 Solution

With $n$ individuals, each of the $n$ can reproduce at rate $\lambda$ and independently die at rate $\lambda$. Thus the total birth and death rates are $q_n=r_n=\lambda n$. The gain into state $n$ comes from a birth in state $n-1$ and a death in state $n+1$, while its loss rate is $2\lambda n$. ThereforeAt $n=0$, the birth inflow from negative states is absent and ${\dot{P}}_0=\lambda P_1$. The empty state is absorbing: once the population reaches zero, neither process can restart it. Equal birth and death rates make this a critical branching process.