Partition quadratic stochastic operators converge to the empty body

An $F$-quadratic stochastic operator describes a population of two sexes together with an empty body $0$: parents of the same sex, or a pair containing $0$, produce $0$. Rozikov and Zhamilov proved that every such operator has the unique fixed point $e_0$ and that every trajectory converges to it exponentially fast, and Ganikhodzhaev and Rozikov asked whether the same holds when the two sexes are replaced by the blocks of an arbitrary partition. We show that it does. For a partition into $q$ blocks the mass $y$ outside the empty body satisfies $y'\le(1-1/q)\,y^2$, so every trajectory converges to $e_0$ doubly exponentially fast, and there are operators and trajectories along which this inequality is an equality at every step. The requirement that pairs containing $0$ produce $0$ cannot be dropped.