  In this paper, we study the initial-boundary value problem and its asymptotic
behavior for a repulsive chemotaxis model with logarithmic sensitivity and
logistic growth. We establish global well-posedness of strong solutions for
large initial data with Neumann boundary conditions and, moreover, establish
the qualitative result that both the population density and chemical
concentration asymptotically converge to constant states with the population
density specifically converging to its carrying capacity. We additionally prove
that the vanishing chemical diffusivity limit holds in this regime. Lastly, we
provide numerical confirmation of the rigorous qualitative results, as well as
numerical simulations that demonstrate a separation of scales phenomenon.
