Integrals of branching processes

For age-dependent branching processes X(t) integrals of the type Y(t)∫0tX(s)dsX(s)ds are studied. These may be looked upon as a measure of the amount of toxins produced in time (0, t) by a colony of bacteria. An integral equation for the moment generating function of {X(t), Y(t)} is deduced and used to analyze the asymptotic behaviour of the two first moments of the vector. A functional equation is found that determines the limit distribution of Y(t) uniquely, and an iterative method of solving the equation is given. An application is made to the case of binary splitting with exponential life-lengths, and last the convergence of the process [X(t)/E{X(t)}, Y(t)/E{Y(t)}] is studied.