Consider the famous equation associated to the logistic map: $$x_{n+1}=r x_n\left(1-x_n\right),\quad r>1 $$
It is well known that there is no closed formula for $x_n$ as a function of $n$. However, consider the following alternative:
$$y_{n+1}=r y_n\left(1-\frac{1}{N}\sum_{m=0}^n y_m\right)$$
Where $N$ is the size of the system: $y_n$ can be interpreted as the number of new offsprings born at step $n$ and the total size of the population at step $n$ is always $\sum_{m=0}^n y_m\leq N$. $r$ is interpreted as the reproduction number. Because of the limited capacity, the "effective" $r$ is slightly different. Assume $y_0=1$.
Is it possible to find a closed form expression for $y_n$? When the total number of individuals is much smaller than $N$, then we have $y_n\approx r^n$. Can we at least determine an intermediate regime? i.e. when $n\gg 1$ yet $\sum_{m=0}^n y_m \ll N$.
Has this been done somewhere? Surely, but I cannot find a source.