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Individual Losers and Collective Winners MICRO – individuals with arbitrary high death rate INTER – arbitrary low birth rate; arbitrary low density of catalisers MACRO –always resilient collective patches The importance of being discrete: Life always wins on the surface N M. Shnerb, Y Louzoun, E Bettelheim, and S Solomon Proc. Natl. Acad. Sci. USA, 97/ 19, , Sep 12, Proliferation and Competition in Discrete Biological Systems YLouzoun S Solomon, H Atlan and I R. Cohend Bulletin of Mathematical Biology Volume 65, Issue 3, May 2003, P AUTOCATALYTIC DYNAMICS

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b-> 0; a+b-> b+a+b The Importance of Being Discrete; Life Always Wins on the Surface

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A Diffusion of A at rate D a

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A

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B Diffusion of B at rate D b

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B A A+B A+B+B; Birth of new B at rate

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A B

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ABAB

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AB A+B A+B+B; Birth of new B at rate

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AB B A+B A+B+B; Birth of new B at rate

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ABB A+B A+B+B; Birth of new B at rate

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ABB BB A+B A+B+B; Birth of new B at rate

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ABB BB A+B A+B+B; Birth of new B at rate

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B Another Example A

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A+B A+B+B; Birth of new B at rate Another Example A B

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A+B A+B+B; Birth of new B at rate Another Example A B

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A+B A+B+B; Birth of new B at rate Another Example A AA A B

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A+B A+B+B; Birth of new B at rate Another Example A BBBBBB

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A+B A+B+B; Birth of new B at rate Another Example A BBB

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B B B Death of B at rate

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B B B

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B B B

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B B

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B B B+B B; Competition of B’s at rate

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B B

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B B

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B B

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B

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contemporary estimations= doubling of the population every 30yrs Malthus : autocatalitic proliferation: db/dt = a b with a =birth rate - death rate exponential solution: b(t) = b(0) e a t

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WELL KNOWN Logistic Equation (but usually ignored spatial distribution, discreteness and randomeness ! b. = ( a - ) b + D b b – b 2 -

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almost all the social phenomena, except in their relatively brief abnormal times obey the logistic growth. Elliot W Montroll: Social dynamics and quantifying of social forces (1978 or so) Volterra Montroll 'I would urge that people be introduced to the logistic equation early in their education… Not only in research but also in the everyday world of politics and economics … Sir Robert May Nature

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almost all the social phenomena, except in their relatively brief abnormal times obey the logistic growth. Elliot W Montroll: Social dynamics and quantifying of social forces (1978 or so) Volterra Montroll d X = (a - c X) X

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d X i = (a i + c (X.,t)) X i + j a ij X j Volterra Lotka Montroll Eigen almost all the social phenomena, except in their relatively brief abnormal times obey the logistic growth. Elliot W Montroll: Social dynamics and quantifying of social forces (1978 or so) d X = (a - c X) X

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Insert the going down / going up alternative

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Yet the agents b always win ! b. = ( a - ) b + D b b => b (x,t) ~ e ( a 0 – ) t b-> 0; a+b-> b+a+b AUTOCATALYTIC The Importance of Being Discrete; Life Always Wins on the Surface

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- On a large enough 2 dimensional surface, witout competition the B population always grows! - In higher dimensions, D a always suffices D b ! one can prove rigorously (RG flow, Branching Random Walks Theorems) that : In fact for A death rate a : D a + a suffices !

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Insert here the single A movie the directed percolation slide the jumping fence movie The polish Animation

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- spatial patches = first self-sustaining proto-cells. Interpretations in Various Fields: - individuals =chemical molecules, Origins of Life: Speciation: - Sites: various genomic configurations. - B= individuals ; Jumps of B= mutations. - A= advantaged niches (evolving fitness landscape). - emergent adaptive patches= species Immune system: - B cells; A antigen B cells that meet antigen with complementary shape multiply. (later in detail the AIDS analysis)

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“ continuum ” Solution: uniform in space and time: a < 0 b a TIME birth rate > death rate

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– c b 2 = competition for resources and other the adverse feedback effects saturation of the population to the value b= a / c Verhulst way out of it: db/dt = a b – c b 2 Solution: exponential ========= saturation at b= a /c a < 0 a b

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For humans data at the time could not discriminate between exponential growth of Malthus and logistic growth of Verhulst But data fit on animal population: sheep in Tasmania: exponential in the first 20 years after their introduction and saturated completely after about half a century.

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Confirmations of Logistic Dynamics pheasants turtle dove humans world population for the last 2000 yrs and US population for the last 200 yrs, bees colony growth escheria coli cultures, drossofilla in bottles, water flea at various temperatures, lemmings etc.

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almost all the social phenomena, except in their relatively brief abnormal times obey the logistic growth. “ Social dynamics and quantifying of social forces ” Elliott W. Montroll US National Academy of Sciences and American Academy of Arts and Sciences 'I would urge that people be introduced to the logistic equation early in their education … Not only in research but also in the everyday world of politics and economics …” Nature Robert McCredie, Lord May of Oxford, President of the Royal Society

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Logistic Equation usually ignored spatial distribution, Introduce discreteness and randomeness ! b. = ( conditions x birth rate - death x b + diffusion b - competition b 2 conditions is the result of many spatio-temporal distributed discrete individual contributions rather then totally uniform and static

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Instead: emergence of singular spatio-temporal localized collective islands with adaptive self-serving behavior => resilience and sustainability even for << 0 ! Multi-Agent Complex Systems Implications: one can prove rigorously that the DE prediction: Time Differential Eqations ( continuum << 0 approx ) Multi-Agent stochastic a prediction Is ALWAYS wrong !

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wealth, life surprize decay snapshots

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Angels and Mortals movie by my student Gur Ya ’ ari

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EXAMPLE of Theoretical Applied Science APPLICATION: Liberalization Experiment Poland Economy after MICRO growth ___________________ => MACRO growth 1990 MACRO decay (90) 1992 MACRO growth (92) 1991 MICRO growth (91) GNP THEOREM (RG, RW) one of the fundamental laws of complexity Global analysis prediction Complexity prediction Education 88 MACRO decay Maps Andrzej Nowak ’ s group (Warsaw U.), CO 3 collaboration

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one can prove rigorously (Renormalization Group (2000), Branching Random Walks Theorems (2002)) that : - In all dimensions d: D a > 1-P d always suffices P d = Polya ’ s constant ; P 2 = 1 -On a large enough 2 dimensional surface, the B population always grows! No matter how fast the death rate , how low the A density, how small the proliferation rate

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A The Role of DIFFUSION The Emergence of Adaptive B islands Take just one A in all the lattice:

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A

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A B diffusion

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A Growth stops when A jumps to a neighboring site

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AA

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A Growth will start on the New A site B population on the old A site will decrease

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A AA

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A AA B diffusion

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A A A A Growth stops again when A jumps again (typically after each time interval 1/D A )

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A A A AA

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A A AAA

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TIME S P C A E ln b TIME ln b at current A location (A location (unseen) and b distribution)

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