Modern aspects of classical Modern aspects of classical branching processesbranching processes

Vladimir VatutinVladimir Vatutin(Steklov Mathematical Institute, (Steklov Mathematical Institute,

Moscow)Moscow)

RURK DYNASTY (8??-1598)RURK DYNASTY (8??-1598)

Viking RURK

...

Ivan IV (The Terrible)

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

RURK DYNASTY (8??-1598)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 (1613-ROMANOV DYNASTY (1613-1917)1917)

Mikhail Romanov

....

ROMANOV DYNASTY (1613-ROMANOV DYNASTY (1613-1917)1917)

Peter I

Mikhail Romanov

....

Catherine The Great

ROMANOV DYNASTY (1613-1917)

Peter I

Alexandr II

Mikhail Romanov

....

Catherine The Great

Nikolay II

Alexandr III

A PART OF QUEEN VICTORIA A PART OF QUEEN VICTORIA DYNASTYDYNASTY

Queen Victoria

A PART OF QUEEN VICTORIA DYNASTY

Queen Victoria

PrincessAlice

A PART OF QUEEN VICTORIA A PART OF QUEEN VICTORIA DYNASTYDYNASTY

Queen Victoria

PrincessAlice

PrincessAlix

Ludvig IVGrand Duke

Of HesseAnd by Rhine

.........

Prince Philip, Duke of Edinburgh

Queen Elizabeth II

ROMANOV-GOTTORP DYNASTY (1613-1917)

Peter I

Alexandr II

Mikhail Romanov

....

Catherine The Great

Alexei

Nikolay II

Alexandr III

Queen Victoria

(haemophilia)

PrincessAlice

PrincessAlix

ROMANOV-GOTTORP DYNASTY (1613-1917)

Peter I

Alexandr II

Mikhail Romanov

....

Catherine The Great

Alexei

Nikolay II

Alexandr III

Queen Victoria

(haemophilia)

PrincessAlice

PrincessAlix

Rasputin

T. R. Malthus T. R. Malthus : An Essay on the Principle of : An Essay on the Principle of Population as it Affects the Future of Population as it Affects the Future of

Society, 1798)Society, 1798)

379379 out of out of 487487 bourgeois families in bourgeois families in the city Berne extinct between the city Berne extinct between 1583 1583 and and 17831783

The proportion of extinct families is The proportion of extinct families is 379/487379/487, that is more than , that is more than 0.750.75 (!) in (!) in approximately approximately 66 generations generations

Galton-Watson processGalton-Watson process

0

Galton-Watson processGalton-Watson process

0

1

1

Galton-Watson processGalton-Watson process

0

1

1

2

2

2

Galton-Watson processGalton-Watson process

0

1

1

2

3

2

2

3

3

Galton-Watson processGalton-Watson process

0

1

1

2

3

2

2

3

3

4

4

4

Growth and ExtinctionGrowth and Extinction

AA= the average number of children per individual= the average number of children per individual

Growth and Extinction Growth and Extinction

AA= the average number of children per individual = the average number of children per individual

A<1A<1 – – subsubcritical process – always critical process – always diesdies

Growth and ExtinctionGrowth and Extinction

AA= the average number of children per individual = the average number of children per individual

A<1A<1 – – subsubcritical process – always critical process – always diesdies

A=1A=1 – critical process -- always – critical process -- always diesdies

Growth and ExtinctionGrowth and Extinction

AA= the average number of children per individual = the average number of children per individual

A<1A<1 – – subsubcritical process – always critical process – always diesdies

A=1A=1 – critical process -- always – critical process -- always diesdies

A>1A>1 – – supersupercritical process – survives with a positive critical process – survives with a positive probabilityprobability

North Atlanitic Right WhaleNorth Atlanitic Right Whale

ExtinctionExtinction of North Atlantic of North Atlantic WhalesWhales

• `The average number of female children `The average number of female children = A= A• The current size of the female population approx. The current size of the female population approx. 150150• (Data for A from Caswell H., Fujimagara, M., Brault S.,1999)(Data for A from Caswell H., Fujimagara, M., Brault S.,1999)

395796n<Extinction with probability >0.99 in n years

177 357n>Survival with probability >0.99 for at least n years

0.9760.988A<1 (!!!)

The average size of the North The average size of the North Atlantic Right Whales Atlantic Right Whales populationpopulation• Current size approximately Current size approximately

150150 female individuals female individuals

• Upper graph: Upper graph: A=0.988A=0.988

• Lower graph: Lower graph: A=0.976A=0.976

5003752501250

150

125

100

75

50

25

x

y

x

y

5003752501250

150

125

100

75

50

25

0

x

y

x

y

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