The structure of genealogical trees of a subcritical branching process in random environment

Vladimir Vatutin

Steklov Mathematical Institute, Moscow, Russia

(jointly with K.Fleishmann, Wias, Berlin)

Abstract

Branching processes are used rather often to describe different phenomena in biology, physics and chemistry. During the last two decades it became clear that the so called general branching processes, or, to be more precise, Crump-Mode-Jagers branching processes can be used in studying various characteristics of evolving populations. It is known that such processes either become extinct or grow exponentially. This fact shows that there is no stability in the number of particles in populations and seems do not allow us to apply branching processes in studying dynamics of populations. For this reason some researchers (P.Jagers and his school and some others) suggested to analyze not the number of particles in the supercritical processes but the composition of the population generated by such processes. This approach occurs to be very useful and gives reasonable explanations to some real phenomena. There is another possibility within the framework of branching processes which does not contradict to biological facts: to consider subcritical branching processes conditioned on non-extinction. In this case we have a stationary distribution of the number of particles in the populations (at least for the Markovian branching processes). However, this idea still does not allow us to explain such phenomenon as the large amplitude of the number of species in some populations of birds or see animals during the evolution of populations. In our talk we consider one more model of branching processes, namely, subcritical branching processes in random environment conditioned on non-extinction and show that the reduced tree of such processes (or, in the other words, the ancestral tree constructed for the species living at a fixed moment) possesses a number of interesting properties which provide (at least on the theoretical level) certain explanation to such phenomena.