Logistic cell-division process
= Logistic cell-division process
{title2=$q_N(k,k+1)=k(1-k/N)$}
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