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Individual Losers and Collective Winners MICRO – individuals with arbitrary high death rateINTER – arbitrary low birth rate; arbitrary low density of catalisersMACRO –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, 10322-10324, Sep 12, 2000 http://xxx.lanl.gov/abs/adap-org/9912005
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 375-396
AUTOCATALYTIC DYNAMICS
b-> 0;a+b-> b+a+b
The Importance of Being Discrete; Life Always Wins on the Surface
A
Diffusion of A at rate Da
A
Diffusion of A at rate Da
A
Diffusion of A at rate Da
A
Diffusion of A at rate Da
B
Diffusion of B at rate Db
B
Diffusion of B at rate Db
B
Diffusion of B at rate Db
B
Diffusion of B at rate Db
B
A
A+B A+B+B; Birth of new B at rate
A B
A+B A+B+B; Birth of new B at rate
AB
A+B A+B+B; Birth of new B at rate
AB
A+B A+B+B; Birth of new B at rate
ABB
A+B A+B+B; Birth of new B at rate
ABB
A+B A+B+B; Birth of new B at rate
ABBBB
A+B A+B+B; Birth of new B at rate
ABBBB
A+B A+B+B; Birth of new B at rate
B
A+B A+B+B; Birth of new B at rate Another Example
A A
A+B A+B+B; Birth of new B at rate Another Example
A AB
A+B A+B+B; Birth of new B at rate Another Example
A AB
A+B A+B+B; Birth of new B at rate Another Example
A AB
A+B A+B+B; Birth of new B at rate Another Example
A ABBB
A+B A+B+B; Birth of new B at rate Another Example
A ABBB
B
B Death of B at rate
B
B Death of B at rate
B
B Death of B at rate
B Death of B at rate
B
B
B+B B; Competition of B’s at rate
B
B
B+B B; Competition of B’s at rate
B B
B+B B; Competition of B’s at rate
B B
B+B B; Competition of B’s at rate
B
B+B B; Competition of B’s at rate
contemporary estimations= doubling of the population every 30yrs
Malthus : autocatalitic proliferation: db/dt = ab
with a =birth rate - death rateexponential solution: b(t) = b(0)e a t
WELL KNOWN Logistic Equation(but usually ignored
spatial distribution, discreteness and randomeness!
b.
= ( a - )b + Db b – b 2
-
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
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
d Xi = (ai + c (X.,t)) Xi +j aij Xj
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
•Insert the going down / going up alternative
Yet the agents b always win !
b.
= ( a - )b + Db b
=> b (x,t) ~ e (a0 – ) t
b-> 0;a+b-> b+a+b AUTOCATALYTIC
The Importance of Being Discrete; Life Always Wins on the Surface
- On a large enough 2 dimensional surface, witout competition the B population always grows!
- In higher dimensions, Da
always suffices Db< a(x,t)
>!
one can prove rigorously (RG flow, Branching Random Walks Theorems)
that:
In fact for A death rate a : Da + a suffices !
•Insert here the single A movie
• the directed percolation slide
• the jumping fence movie
•The polish Animation
- 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)
“continuum” Solution: uniform in space and time:
a < 0
b a
TIME
birth rate > death rate
birth rate > death rate
– c b2 = 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 b2
Solution: exponential =========saturation at b= a /c
a < 0
ab
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.
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.
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
Logistic Equation usually ignored spatial distribution, Introduce discreteness and randomeness!
b.
= ( conditions x birth rate - deathx b + diffusion b - competition b2
conditions is the result of many spatio-temporal distributed discrete individual contributions rather then totally uniform and static
Instead: emergence of singular spatio-temporal localized collective islands with adaptive self-serving behavior
=> resilience and sustainability
even for <a> << 0!
Multi-Agent Complex Systems Implications: one can prove rigorously that the DE prediction:
Time
Differential Eqations
(continuum <a> << 0 approx)
Multi-Agent stochastic
a
prediction
Is ALWAYS wrong !
wealth, life
surprize
decay
snapshots
Angels and Mortals movie by my student Gur Ya’ari
EXAMPLE of Theoretical Applied Science
APPLICATION: Liberalization Experiment Poland Economy after 1989
+ MICRO growth___________________
=> MACRO growth
1990 MACRO decay (90)
1992 MACRO growth (92)
1991 MICRO growth (91)
GNP
89 90 91 92
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.), CO3 collaboration
one can prove rigorously (Renormalization Group (2000) , Branching Random Walks Theorems (2002))that:
- In all dimensions d: Da > 1-Pd
always suffices
Pd = Polya’s constant ; P2 = 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
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14
The Role of DIFFUSION The Emergence of Adaptive B islands
Take just one A in all the lattice:
A
A
A
B diffusion
A
A
Growth stops when A jumps to a neighboring site
A A
A
Growth will start on the New A site
B population on the old A site will decrease
A
A
AAA
AAA
AAA
AAA
B diffusion
AAA
B diffusion
AAA
AAA
AA AA
Growth stops again when A jumps again
(typically after each time interval 1/DA)
AA AAA
A AA A A
TIME SPCA
E
ln b
TIME
ln b at current A location
(A location (unseen) and b distribution)
