1. Finite population. - N - number of individuals - N A and N a – numbers of alleles A and a in population Two different parameters: one locus and two allels A,a. Non-overleaping generation. Then there are exist two problem – modeling behavior N and behavior N A. Two different trajectories in this population system.

2. US Population: 1650-1800 Exponential) growth

3. In 1798, the greatest social scientist Thomas Malthus proposed a model for human populations. His model was based on the observation that the time required for human popu- lations to double was essentially constant (about 25 years at the time), regardless of the initial population size. The Malthus Model

4. This means: Rate of change of population = CBR – CDR CBR = crude birth rate CDR = crude death rate Population grows when births exceed deaths (when there’s no migration). Change in pop = births – deaths Divide this equation by the population:

5. Discrete-in-time Model –0, 1, 2, …, n,…: equally-spaced times at which the population is determined –N 0, N 1, N 2, …, N n : corresponding populations at times 0, 1, 2, …, n,… –b and d: birth and death rates; r = b – d, is the effective growth rate. N 0 N 1 N 2 … N n |---------|---------|----------------|-----> t 0 1 2 … n

6. The Malthus Model Mathematical Equation: (N i + 1 - N i ) / N i = r r = b - d or N i + 1 = N i + r * N i = (1+r)*N i, i = 0, 1,... The initial population, N 0, is given at the initial time, 0.

7. N i + 1 = (1+r)*N i, i = 0, 1,... If r>0 then exponential grows For example for US population in 1900 r = 0.013.

8. HUMAN POPULATION GROWTH

11. Forecasted probability that population will start to decline at or before the indicated date. Lutz et al. Nature 412, 543 - 545 (2001)

12. Grows for different population is not exponential. Why?

13. Logistics Model As a result: N i+1 = N i + [r 0 * (1 - N i /K)] * N i Malthus: N i + 1 = (1+r)*N i, and r-constant. In 1838, Belgian mathematician Pierre Verhulst modified Malthus’ model to allow growth rate to depend on population: Verhulst : r not constant, r = [r 0 * (1 – N/K)], r 0 is maximum possible population growth rate. K is called the population carrying capacity.

14. Feigenbaum! Chaotic behavior for r 0 >2,570 Simulation

15. Here is the curve when P t =0.1, r =1.3 and K=1,000

16. Logistics Model r o controls population growth rate, if reproduction is slow and mortality is fast, the logistic model will not work. K has biological meaning for populations with strong interaction among individuals that control their reproduction: birds have territoriality, plants compete for space and light.

17. Growth of Yeast Cells Population of yeast cells grown under laboratory conditions: P 0 = 10, K = 665, r 0 =.54, Δt = 0.02

18. hare-lynx cycles humans in Egypt

19. Let N- constant, A, a - alleles For N infinity evolutionary operator on zygote level p`= p2 +pq; q` = q2 +pq evolutionary operator on gamete level p`= p; q` = q. For N finite evolutionary operator on gamete level p`= r(p); q` = 1-r(p) or N` A =r(N A ). Wright-Fisher Model

22. Genetic drift: Evolution without selection or mutation A population of fixed size. Each individual has its own type (color). Individuals reproduce clonally. Each individual has the same reproductive success on average: Each offspring individual in the new generation has a parent chosen at random from the parent generation. Parent generation Offspring generation

23. Genetic drift: Evolution without selection or mutation A population of fixed size. Each individual has its own type (color). Individuals reproduce clonally. Each individual has the same reproductive success on average: Each offspring individual in the new generation has a parent chosen at random from the parent generation. Parent generation Offspring generation

24. Genetic drift: Evolution without selection or mutation A population of fixed size. Each individual has its own type (color). Individuals reproduce clonally. Each individual has the same reproductive success on average: Each offspring individual in the new generation has a parent chosen at random from the parent generation. Parent generation Offspring generation

25. Genetic drift: Evolution without selection or mutation A population of fixed size. Each individual has its own type (color). Individuals reproduce clonally. Each individual has the same reproductive success on average: Each offspring individual in the new generation has a parent chosen at random from the parent generation. Parent generation Offspring generation

26. Genetic drift: Evolution without selection or mutation A population of fixed size. Each individual has its own type (color). Individuals reproduce clonally. Each individual has the same reproductive success on average: Each offspring individual in the new generation has a parent chosen at random from the parent generation. Parent generation Offspring generation Some have no offspring!

27. Genetic drift: Eventually one color will take over 2 N generations. For a clonally reproducing population of size N. After on average 2 N generations all but 1 lineage will go extinct. More complex for sexually reproducing entities but qualitative the same idea: Almost all genetic material stems from a very small fraction of the ancestral population more than 2 N generations ago.

28. Genetic drift with mutation Each time an individual reproduces there is a probability μ that it ‘mutates’ and introduces a new color.

29. Genetic drift with mutation Each time an individual reproduces there is a probability μ that it ‘mutates’ and introduces a new color. (1-μ)

30. Genetic drift with mutation Each time an individual reproduces there is a probability μ that it ‘mutates’ and introduces a new color. (1-μ) μ

31. Genetic drift with mutation Each time an individual reproduces there is a probability μ that it ‘mutates’ and introduces a new color. In the limit of large time the number of different colors will on average equal: C = 1 + 2 N μ. In the example above μ = 1/8. > 2 N generations.

33. Fig. 22.7, Temporal variation in a prairie vole (Microtus ochrogaster) esterase gene.

36. Simulation For N=20 number generation is 200 For N=50 number generation is 400

37. Time until fixation/loss of an allele p = intial allele frequency n = population size (Kimura und Otha 1969) number of generations (x n) initial frequency time to loss time to fixation mean persistence

38. Zygote level, genotypes AA, Aa, aa Random mating: N parents pair Next generation

39. No random mating Maximal possible parents pairs (AA, aa)

40. Selection Random mating: N parents pair Next generation Ordering by fitness W AA,W Aa,W aa