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In this paper, we study the minimal Keller-Segel model with a logistic sourceand obtain quantitative and qualitative descriptions of the competition betweenlogistic damping and other ingredient, especially, chemotactic aggregation toguarantee boundedness and convergence. More specifically, we establish howprecisely strong a logistic source can prevent blow-up, and then we obtain anexplicit relationship between logistic damping and other ingredient,especially, chemotactic aggregation so that convergences are ensured and theirrespective convergence rates are explicitly calculated out. Known results inthe literature are completed and refined.  Furthermore, our findings provide clues on how to produce blowup solutionsfor KS chemotaxis models with logistic sources.

How strong a logistic damping can prevent blow-up for the minimal Keller-Segel chemotaxis system?