Consider a simple branching process $Z\_0,Z\_1,Z\_2...$ such that at every discrete step, a particle splits into $k\geq1$ particles where $k$ follows a discrete distribution with probability mass $p(k)$.


From the theory of branching processes, a basic result is that the average number of particles at the $n$th step is given by:


$$\mu:=\mathbb{E}[Z\_n|Z\_0=1]=m^n,$$
where $m=\sum\_{k=1}^\infty k p(k)$ is the mean of the offspring distribution.


### Question


What happens to $\mu$ when the offspring distribution depends on the number of steps? That is, $p(k)\to p\_n(k)$ where $n$ is a parameter that represents the number of steps.


What is the typical vocabulary to describe this generalization? I could not find this with some naive searches on google scholar.


Finally, to illustrate it with an example: Let
$$p\_n(k)= p(k) e^{-\frac{k}{n}}c\_n,$$


where $c\_n$ is to normalise $p\_n(k)$ and $p\_0(k):=p(k)$. Is it still possible to express $\mu$ with a close formula?


(This question is motivated by population dynamics, where as a population grows, the number of new-borns decreases due to limited space, competition etc.)



Branching process with varying offspring distribution at each step
As far as terminology goes, I have seen this called "Branching process in a varying environment" and also "inhomogeneous Galton-Watson process".


Starting from one individual, the mean of $Z\_n$ is simply $\prod\_{i=1}^n m\_i$ where $m\_i$ is the mean of the offspring distribution for the $i$th generation. For example you can get this by induction using the conditional expectation $E(Z\_n|Z\_{n-1})=m\_n Z\_{n-1}$.


