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EMERGENCE OF ASYMMETRY IN EVOLUTION

EMERGENCE OF ASYMMETRY IN EVOLUTION. PÉTER VÁRKONYI . BME, BUDAPEST. IN COOPERATION WITH. GÁBOR DOMOKOS . BME, BUDAPEST. GÉZA MESZÉNA . ELTE, BUDAPEST. C. A. B. evolutionary time. evolutionary time. evolutionary time. x. x. x. symmetrical form. symmetrical form.

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EMERGENCE OF ASYMMETRY IN EVOLUTION

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  1. EMERGENCE OF ASYMMETRY IN EVOLUTION PÉTER VÁRKONYI BME, BUDAPEST IN COOPERATION WITH GÁBOR DOMOKOS BME, BUDAPEST GÉZA MESZÉNA ELTE, BUDAPEST

  2. C A B evolutionary time evolutionary time evolutionary time x x x symmetrical form symmetrical form symmetrical form EMERGENCE OF ASYMMETRY definition of symmetry time-dependent model

  3. I. TWO KINDS OF SYMMETRY IN ADAPTIVE DYNAMICS - simple reflection symmetry - „special” symmetry II. A TIME-DEPENDENT VERSION OF THE MODEL III. BRANCHING IN THE TIME-DEPENDENT MODEL - without symmetry - with reflection symmetry - with „special” symmetry IV. AN EXAMPLE

  4. x0 is symmetrical strategy if for arbitrary x and y x (slope of the spiral) x0-x x0=0 x0+x x0 x0 I. SYMMETRY IN ADAPTIVE DYNAMICS EXAMPLES IN LITERATURE: Geritz, S. A. H., Kisdi É., Meszéna G., Metz., J. A. J.: Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evol. Ecol. 12:35-57. (1998) Meszéna G., Geritz S.A.H., Czibula I.: Adaptive dynamics in a 2-patch environment: asimple model for allopatric and parapatric speciation. J. of Biological Systems Vol. 5, No. 2 265-284. (1997)

  5. x (slope of the spiral) x0-x x0=0 x0+x I. SYMMETRY IN ADAPTIVE DYNAMICS x0 is symmetrical strategy if for arbitrary x and y x0 x0 • a symmetrical strategy is always singular • all the 8 typical configurations may appear

  6. x0 is special symmetrical strategy if for arbitrary x and y x0 x0 • a special symmetrical strategy • is always singular • there are two typical configurations I. A MORE SPECIAL SYMMETRY IN ADAPTIVE DYNAMICS If x0 is a symmetrical strategy, the asymmetrical individuals and their reflections are often completely equivalent.

  7. assumption: change of the model is slow, compared to evolution: evolutionary equilibrium in quasi-permanent model 3 separate time-scales: * - population dynamics (fast) * t - evolution (slower) * T- environmental change (slowest) problem is reduced to analysis of non time-dependent model at different values of a (T) parameter II. TIME-DEPENDENT MODEL T )

  8. at T=0: * x0 is a CSS and ESS singular strategy * population of x0 strategists in evolutionary equilibrium slow evolution later: * x0(T) slowly changes * a10(T) and a01(T) slowly change III. BRANCHING IN TIME-DEPENDENT MODEL (WITHOUT SYMMETRY)

  9. at T=0: * x0 is a CSS and ESS singular strategy * population of x0 strategists in evolutionary equilibrium a10 later: * x0(T) slowly changes * a10(T) and a01(T) slowly change T=0 a01 III. BRANCHING IN TIME-DEPENDENT MODEL (WITHOUT SYMMETRY)

  10. at T=0: * x0 is a CSS and ESS singular strategy * population of x0 strategists in evolutionary equilibrium later: * x0(T) slowly changes * a10(T) and a01(T) slowly change CSS 2 non-CSS 3 1 a01 ESS non-ESS III. BRANCHING IN TIME-DEPENDENT MODEL (WITHOUT SYMMETRY) a10

  11. CSS 2 non-CSS 3 1 a01 ESS non-ESS III. BRANCHING IN TIME-DEPENDENT MODEL (WITHOUT SYMMETRY) 1: the singular strategy becomes non-ESS T a10 t x x0

  12. CSS 2 non-CSS 3 1 a01 ESS non-ESS III. BRANCHING IN TIME-DEPENDENT MODEL (WITHOUT SYMMETRY) 2: T a10 t x x0

  13. CSS 2 non-CSS 3 1 a01 ESS non-ESS III. BRANCHING IN TIME-DEPENDENT MODEL (WITHOUT SYMMETRY) 3: the singular strategy becomes degenerate a10 this is atypical

  14. at T=0: * x0 is a CSS and ESS symmetrical strategy * population of x0 strategists in equilibrium T T t t A B x x0 x x0 III. BRANCHING IN TIME-DEPENDENT MODEL WITH REFLECTION SYMMETRY

  15. at T=0: * x0 is a CSS and ESS special symmetrical strategy * population of x0 strategists in equilibrium later: * x0 does not change * a00 changes slowly a00 T=0 III. BRANCHING IN TIME-DEPENDENT MODEL WITH SPECIAL SYMMETRY

  16. III. BRANCHING IN TIME-DEPENDENT MODEL WITH SPECIAL SYMMETRY at T=0: * x0 is a convergence stable and ESS special symmetrical strategy * population of x0 strategists in equilibrium later: * x0 does not change * a00 changes slowly 1 a00

  17. the m size of mutational steps is small but not infinitely small there is a time-interval when |a00|<<m2 The model behaves as in the case of a00=0; the character of the singular strategy is determined by higher order terms. III. BRANCHING IN TIME-DEPENDENT MODEL WITH SPECIAL SYMMETRY 1: the singular strategy becomes non-ESS and non-CSS 1 a00

  18. a10 CSS non-CSS if a000 a01 ESS non-ESS III. BRANCHING IN TIME-DEPENDENT MODEL WITH SPECIAL SYMMETRY 1: the singular strategy becomes non-ESS and non-CSS 1 a00

  19. T a00<0 a000 t a00>0 x x0 A III. BRANCHING IN TIME-DEPENDENT MODEL WITH SPECIAL SYMMETRY a10 CSS non-CSS a01 ESS non-ESS

  20. III. BRANCHING IN TIME-DEPENDENT MODEL WITH SPECIAL SYMMETRY T a10 a00<0 CSS t non-CSS a000 a01 ESS non-ESS a00>0 x x0 A

  21. III. BRANCHING IN TIME-DEPENDENT MODEL WITH SPECIAL SYMMETRY T a10 a00<0 CSS non-CSS a000 a01 ? ESS t non-ESS a00>0 x x0

  22. s(y) 4th order x0 y s(y) x0 y s(y) x0 y III. BRANCHING IN TIME-DEPENDENT MODEL WITH SPECIAL SYMMETRY a10 CSS non-CSS a01 ESS non-ESS

  23. III. BRANCHING IN TIME-DEPENDENT MODEL WITH SPECIAL SYMMETRY T a10 a00<0 CSS non-CSS a000 a01 ESS t non-ESS a00>0 x x0 C

  24. * x0=0 is a special symmetrical strategy * 2 patches with capacities c1, c-c1 with optimal strategies 0 and 1 * resident population: * the number of the offspring is proportional to the frequency of the given strategy; the offspring is randomly distributed in the patches * the chance of the xj strategists surviving the 1st period in the ith patch is proportional to fi(xj) * in the 2nd period, the living space in the patches is allocated randomly among survivors. IV. AN EXAMPLE Geritz, S. A. H., Kisdi É., Meszéna G., Metz., J. A. J.: Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evol. Ecol. 12:35-57. (1998)

  25. x x -1 1 y 1 + + + + + 0 + -1 x -1 0 1 c1/c=0,5 c1/c=0,6 c1/c=0,4 AN EXAMPLE IV. AN EXAMPLE

  26. x 1 0,953 0,643 T CSS, ESS c1/c non CSS, non ESS 0,355 0,5 0,858 1 CSS, non ESS CSS, ESS coalitions AN EXAMPLE IV. AN EXAMPLE y 1 + + + + + 0 + -1 x -1 0 1 c1/c=0,5 c1/c=0,6 c1/c=0,4

  27. x 1 0,643 T CSS, ESS c1/c non CSS, non ESS 0,5 0,858 1 CSS, non ESS t Time-dependent model produces branching where asymmetrical mutants spread, but symmetrical strategists also survive c1/c=0,5 0,953 x AN EXAMPLE IV. AN EXAMPLE 0,953 0,355 CSS, ESS coalitions

  28. A B C T T T x x x occurs at symmetrical strategies occurs at special symmetrical strategies SUMMARY * two classes of symmetrical strategies * in slowly changing environment Can we observe this kind of branching in Nature ?

  29. BRANCHING IN TIME-DEPENDENT MODEL WITH REFLECTION SYMMETRY at t=0: * x0 is a convergence stable and ESS symmetrical strategy * population of x0 strategists in equilibrium later: * x0 does not change * a10 and a01 slowly change

  30. BRANCHING IN TIME-DEPENDENT MODEL WITH REFLECTION SYMMETRY at t=0: * x0 is a convergence stable and ESS singular strategy * population of x0 strategists in equilibrium a10 CS later: * x0 does not change * a10 and a01 slowly change non-CS 2 4 1 a01 ESS non-ESS 3

  31. BRANCHING IN TIME-DEPENDENT MODEL WITH REFLECTION SYMMETRY 1: the singular strategy becomes non-ESS a10 CS non-CS 2 evolutionary time 4 1 a01 ESS non-ESS 3 x x0

  32. BRANCHING IN TIME-DEPENDENT MODEL WITH REFLECTION SYMMETRY 2: the singular strategy becomes non-CS a10 CS non-CS 2 evolutionary time 4 1 a01 ESS non-ESS 3 x0 x

  33. BRANCHING IN TIME-DEPENDENT MODEL WITH REFLECTION SYMMETRY 3: the (non-CS) singular strategy becomes non-ESS a10 CS CS non-CS non-CS 2 evolutionary time 4 1 a01 ESS non-ESS 3 x x0

  34. BRANCHING IN TIME-DEPENDENT MODEL WITH REFLECTION SYMMETRY 4: the singular strategy becomes degenerate a10 this is atypical CS non-CS 2 4 1 a01 ESS non-ESS 3

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