Morse, Arnold s a Keljfeljancsi

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EMERGENCE OF ASYMMETRY IN EVOLUTION
PÉTER VÁRKONYI
BME, BUDAPEST
IN COOPERATION WITH
GÁBOR DOMOKOS
BME, BUDAPEST
GÉZA MESZÉNA
ELTE, BUDAPEST
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EMERGENCE OF ASYMMETRY
definition of symmetry
time-dependent model
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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
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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.
1235-57. (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. 2 265-284. (1997)
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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

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I. A MORE SPECIAL SYMMETRY IN ADAPTIVE DYNAMICS
If x0 is a symmetrical strategy, the asymmetrical
individuals and their reflections are often
completely equivalent.
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II. TIME-DEPENDENT MODEL
T )
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III. BRANCHING IN TIME-DEPENDENT MODEL
(WITHOUT SYMMETRY)
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III. BRANCHING IN TIME-DEPENDENT MODEL
(WITHOUT SYMMETRY)
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III. BRANCHING IN TIME-DEPENDENT MODEL
(WITHOUT SYMMETRY)
a10
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III. BRANCHING IN TIME-DEPENDENT MODEL
(WITHOUT SYMMETRY)
1 the singular strategy becomes non-ESS
T
a10
t
x
x0
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III. BRANCHING IN TIME-DEPENDENT MODEL
(WITHOUT SYMMETRY)
2
T
a10
t
x
x0
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III. BRANCHING IN TIME-DEPENDENT MODEL
(WITHOUT SYMMETRY)
3 the singular strategy becomes degenerate
a10
this is atypical
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III. BRANCHING IN TIME-DEPENDENT MODEL WITH
REFLECTION SYMMETRY
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III. BRANCHING IN TIME-DEPENDENT MODEL WITH
SPECIAL SYMMETRY
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III. BRANCHING IN TIME-DEPENDENT MODEL WITH
SPECIAL SYMMETRY
at T0 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
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III. BRANCHING IN TIME-DEPENDENT MODEL WITH
SPECIAL SYMMETRY
1 the singular strategy becomes non-ESS and
non-CSS
1
a00
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III. BRANCHING IN TIME-DEPENDENT MODEL WITH
SPECIAL SYMMETRY
1 the singular strategy becomes non-ESS and
non-CSS
1
a00
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III. BRANCHING IN TIME-DEPENDENT MODEL WITH
SPECIAL SYMMETRY
a10
CSS
non-CSS
a01
ESS
non-ESS
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III. BRANCHING IN TIME-DEPENDENT MODEL WITH
SPECIAL SYMMETRY
T
a10
a00lt0
CSS
t
non-CSS
a00?0
a01
ESS
non-ESS
a00gt0
x
x0
A
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III. BRANCHING IN TIME-DEPENDENT MODEL WITH
SPECIAL SYMMETRY
T
a10
a00lt0
CSS
non-CSS
a00?0
a01
?
t
ESS
non-ESS
a00gt0
x
x0
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III. BRANCHING IN TIME-DEPENDENT MODEL WITH
SPECIAL SYMMETRY
a10
CSS
non-CSS
a01
ESS
non-ESS
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III. BRANCHING IN TIME-DEPENDENT MODEL WITH
SPECIAL SYMMETRY
T
a10
a00lt0
CSS
non-CSS
a00?0
a01
t
ESS
non-ESS
a00gt0
x
x0
C
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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. 1235-57. (1998)
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AN EXAMPLE
IV. AN EXAMPLE
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AN EXAMPLE
IV. AN EXAMPLE
y
1





0

-1
x
0
1
-1
c1/c0,5
c1/c0,6
c1/c0,4
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AN EXAMPLE
IV. AN EXAMPLE
0,953
0,355
CSS, ESS coalitions
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SUMMARY
two classes of symmetrical strategies in
slowly changing environment
Can we observe this kind of branching in Nature ?
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(No Transcript)
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BRANCHING IN TIME-DEPENDENT MODEL WITH REFLECTION
SYMMETRY
at t0 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
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BRANCHING IN TIME-DEPENDENT MODEL WITH REFLECTION
SYMMETRY
at t0 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
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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
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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
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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
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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