Global stability in a negative chemotaxis system with chemically induced lethality

In this paper, we investigate the long-time dynamics of a repulsive Keller-Segel chemotaxis system. The model features negative chemotaxis, logistic growth and a cell death term, accounting for a lethal chemorepellent that is self-produced by the cells and externally supplied. We prove that, for constant chemorepellent supplies, depending on their magnitude with respect to the logistic growth rate, solutions converge in $L^{\mathrm{\infty }}$ norm toward extinction of the population, or equilibrate toward a nontrivial spatially homogeneous steady state.