Removal-Only Actuation in Age-Structured Branching Populations: Fundamental Limits of Equilibrium Placement

Abstract

A subcritical age-structured branching population dies out almost surely. Conditioned on survival, it converges to its Yaglom limit, a quasi-stationary equilibrium that we take as the operating point for control. We model preventive removal (culling) as an age-dependent actuator that raises the mortality rate and leaves the offspring law untouched, and we show that its authority over this equilibrium is bounded for structural reasons. Two facts drive the result. First, the input is matched to the killing rate but unmatched with respect to the Foster--Lyapunov drift, so the transmission barrier $\Lbar$ set by reproduction alone is invariant under such actuation. Second, and this does not follow from invariance alone,} the supremum of the reachable decay rates is $\Lbar+ν^\star$, where $ν^\star\le0$ is the Malthusian parameter of the lineage conditioned never to die childless; the gap $|ν^\star|$ is given in closed form and vanishes exactly when no individual has two or more offspring. Consequently no removal law of this class reaches the barrier, and along the admissibility boundary the achievable decay rate is governed by the shape of the actuator rather than by its size. We illustrate these results on a model calibrated to the 2001 Cumbrian foot-and-mouth outbreak.