Logistic cell-division process ($q_N(k,k+1)=k(1-k/N)$)

For a carrying capacity $N$, each of $k$ cells splits into two at rate $1-k/N$, so the population increases by one at the displayed total rate and is absorbed at $N$. Starting from population density $p\in (0,1)$, its normalized density converges on finite horizons to $p{e}^t/(1-p+p{e}^t)$ by the Kurtz fluid limit theorem.

// Target: analysis.bigb