We consider a population in which each organism can generate another organism and in turn can die as a result of competing with the other members of the population. Moreover, there is a constant flux of new organisms which come from outside. We represent respectively by $\lambda$, $\mu$ and $\beta$ the corresponding rates in each case. Find the master equation of the process.
If we don't consider the fact that there is a constant flux of new organisms which come from outside, the master equation will be
$$p_n(t+dt)=\lambda\,dt\, p_{n-1}(t)+\mu(n+1)\,dt\, p_{n+1}(t)+(1-(\lambda+\mu n)dt)p_n(t),$$
where $p_n(t)$ is the probability of finding $n$ organisms at a time $t$. However, I don't know how to introduce this variation. Any help would be appreciated.