On Age-Dependent Binary Branching Processes

We are interested in the following problem which is of possible biological, chemical, and physical interest. A particle existing at time 0 is assumed to have a life-length whose cumulative probability distribution is given by a function G(t). At the end of its life it is transformed into n similar particles with probability qn , n _ 0. These new particles are taken to have the same life-length distribution and transformation probabilities as the original one, and the process now continues. Under the hypothesis that the life-length distribution and transformation probabilities for each particle are independent of its time of birth and the number of other particles existing at the time, the problem is to determine the distribution of the number of particles existing at time t, which we call Z(t). If G(t) = 1 - ea, where a is constant, we have the Markovian case where the state of the system at t depends only upon the number of particles present and is independent of their ages. The probability that a particle existing at t is transformed between t and t + At is aAt + o(At), independently of age and absolute time. In this case the integral equation obtained below for the generating function of Z(t) reduces to a first order partial differential equation. If G(t) is a step-function with one step, we have the Galton-Watson familytree model. The case where G(t) is a convolution of k distributions of the form 1 - e-a was treated by D. G. Kendall, [6]. In this case, the particle goes through k stages before it is transformed, and by considering jointly the numbers present in each stage at a given time the process is made Markovian. The problem of the age-structure of the particles at a given time is not treated in the present paper but can be approached by similar methods; see [5]. This problem has been treated by Kendall in [7], using different methods, and more