## Problem 137 — rank:146

**Title:** Stem-cell niche control

**Added binding assumptions (LaTeX):**

```latex
Define the full daughter-number distribution, not only a mean self-renewal probability, and supply the niche-state dynamics. A nonlinear birth--death model does not generally yield a closed deterministic equation for the mean without an approximation and error regime. In a finite population without immigration, a stable positive mean-field equilibrium can coexist with eventual extinction; distinguish finite-horizon maintenance, quasi-stationarity and true invariant persistence.
```

**Direct edits:** 0. **Fresh external checks:** None; supplied sources retained as provenance.