Section 5.5

Age-dependent branching processes

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Problem 5.5.1

Let $Z(t)$ be the population-size at time $t$ in an age-dependent branching process, the lifetime distribution of which is exponential with parameter $\lambda$ . If $Z(0) = 1$ , show that the probability generating function $G_{t}(s)$ of $Z(t)$ satisfies

\frac{\partial }{\partial t} G_{t}(s) = \lambda \left\{ G\left(G_{t}(s)\right)-G_{t}(s)\right\}

where $G$ is the probability generating function of a typical family-size. Show in the case of 'exponential binary fission', when $G(s) = s^{2}$ , that

G_{t}(s) = \frac{s e^{-\lambda t}}{1-s\left(1-e^{-\lambda t}\right)}

and hence derive the probability mass function of the population size $Z(t)$ at time $t$ .

Problem 5.5.2

Solve the differential equation of Exercise (5.5.1) when $\lambda = 1$ and $G(s) = \frac{1}{2}\left(1+s^{2}\right)$ , to obtain

G_{t}(s) = \frac{2 s+t(1-s)}{2+t(1-s)}

Hence find $\mathbb {P}\left(Z(t) \geq k\right)$ , and deduce that

\mathbb {P}\left(Z(t) / t \geq x \mid Z(t) > 0\right) \rightarrow e^{-2 x} \quad \text{ as } t \rightarrow \infty