Individual Losers and Collective Winners

2. 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 YLouzounS Solomon, H Atlan and I R. Cohend Bulletin of Mathematical Biology Volume 65, Issue 3 , May 2003, P 375-396 AUTOCATALYTIC DYNAMICS

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

4. Diffusion of A at rate Da

5. Diffusion of A at rate Da

6. Diffusion of A at rate Da

7. Diffusion of A at rate Da

8. Diffusion of B at rate Db

9. Diffusion of B at rate Db

10. Diffusion of B at rate Db

11. Diffusion of B at rate Db

12. A+B A+B+B; Birth of new B at rate l

13. A+B A+B+B; Birth of new B at rate l

14. A+B A+B+B; Birth of new B at rate l

15. A+B A+B+B; Birth of new B at rate l

16. A+B A+B+B; Birth of new B at rate l

17. A+B A+B+B; Birth of new B at rate l

18. A+B A+B+B; Birth of new B at rate l

19. A+B A+B+B; Birth of new B at rate l

20. A+B A+B+B; Birth of new B at rate l B Another Example A A

21. A+B A+B+B; Birth of new B at rate l Another Example B A A

22. A+B A+B+B; Birth of new B at rate l Another Example B A A

23. A+B A+B+B; Birth of new B at rate l Another Example B AA

24. A+B A+B+B; Birth of new B at rate l Another Example BBB A A

25. A+B A+B+B; Birth of new B at rate l Another Example BBB A A

26. Death of B at rate m B

27. Death of B at rate m B

28. Death of B at rate m B

29. Death of B at ratem B

30. B+B B; Competition of B’s at rate g

31. B+B B; Competition of B’s at rate g

32. B+B B; Competition of B’s at rate g

33. B+B B; Competition of B’s at rate g

34. B+B B; Competition of B’s at rate g

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

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

37. 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) '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 Volterra Montroll

38. 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) dX=(a-c X) X Volterra Montroll

39. 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) Lotka Volterra Montroll Eigen dX=(a-c X) X dXi =(ai + c (X.,t))Xi +j aij Xj

40. Insert the going down / going up alternative

41. The Importance of Being Discrete; Life Always Wins on the Surface b-> 0;a+b-> b+a+b AUTOCATALYTIC b. = ( a l- m)b + Db D b => b (x,t) ~ e (la0 – m) t Yet the agents balways win !

42. one can prove rigorously (RG flow, Branching Random Walks Theorems)that: - On a large enough 2 dimensional surface, witout competition the B populationalways grows! - In higher dimensions,l > Daalways suffices m, Db, < a(x,t) >! In fact for A death rate ma: l > Da+ masuffices !

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

44. Interpretations in Various Fields: Origins of Life: - individuals =chemical molecules, - spatial patches =firstself-sustaining proto-cells. 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; AantigenB cells that meet antigen with complementary shape multiply. (later in detail the AIDS analysis)

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

46. Verhulst way out of it: db/dt = ab– c b2 Solution: exponential=========saturation at b= a /c a > 0 a < 0 – c b2= competition for resources and other the adverse feedback effects saturation of the population to the value b= a / c b

47. 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.

48. 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.

49. 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

50. Logistic Equation usually ignored spatial distribution,Introducediscretenessandrandomeness ! b. = ( conditions x birth rate - death) xb+diffusion b - competition b2 conditions is the result ofmany spatio-temporal distributed discrete individual contributionsrather then totally uniform and static