Modern aspects of classical branching processes Vladimir Vatutin (Steklov Mathematical Institute, Moscow)

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Presentation transcript:

Modern aspects of classical branching processes Vladimir Vatutin (Steklov Mathematical Institute, Moscow)

RURK DYNASTY (8??-1598) Viking RURK... Ivan IV (The Terrible)

Ivan the Terrible is killing his son (Painter - Il’ya Repin)

RURK DYNASTY (8??-1598) Viking RURK... Ivan IV (The Terrible) Fedor died in 1598 Ivan (killed by his father in 1581) Dmitrii (died in 1591 of an epileptic fit)

ROMANOV DYNASTY ( ) Mikhail Romanov....

ROMANOV DYNASTY ( ) Peter I Mikhail Romanov.... Catherine The Great

ROMANOV DYNASTY ( ) Peter I Alexandr II Mikhail Romanov.... Catherine The Great Nikolay II Alexandr III

A PART OF QUEEN VICTORIA DYNASTY Queen Victoria

A PART OF QUEEN VICTORIA DYNASTY Queen Victoria Princess Alice

A PART OF QUEEN VICTORIA DYNASTY Queen Victoria Princess Alice Princess Alix Ludvig IV Grand Duke Of Hesse And by Rhine Prince Philip, Duke of Edinburgh Queen Elizabeth II

ROMANOV-GOTTORP DYNASTY ( ) Peter I Alexandr II Mikhail Romanov.... Catherine The Great Alexei Nikolay II Alexandr III Queen Victoria (haemophilia) Princess Alice Princess Alix

ROMANOV-GOTTORP DYNASTY ( ) Peter I Alexandr II Mikhail Romanov.... Catherine The Great Alexei Nikolay II Alexandr III Queen Victoria (haemophilia) Princess Alice Princess Alix Rasputin

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

Galton-Watson process 0

0 1 1

Growth and Extinction A= the average number of children per individual

Growth and Extinction A= the average number of children per individual A<1 – subcritical process – always dies

Growth and Extinction A= the average number of children per individual A<1 – subcritical process – always dies A=1 – critical process -- always dies

Growth and Extinction A= the average number of children per individual A<1 – subcritical process – always dies A=1 – critical process -- always dies A>1 – supercritical process – survives with a positive probability

North Atlanitic Right Whale

Extinction of North Atlantic Whales `The average number of female children = A `The average number of female children = A The current size of the female population approx. 150 The current size of the female population approx. 150 (Data for A from Caswell H., Fujimagara, M., Brault S.,1999) (Data for A from Caswell H., Fujimagara, M., Brault S.,1999) n<Extinction with probability >0.99 in n years n>Survival with probability >0.99 for at least n years A<1 (!!!)

The average size of the North Atlantic Right Whales population Current size approximately 150 female individuals Current size approximately 150 female individuals Upper graph: A=0.988 Upper graph: A=0.988 Lower graph: A=0.976 Lower graph: A=0.976

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