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Bounded Populations. Extinction
Time and Time of Supplanting All
Particles by Particles of One Type.
Klokov S.A., Topchii V.A.
Omsk Branch of Sobolev Institute of Mathematics
SB RAS
Object
• Population dynamics and population evolution
analysis based on changes of the DNA.
• Discrete time Markov models.
• Haploid models without mutations.
• Population size is fixed (the birth and death
distribution is such that the total population size
does not change) or bounded.
• Random number of offspring.
• Evolution of families.
• Extinction time.
The most popular models of the given class were
introduced by Wright and Fisher, Moran,
Karlin, and McGregor. In the Wright–Fisher
model, the numbers of particles of different
types in a generation are the parameters of a
polynomial distribution of the birth of
offspring. In the Moran model, only one
particle dies replacing one or several others
with its offspring. In the Karlin and McGregor
model, the process branches and the
distribution is redefined by the condition of
fixed total number of offspring.
Wright–Fisher Model
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k
N-k
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All parents are equally
likely
j
N-j
Galton-Watson. Common ancestor.
t(n)
Z0
Z1
Z2
EZn=m^n
Zubkov 1975 Galton-Watson
Z3
Z4
Z n- t(n)
Z n-1
m=Ex(<1)(=1)(>1) ;
Zn
Zn >0
t(n)g m<1; t(n)n-n m>1; t(n) nh h uniformly on [0,1] m=1.
Common Ancestor
• Branching models
O’Connell Neil (1995)
750000-1500000 years ago
• Wright–Fisher model (Mitochondrial Eve)
unisexual Cann (1987) 100000-200000 years ago
In Africa
bisexual Сhang (1999) 500 years ago , 32
generations
Fixation. Example 1.
Population size 5; Maximal number of offspring 3; 4 generations passed.
Fixation. Example 1. s=2.
O
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Z n-3
Z n-2
Zn-1
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Zn
t 3
Empirical Results
Theorem 1.
Let t be fixation time, N population size, k
number of type I individuals initially, s>1
splitting factor, then
s-N
Ekt  (2   N )
 s -1
-1
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N ln N - k ln k - ( N - k ) ln(N - k )
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Empirical Results
Suppose that we have a fixed size population consisting
of N particles each of which can belong to one of N
types. The composition of the population changes at
the integer time moments. If, at the present time
moment, the number of particles of each type is
defined by the vector
then, at the next time moment, the
number of particles of each type is described by the
random vector
We assume that the last random vectors have
exchangeable distributions (i.e., the birth and death
law does not depend on their type) are independent
and identically distributed
We are interested in the random variable
, the
fixation time of the population (i.e., supplanting
all particles by particles of one type or the first
attainment of the set of absorbing states
)
and a bound for its expectation in terms of the moments
of the distribution
Along with , we consider the random variable
equal to the number of generations starting
from which the population first consists of
particles of a single type.
Theorem 2.
Extinction
Population size
Upper bound for population size
Initial
Population size
…
0
time
Population size trajectory. Time to extinction is Huge
Extinction
Let
process,
be supercritical branching
,
- probabilistic generating
function.
Let
and
- extinction probability.
-
- population
bounded on level l. Let
Theorem 3.
Let
then
For crush period
and
,