …)= probability of having N 1 particles in the first node, etc…

19. 19 MODELS WITH CONSERVED NUMBER OF PARTICLES A system of N interacting Brownian dynamics Interaction potential Gaussian White noise DENSITY

20. Is it really neccesary to take into account the individual character of the system in the continuum descriptions?

21. One of the simplest ABM: Brownian bug model. Birth-death model with non-conserved total number of particles - 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 (per unit of time), or die with probability . Young, Roberts and Stuhne, Reproductive pair correlations and the clustering of organisms, Nature 412, 328 (2001). * The physical phenomenon that minute particles, immersed in a fluid, move about randomly. EXAMPLE

22. C → 2C autocatalisis, or reproduction C → 0 death  Modeling in terms of continuous concentration field: LET’S WRITE DOWN A MEAN-FIELD LIKE CONTINUUM EQUATION

23.    Total number of particles If  : explosion If  : extinction If  simple diffusion

24. At the critical point (  ), fluctuations are strong and lead to clustering Very simple mechanism: Reproductive correlations: Newborns are close to parents. This is missed in a continuous deterministic description in which birth is homogeneous NOT SIMPLE DIFFUSION

25. Making the continuum limit PROPERLY Fluctuations play a very important role and a proper continuum limit must be performed. Demographic noise

26. Some bibliography: -Van Kampen, -Gardiner, -Birolil