Some new results on an allelopathic competition model with quorum sensing and delayed toxicant production

A 4-equation delay differential system representing a bacterial allelopathic competition is analyzed. A distributed delay term models a linear quorum-sensing mechanism which regulates the delayed allelochemicals’ production process. The proved qualitative properties of the solutions are positivity, boundedness, global existence in the future, and uniqueness. Sufficient conditions for local asymptotic stability properties of biologically meaningful steady-state solutions are given in terms of the parameters of the system. The global asymptotic stability of a biologically meaningful steady-state solution is proved by constructing a suitable Lyapunov functional.