Transcript lecture5 - Kirzhner Valery

Types of selection
Cyclical environment
Environmenta
l state 1
(summer)
Environmenta
l state 2
(winter)
Directional Stabilizing Disruptive
Interaction between species
Host-parasite case
Mutualistic case
constant fertility selection
AA  x, Aa  y, aa  z.
if
ab0
2
t
dy
xt 1 
,
4f
(1/ 2) dy  2cxt (1  xt  yt )
yt 1 
,
f
2
t
f  2cxt (1  xt  yt )  dy
2
t
Simulation
Frequency depended selection
WAA p 2  WAa pq
Waa q 2  WAa pq
p' 
; q' 
;
W
W
where WAA , WAa , Waa depended from p, q.
Simple lest case of frequency dependence: haploid selection
WA  p,Wa  q
WAA  WA WA , WAa  WA Wa , Waa  Wa Wa .
WAA p 2  WAa pq
Waa q 2  WAa pq
p' 
; q' 
;
W
W
W  (WAA p 2  2WAa pq  Waa q 2 )
W  (WA2 p 2  2WA Wa pq  Wa2q 2 )
W  (WA p  Wa q ) 2
Let WA = 1 + t - sp,Wa = 1.
WA2 p 2  WA Wa pq
p' 
, W  (WA p  Wa q) 2 .
W
WA2 p 2  WA Wa p(1  p)
p' 
, W  (WA p  Wa (1  p)) 2 .
W
(t  sp  1)
p'  p
.
(t  sp)p  1
(p(1 + t - sp) - [1 + (t - sp)p])
p = p - p =
(1 + (t - sp)p)
p(1 - p)(t - sp)
=
.
(1 + (t - sp)p)
Mean fitness is not maximize at this is stable le point
W  (WA p  Wa (1  p)) 2
W  (1 + (t - sp)p) 2 .
t
t
t2 2
p  , W  1, p  , W  (1  )  1.
s
2s
4s
Simulation
Diploid frequency-dependent selection
WAA p 2  WAa pq
Waa q 2  WAa pq
p' 
; q' 
;
W
W
where
WAA  1  3pq  3q 2 ,
WAa  1  spq,
Waa  1  3pq  3p 2 .
Simulation
Deterministic chaos
In 1976 Sir Robert May, then a professor of biology at Princeton,
pointed out that the logistic map led to chaotic dynamics.
The logistic mapping g is defined by
xn+1 = g(xn) = rxn(1 - xn).
xn1  4rxn (1  xn ), 0  r  1.
Why Chaos?
simulation
An bifurcation diagram
What is deterministic chaos?
Lyapunov's exponents:
Non-chaotic
Chaotic divergence of the
trajectories, started in closed
points
Lorenz Attractor
Tamari Attractor
• KAM (Kolmogorov-Arnold-Moser) attractor
Fractal-geometrical chaos
Fractals in Nature
Mandelbroth set z
= z^2 + c
c [-2;0,25]
Notocactus-Magnificus
Chaos in Weather
Chaos in Weather
• Fractals reproducing realistic shapes, such as mountains, clouds, or
plants, can be generated by the iteration of one or more affine
transformations. An affine transformation is a recursive
transformation of the type
•
• Each affine transformation will generally yield a new attractor in the
final image. The form of the attractor is given through the choice of
the coefficients a, b, c, d, e, and f, which uniquely determine the
affine transformation. To get a desire shape, the collage of several
attractors may be used (i.e. several affine transformations). This
method is referred to as an Iterated Function System (IFS).
• An example of an iterated function system is the black spleenwort
fern. It is constructed through the use of four affine transformations
(with weighted probabilities):
•
Inbreeding
•
•
•
•
Non-random mating (between related individuals)
Leads to correlation between genotypes of mates
Frequencies are no longer products of allele
frequencies
Leads to reduction in heterozygosity (measured by F)
AA
p2 + pqF
•
Aa
2pq - 2pqF
aa
q2 + pqF
Can rederive evolutionary equations using these new
genotype frequencies