Generalization of pure birth process

Question

The pure birth process is the generalization of the Poisson process where instead of the transition rate being $\lambda$ we write $\lambda_n$ to account for dependence of the rate on the state. What if we added as well dependence on time, meaning we considered $\lambda_n(t)$? What is this kind of process called?

Answer 1

Non-homogeneous birth process.

Pure birth processes are described by the following general equation: $$P'_r(t)=-\lambda_r(t)P_r(t)+\lambda_{r-1}(t)P_{r-1}(t)$$ where $P_r(t)$ is the probability of having $r$ individuals in the population at time $t$. The Homogeneous Poisson Process arises when $\lambda_r(t)$ is constant in time $\lambda_r(t)= \lambda$ and the homogeneous birth process is when $\lambda_r(t)= \lambda_r$