(i) Explain what a $Q$-matrix is. Let $Q$ be a $Q$-matrix. Define the notion of a Markov chain $\left(X_{t}, t \geqslant 0\right)$ in continuous time with $Q$-matrix given by $Q$, and give a construction of $\left(X_{t}, t \geqslant 0\right)$. [You are not required to justify this construction.]

(ii) A population consists of $N_{t}$ individuals at time $t \geqslant 0$. We assume that each individual gives birth to a new individual at constant rate $\lambda>0$. As the population is competing for resources, we assume that for each $n \geqslant 1$, if $N_{t}=n$, then any individual in the population at time $t$ dies in the time interval $[t, t+h)$ with probability $\delta_{n} h+o(h)$, where $\left(\delta_{n}\right)_{n=1}^{\infty}$ is a given sequence satisfying $\delta_{1}=0, \delta_{n}>0$ for $n \geqslant 2$. Formulate a Markov chain model for $\left(N_{t}, t \geqslant 0\right)$ and write down the $Q$-matrix explicitly. Then find a necessary and sufficient condition on $\left(\delta_{n}\right)_{n=1}^{\infty}$ so that the Markov chain has an invariant distribution. Compute the invariant distribution in the case where $\delta_{n}=\mu(n-1)$ and $\mu>0$.