## Problem 150 — rank:151

**Title:** Formation of cooperative groups

**Added binding assumptions (LaTeX):**

```latex
For a multitype branching approximation, specify the offspring generating functions as well as the mean matrix. Spectral radius above one allows positive survival under regularity, but does not ensure establishment probability at least $1-\delta$ from a small founder population. Compute the extinction fixed point and initial-type dependence, then distinguish branching-phase establishment from later density-dependent persistence and finite-horizon assay success.
```

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