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

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

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

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: (1998) Meszéna G., Geritz S.A.H., Czibula I.: Adaptive dynamics in a 2-patch environment: a simple model for allopatric and parapatric speciation. J. of Biological Systems Vol. 5, No (1997) x 0 =0 x (slope of the spiral) x0+xx0+xx0-xx0-x x 0 is symmetrical strategy if for arbitrary  x and  y x0x0 x0x0 I. SYMMETRY IN ADAPTIVE DYNAMICS

-a symmetrical strategy is always singular -all the 8 typical configurations may appear I. SYMMETRY IN ADAPTIVE DYNAMICS x 0 is symmetrical strategy if for arbitrary  x and  y x0x0 x0x0 x 0 =0 x (slope of the spiral) x0+xx0+xx0-xx0-x

I. A MORE SPECIAL SYMMETRY IN ADAPTIVE DYNAMICS x0x0 x0x0 x 0 is special symmetrical strategy if for arbitrary  x and  y If x 0 is a symmetrical strategy, the asymmetrical individuals and their reflections are often completely equivalent. - a special symmetrical strategy is always singular - there are two typical configurations

II. TIME-DEPENDENT MODEL 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 T )

at T=0: * x 0 is a CSS and ESS singular strategy * population of x 0 strategists in evolutionary equilibrium slow evolution later: * x 0 (T) slowly changes * a 10 (T) and a 01 (T) slowly change III. BRANCHING IN TIME-DEPENDENT MODEL (WITHOUT SYMMETRY)

III. BRANCHING IN TIME-DEPENDENT MODEL (WITHOUT SYMMETRY) at T=0: * x 0 is a CSS and ESS singular strategy * population of x 0 strategists in evolutionary equilibrium later: * x 0 (T) slowly changes * a 10 (T) and a 01 (T) slowly change a 01 a 10 T=0

III. BRANCHING IN TIME-DEPENDENT MODEL (WITHOUT SYMMETRY) at T=0: * x 0 is a CSS and ESS singular strategy * population of x 0 strategists in evolutionary equilibrium later: * x 0 (T) slowly changes * a 10 (T) and a 01 (T) slowly change ESS non-ESS CSS non-CSS a 01 a 10

1: the singular strategy becomes non-ESS T x x0x0 III. BRANCHING IN TIME-DEPENDENT MODEL (WITHOUT SYMMETRY) t ESS non-ESS CSS non-CSS a 01 a 10

T t 2: x x0x0 III. BRANCHING IN TIME-DEPENDENT MODEL (WITHOUT SYMMETRY) ESS non-ESS CSS non-CSS a 01 a 10

3: the singular strategy becomes degenerate this is atypical III. BRANCHING IN TIME-DEPENDENT MODEL (WITHOUT SYMMETRY) a 10 ESS non-ESS CSS non-CSS a 01

III. BRANCHING IN TIME-DEPENDENT MODEL WITH REFLECTION SYMMETRY at T=0: * x 0 is a CSS and ESS symmetrical strategy * population of x 0 strategists in equilibrium x x0x0 x0x0 x AB T tt T

III. BRANCHING IN TIME-DEPENDENT MODEL WITH SPECIAL SYMMETRY at T=0: * x 0 is a CSS and ESS special symmetrical strategy * population of x 0 strategists in equilibrium later: * x 0 does not change * a 00 changes slowly a 00 T=0

at T=0: * x 0 is a convergence stable and ESS special symmetrical strategy * population of x 0 strategists in equilibrium later: * x 0 does not change * a 00 changes slowly a 00 1 III. BRANCHING IN TIME-DEPENDENT MODEL WITH SPECIAL SYMMETRY

1: the singular strategy becomes non-ESS and non-CSS The model behaves as in the case of a 00 =0; the character of the singular strategy is determined by higher order terms. a 00 1 III. BRANCHING IN TIME-DEPENDENT MODEL WITH SPECIAL SYMMETRY the m size of mutational steps is small but not infinitely small there is a time-interval when |a 00 |<<m 2

a 01 a 10 ESS non-ESS CSS non-CSS 1: the singular strategy becomes non-ESS and non-CSS a 00 1 if a 00  0 III. BRANCHING IN TIME-DEPENDENT MODEL WITH SPECIAL SYMMETRY

a 01 a 10 ESS non-ESS CSS non-CSS III. BRANCHING IN TIME-DEPENDENT MODEL WITH SPECIAL SYMMETRY t x x0x0 a 00 <0 a 00  0 a 00 >0 T A

t T a 01 a 10 ESS non-ESS CSS non-CSS x x0x0 a 00 <0 a 00  0 a 00 >0 III. BRANCHING IN TIME-DEPENDENT MODEL WITH SPECIAL SYMMETRY A

t T a 01 a 10 ESS non-ESS CSS non-CSS x x0x0 a 00 <0 a 00  0 a 00 >0 ? III. BRANCHING IN TIME-DEPENDENT MODEL WITH SPECIAL SYMMETRY

a 01 a 10 ESS non-ESS CSS non-CSS s(y) 4 th order x0x0 y x0x0 y s(y) x0x0 y III. BRANCHING IN TIME-DEPENDENT MODEL WITH SPECIAL SYMMETRY

t T a 01 a 10 ESS non-ESS CSS non-CSS x x0x0 a 00 <0 a 00  0 a 00 >0 III. BRANCHING IN TIME-DEPENDENT MODEL WITH SPECIAL SYMMETRY C

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: (1998) * 2 patches with capacities c 1, c-c 1 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 x j strategists surviving the 1 st period in the i th patch is proportional to f i (x j ) * in the 2 nd period, the living space in the patches is allocated randomly among survivors. * x 0 =0 is a special symmetrical strategy

x x 1 c 1 /c=0,4 c 1 /c=0,5c 1 /c=0,6 x y AN EXAMPLEIV. AN EXAMPLE

c 1 /c=0,4 c 1 /c=0,5c 1 /c=0,6 x y ,8580,51 0,643 1 CSS, ESS non CSS, non ESS CSS, non ESS c 1 /c x AN EXAMPLE T 0,953 CSS, ESS coalitions 0,355 IV. AN EXAMPLE

0,953 CSS, ESS coalitions 0,355 0,8580,51 0,643 1 CSS, ESS non CSS, non ESS CSS, non ESS c 1 /c x AN EXAMPLE t x 0,953 c 1 /c=0,5 Time-dependent model produces branching where asymmetrical mutants spread, but symmetrical strategists also survive IV. AN EXAMPLE T

SUMMARY occurs at symmetrical strategies occurs at special symmetrical strategies * two classes of symmetrical strategies * in slowly changing environment Can we observe this kind of branching in Nature ? x x TT x T ABC

BRANCHING IN TIME-DEPENDENT MODEL WITH REFLECTION SYMMETRY at t=0: * x 0 is a convergence stable and ESS symmetrical strategy * population of x 0 strategists in equilibrium later: * x 0 does not change * a 10 and a 01 slowly change

at t=0: * x 0 is a convergence stable and ESS singular strategy * population of x 0 strategists in equilibrium a 01 a 10 ESS non-ESS CS non-CS later: * x 0 does not change * a 10 and a 01 slowly change 4 BRANCHING IN TIME-DEPENDENT MODEL WITH REFLECTION SYMMETRY

1: the singular strategy becomes non-ESS evolutionary time x x0x0 a 01 ESS non-ESS CS non-CS a 10 BRANCHING IN TIME-DEPENDENT MODEL WITH REFLECTION SYMMETRY

2: the singular strategy becomes non-CS evolutionary time x x0x0 a 01 ESS non-ESS CS non-CS a 10 BRANCHING IN TIME-DEPENDENT MODEL WITH REFLECTION SYMMETRY

a 10 CS non-CS 3: the (non-CS) singular strategy becomes non-ESS evolutionary time x x0x0 a 01 ESS non-ESS CS non-CS BRANCHING IN TIME-DEPENDENT MODEL WITH REFLECTION SYMMETRY

4: the singular strategy becomes degenerate this is atypical a 10 a 01 ESS non-ESS CS non-CS BRANCHING IN TIME-DEPENDENT MODEL WITH REFLECTION SYMMETRY