= 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 $2\lambda n$. Therefore
$$
\boxed{\dot P_n=\lambda(n-1)P_{n-1}+\lambda(n+1)P_{n+1}-2\lambda nP_n.}
$$
At $n=0$, 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>.
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