Agent-based models applied to Biology and Economics. Gur Yaari Phd - - PowerPoint PPT Presentation

Agent-based models applied to Biology and Economics.

Gur Yaari Phd student, under the supervision of Prof. Sorin Solomon Racah Institute of Physics in the Hebrew University of Jerusalem, Israel Multi-Agent Systems Division. ISI, Torino, Italy

“G.L.V” Generalized Lotka Volterra

Stochastic Lotka-Volterra Systems of Competing Auto-catalytic Agents lead Generically to Truncated Pareto Power Wealth Distribution, Truncated Levy Distribution of Market Returns, Clustered Volatility, Booms and Crashes.

• S. Solomon,

in Decision Technologies for Computational Finance, edited by A.-P. Refenes, A. N. Burgess, and J. E. Moody (Kluwer Academic Publishers, 1998). http://xxx.lanl.gov/abs/cond-mat/9803367 Generalized Lotka Volterra (GLV) Models of Stock Markets

• S. Solomon, pp 301-322

in "Applications of Simulation to Social Sciences" , Eds: G Ballot and G Weisbuch; Hermes Science Publications 2000 http://xxx.lanl.gov/abs/cond-mat/9901250 Power laws in cities population, financimarkets and internet sites (scaling in systems with a variable number of components) Aharon Blank and Sorin Solomon Physica A 287 (1-2) (2000) pp.279-288 http://xxx.lanl.gov/abs/cond-mat/0003240 Power-law distributions and L'evy-stable intermittent fluctuations in stochastic systems of many autocatalytic elements.

• O. Malcai O. Biham and S. Solomon,
• Phys. Rev. E, 60, 1299 (1999). http://xxx.lanl.gov/abs/cond-mat/9907320

Theoretical analysis and simulations of the generalized Lotka-Volterra model Ofer Malcai,1 Ofer Biham,1 Peter Richmond,2 and Sorin Solomon Phys. Rev. E 66, 031102 (2002) URL: http://link.aps.org/abstract/PRE/v66/e031102 doi:10.1103/PhysRevE.66.031102 http://xxx.lanl.gov/PS_cache/cond-mat/pdf/0208/0208514.pdf

“AB Model”

The importance of being discrete: Life always wins on the surface Nadav M. Shnerb, Yoram Louzoun, Eldad Bettelheim, and Sorin Solomon

• Proc. Natl. Acad. Sci. USA, Vol. 97, Issue 19, 10322-10324, September 12, 2000

http://xxx.lanl.gov/abs/adap-org/9912005 Adaptation of Autocatalytic Fluctuations to Diffusive Noise

• N. M. Shnerb, E. Bettelheim, Y. Louzoun, O. Agam, S. Solomon

Phys Rev E Vol 63, No 2, 2001; http://xxx.lanl.gov/abs/cond-mat/0007097 HIV time hierarchy: winning the war while, loosing all the battles Uri Hershberg, Yoram Louzoun, Henri Atlan and Sorin Solomon Physica A: 289 (1-2) (2001) pp.178-190 ; http://xxx.lanl.gov/abs/nlin.AO/0006023 [pdf] The Emergence of Spatial Complexity in the immune System. Louzoun, Y, Solomon. S., Atlan. H., Cohen.I.R. (2001) Physica A, 297 (1-2) pp. 242-252. http://xxx.lanl.gov/html/cond-mat/0008133 Proliferation and Competition in Discrete Biological Systems Y Louzoun S Solomon, H Atlan and I R. Cohend Bulletin of Mathematical Biology Volume 65, Issue 3 , May 2003, P 375-396

Issues that had been explained (at least qualitatively) by these two models and their extensions:

• The distribution of individuals in colonies (cities population)
• The distribution of individuals/companies wealth
• Genetic evolution – The origin of life (??)

The H.I.V war against our Immune system

• The distribution of number of individuals for various species
• Patches of green vegetation in the desert
• Waiting for new Ideas...
• The distribution of word's frequency in a language

“G.L.V” Generalized Lotka Volterra

Thomas Robert Malthus (1766–1834)

contemporary estimations= doubling of the population every 30yrs exponential solution: X(t) = X(0)ea t dx dt =a⋅x autocatalitic proliferation: with a =birth rate - death rate

1798

dX/dt = a X – c X2

Solution: exponential ==========saturation at X= a / c

Pierre François Verhulst (1804-1849) way out exponential explosion:

1838

For humans data at the time could not discriminate between:

• 1. exponential growth of Malthus
• 2. logistic growth of Verhulst

But data fit on animal population: sheep in Tasmania

• exponential in the first 20 years after their introduction and

completely saturated after about half a century. ==> Verhulst

“G.L.V” Generalized Lotka Volterra

proposed independently (Lotka-1925 , Volterra-1926)

The Lotka Volterra Equations.

Describes Predator-Prey Relations:

Alfred James Lotka (March 2, 1880 - December 5, 1949) Vito Volterra (May 3, 1860 - October 11, 1940)

dx dt = x⋅− ⋅y dy dt =− y⋅− ⋅x 

Taking into account self competition (limited

resources) we have:

dx dt = x⋅−1⋅x−12⋅y  dy dt = y⋅−− 1⋅y21⋅x 

Results: Cycles in time in the number of Predator's-Prey's Number of individuals

“G.L.V” Generalized Lotka Volterra

General Frame-work for various processes in Economics and Biology. N agents (fix number) : N individuals (Economics) N spieces (Biology)

The Model

wit1=wit⋅ 1t∑

j=1 N

ai , j⋅w jt−∑

j=1 N

ci , j⋅w jt⋅wit

w jt1=w jt i=1.. N ;i≠ j

Update:

each time step

Choose individual i randomly from 1..N Is a stochastic variable drawn from a distribution with : D≡〈t

2〉−〈t〉 2

t

Stochastic Term!

wit1=wit⋅1ta⋅ wt−c⋅wit⋅ wt

In order To Kis (keep it simple) it:

• ne can choose uniform interaction:

ai , j= a N ci , j= a N

And then the process can be described by:

 wt = 1 N ⋅∑

j=1 N

wit

where:

Another possibility is:

wit1=wit⋅1t

With the restriction:

wit≥c⋅ wt

Xi (t+τ) – Xi (t) = λi (t) Xi (t) + a X (t) – c(X.,t) Xi (t) admits a few practical interpretations Xi (t) = the individual wealth of the agent i then

λi (t) = the random part of the returns that its capital Xi (t) produces

during the time between t and t+τ

a = the autocatalytic property of wealth at the social level

= the wealth that individuals receive as members of the society in subsidies, services and social benefits. This is the reason it is proportional to the average wealth This term prevents the individual wealth falling below a certain minimum fraction of the average.

c(X.,t) parametrizes the general state of the economy:

large and positive correspond = boom periods negative =recessions

A different interpretation: a set of companies i = 1, … , N Xi (t)= shares prices

~ capitalization of the company i

~ total wealth of all the market shares of the company

λi (t) = fluctuations in the market worth of the company

~ relative changes in individual share prices (typically fractions of the nominal share price)

aX = correlation between Xi and the market index w c(X.,t) usually of the form c X → represents competition

Time variations in global resources may lead to lower or higher values of

c →increases or decreases in the total X

Yet another interpretation: investors herding behavior (also cities) Xi (t)= number of traders adopting a similar investment policy or

• position. they comprise herd i
• ne assumes that the sizes of these sets vary autocatalytically according

to the random factor λi (t) This can be justied by the fact that the visibility and social connections

• f a herd are proportional to its size

aX represents the diffusion of traders between the herds c(X.,t) = popularity of the stock market as a whole

competition between various herds in attracting individuals

Important Results: When N is very large and t → ∞ , The “wealth” (w's) distribution approaches a stationary distribution with the well common empirical feature of a power-law “tail”. For the first case one gets: For the second case one gets:

Px=x

−1−⋅exp

1− x

xit=wi t  w t i=1,...n

Px~x

−1−

=1 2⋅a D

= 1 1−c

Computer's Simulations

Fits the well known empirical Pareto's law Or the Zipf's law

Vilfredo Pareto (1848-1923) P(x) ~ x –1-α d x

Income distribution

George Kingsley Zipf , (1902-1950)

Economics Statistics Word's distribution Cities size's distribution

P(n) ~ n-α d n n- rank

red circles: Pareto for 400 richest people is USA the average wealth history 88-2003 and model fit blue squares: simulation results − α

Zipf plot of the wealths of the investors in the Forbes 400 of 2003 vs. their ranks. The corresponding simulation results are shown in the inset.

In addition, the average of the wealth is basically a result of a random walk with steps taken from a power-law distribution → truncated Levy distribution:

Lr=0 ≡ P wt=  wt ~

−1 

r=  wt−  wt  wt A strong prediction of this model is that the exponents of the wealth distribution and the returns of the stock market (the fluctuations of the average wealth ) are connected through a simple connection:

1 ⇒ 1 

The relative probability of the price being the same as a function of the time interval

• M. Levy S.S

β α

Prediction of The Lotka- Volterra- model:

−1/β

And indeed it was found empirically:

The AB Model Two types of agents: A B Three types of processes:

• 1. Diffusion – Locate ourselves in concrete space – until now d-dimensional square

lattice

• 2. Birth -with rate λ :

B(r) + A(r)  B(r) + B(r) + A(r)

• 3. Death - with rate µ :

B(r)  Ø

• 4. Competition -with rate σ :

B(r) + B(r)  B(r)

• with rate σ·<B>r : B(r) + B(r)  B(r)
• with rate σ : B(r) + B(r)  B(r)+B(r+i)

Differential Equation:

dAx ,t dt =Da⋅∇

2⋅Ax ,t

dBx ,t dt =Db⋅∇

2⋅Bx ,t⋅Ax ,t−⋅B x ,t

Differential Equation:

−⋅B

2

r−  ⋅B r⋅〈B r 

Competition dAx ,t dt =Da⋅∇

2⋅Ax ,t

dBx ,t dt =Db⋅∇

2⋅Bx ,t⋅Ax ,t−⋅B x ,t

Naïve approach: when t is very large:

Differential Equation:

A r⇒ nA

⋅nA−0

⋅nA−0

Exponential Growth Exponential Decay Malthus Naïve prediction dBx ,t dt =Db⋅∇

2⋅Bx ,t⋅Ax ,t−⋅B x ,t

⇒ dBx ,t dt =⋅Ax ,t− ⋅Bx ,t

Important Results: If d ≤ 2 Life always win ! For all parameter's values

Computer's Simulations + R.G (Renormalization Group) calculation:

In higher dimensions, λ/Da > 1-Pd

Surprize! Simulations + R.G calculation: outcome of the discreteness nature of the agents,

Only Practical Effect: Islands are of limited height

Island's formation Movie 1

B-islands search, follow, adapt to, and exploit fortuitous fluctuations in A density. This is in apparent contradiction to the “fundamental laws” where individual B don’t follow anybody In Complexity it is very popular to talk about : Emergent Collective Dynamics:

B's Island's search for food.. Movie2

Desert / Vegetation:

• B = plants,
• A= water / light / nutrients
• patches- patterns= stripes, oases

Desert <-> Patchy <-> Full cover

(contact to Judean Desert-Jerusalem mountains studies) ; Reclaim; PRL 90, 38101 (2003)

Interpretations in Various Fields

Finance:

sites= individuals, companies, B = capital A= wealth generating conditions jobs, good location, good managers, customers, education

Genetic Evolution:

• Sites: various genomic configurations.
• B= individuals; Jumps of B= mutations.
• A= advantaged niches
• emergent adaptive patches= species

why are there not a continuum of creatures between snails and salamanders (both are partenogenetic).

Polish spatial economic map since 89 (Andrzej Nowak ).

The analogy between these two models

wit1=wit⋅1ta⋅ wt−c⋅wit⋅ wt

⇒ dwit dt =wit⋅ta⋅ wt−c⋅wit⋅ wt

Stochastic noise Discrete A's

dBx ,t dt =Db⋅∇

2⋅Bx ,t⋅Ax ,t−⋅B x ,t−⋅Bx ,t

⋅〈 Bx ,t〉 dAx ,t dt =Da⋅∇

2⋅Ax ,t

Major differences

⇒ dwit dt =wit⋅ta⋅ wt−c⋅wit⋅ wt

dBx ,t dt =Db⋅∇

2⋅Bx ,t⋅Ax ,t−⋅B x ,t−⋅Bx ,t

⋅〈 Bx ,t〉

Major differences

⇒ dwit dt =wit⋅ta⋅ wt−c⋅wit⋅ wt

But... If

a⋅ wt⇒a⋅ wit

We get similar results for the GLV And

a⋅ wit⇔ ∇

2⋅B

r= 1 N.O.N ⋅∑

j=1 N.O.N

B r  e j

dBx ,t dt =Db⋅∇

2⋅Bx ,t⋅Ax ,t−⋅B x ,t−⋅Bx ,t

⋅〈 Bx ,t〉

Major differences

And

〈i ,t⋅ j ,t'〉=i , j⋅t ,t '≠〈A r ,t⋅A r ' ,t '〉

And But it turns out that the results of the AB Model are still valid even for uncorrelated noise

Future research

Combine these two models in a more quantitative manner Find more empirical evidences for these type of processes Find more empirical evidences for these type of processes Adjust the model to the specific case studied

Thank You very much Grazie mille

הבר הדות

NAGYON KÖSZÖNÖM

dunke schoen

Merci beaucoup