Extinction Moment for a Branching Tree Evolution with Birth Rate and Death Rate Both Depending on Age

Abstract

In this paper, a branching tree evolution is established, in which the birth rate and the death rate are both dependent on node’s age. The extinction probability and the t-pre-extinction (extinct before time $t$ ) probability are studied, by which the distribution of the extinction moment can be given. The analytical formula and the approximation algorithm for the distribution of extinction moment are given; furthermore, the analytical formula and the approximation algorithm of extinction probability are given, and a necessary and sufficient condition of extinction with probability 1 is given. It is the first time to study the distribution of extinction time for the branching process with birth rate and the death rate both depending on node’s age, and the results will do great help in the theory of branching process. It is expected to be applied in the fields of biology, genetics, medicine, epidemiology, demography, nuclear physics, actuarial mathematics, algorithm, and data structures, etc.