Modern aspects of classical branching processes

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Modern aspects of classical branching processes

• Vladimir Vatutin
• (Steklov Mathematical Institute, Moscow)

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RURK DYNASTY (8??-1598)
Viking RURK
...
Ivan IV (The Terrible)
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Ivan the Terrible is killing his son (Painter -
Ilya Repin)
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RURK DYNASTY (8??-1598)
Viking RURK
...
Ivan IV (The Terrible)
Ivan (killed by his father in 1581)
Dmitrii (died in 1591 of an epileptic fit)
Fedor died in 1598
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ROMANOV DYNASTY (1613-1917)
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ROMANOV DYNASTY (1613-1917)
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ROMANOV DYNASTY (1613-1917)
Alexandr III
Nikolay II
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A PART OF QUEEN VICTORIA DYNASTY
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A PART OF QUEEN VICTORIA DYNASTY
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A PART OF QUEEN VICTORIA DYNASTY
Ludvig IV Grand Duke Of Hesse And by Rhine
.........
Prince Philip, Duke of Edinburgh
Queen Elizabeth II
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ROMANOV-GOTTORP DYNASTY (1613-1917)
Alexandr III
Nikolay II
Alexei
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ROMANOV-GOTTORP DYNASTY (1613-1917)
Alexandr III
Rasputin
Nikolay II
Alexei
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T. R. Malthus An Essay on the Principle of
Population as it Affects the Future of Society,
1798)

• 379 out of 487 bourgeois families in the city
  Berne extinct between 1583 and 1783
• The proportion of extinct families is 379/487,
  that is more than 0.75 (!) in approximately 6
  generations

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Galton-Watson process
0
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Galton-Watson process
1
0
1
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Galton-Watson process
2
1
2
0
1
2
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Galton-Watson process
3
2
1
2
0
1
3
2
3
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Galton-Watson process
4
3
4
2
4
1
2
0
1
3
2
3
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Growth and Extinction

• A the average number of children per individual

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Growth and Extinction

• A the average number of children per individual
• Alt1 subcritical process always dies

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Growth and Extinction

• A the average number of children per individual
• Alt1 subcritical process always dies
• A1 critical process -- always dies

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Growth and Extinction

• A the average number of children per individual
• Alt1 subcritical process always dies
• A1 critical process -- always dies
• Agt1 supercritical process survives with a
  positive probability

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North Atlanitic Right Whale
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(No Transcript)
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Extinction of North Atlantic Whales

• The average number of female children A
• The current size of the female population
  approx. 150
• (Data for A from Caswell H., Fujimagara, M.,
  Brault S.,1999)

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The average size of the North Atlantic Right
Whales population

• Current size approximately 150 female individuals
• Upper graph A0.988
• Lower graph A0.976

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In a book store near you from May 19th, 2005