Klokov S.A., Topchii V.A.

1. 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

2. 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.

3. 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.

4.                      .  .  .   .  .  .  .  .  .  .   .  .                  All parents are equally likely k j N-j N-k Wright–Fisher Model

5. t(n)g m<1; t(n)n-n m>1; t(n) nh h uniformly on[0,1] m=1. t(n) Z Z Z Z Z Z Z Z 0 1 2 3 4 n- t(n) n-1 n EZn=m^n Zubkov 1975 Galton-Watson m=Ex(<1)(=1)(>1) ;Z >0 n Galton-Watson. Common ancestor.

6. Common Ancestor • Branching models O’Connell Neil (1995) 750000-1500000 years ago • Wright–Fisher model (Mitochondrial Eve) unisexualCann (1987) 100000-200000 years ago In Africa bisexual Сhang (1999) 500 years ago , 32 generations

7. Fixation. Example 1. Population size 5; Maximal number of offspring 3; 4 generations passed.

8. O O O O O O O   O    O O     O O O O  O     O O O  O O Fixation. Example 1. s=2.

9. Empirical Results

10. Theorem1. Let  be fixation time, N population size, k number of type I individuals initially, s>1splitting factor, then

11. Empirical Results

12. 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

13. 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.

14. Theorem2.

15. Extinction Population size Upper bound for population size … Initial Population size 0 time Population size trajectory. Time to extinction is Huge

16. Extinction Let be supercritical branching process, , - probabilistic generating function. Let and • extinction probability. • - population bounded on level l. Let

17. Theorem3. Let and , then For crush period