Age-dependent branching processes under a condition of ultimate extinction

SUMMARY The processes discussed in this paper are models for populations of independent identical individuals reproducing by fission, with the alternative possibility of death. The probabilities for birth and death are considered to be age-dependent, so that under suitable assumptions the populations have a probability to become extinct, which is neither zero nor one. It is the purpose of the present paper to study conditional probabilities for such populations where the condition is that they do ultimately become extinct. This is done by working in terms of a space of family trees, and the results include conditional probabilities for the life-lengths of individuals and the number of their offspring. An example is given, and an application is made to a model of bacterial toxicity. For a Markov process possessing a set of transient states, and one or more sets of absorbing states, there is a simple method for deriving probabilities conditional on absorption in a given set, from unconditional probabilities for the same process (Breny, 1962; Kemeny & Snell, 1960, p. 64; Waugh, 1958). This method takes a particularly simple form for the Markovian binary-fission-or-death branching process when the condition is that of ultimate extinction, and it has been applied in this form to a model of the toxic effect of a bacterial invasion (Puri, 1966). In the present note, a related method is developed for nonMarkovian age-dependent processes. The results take rather simple forms when expressed in terms of the distributions, conditional and otherwise, of the life-lengths of individuals and of the number of their children. In the process discussed in this note the life-length distributions may take general forms, whereas in the Markovian case they are all negativeexponential. An example for Erlangian life-lengths is given, and the method is used to extend results of Jagers (1967) for the integral of the population size. Incidentally, Kendall (1966) has shown how to obtain conditional probabilities for Markovian birth processes that tend to infinity, where conditioning is on the random variable W = lim {Zt1f'(Z1)}, Zt being the population size at time t. The variable W is discussed