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Abstract: A model of competition between plasmid-bearing and plasmid-free organisms in the chemostat with periodic input of nutrient and two distributed delays is investigated. The delays model the fact that the nutrient is partially recycled after the death of the biomass by bacterial decomposition. It is assumed that there is inter-specific competition between the plasmid-bearing and plasmid-free organisms as well as intra-specific competition within each population. Analysis of the extinction of the organisms, including plasmid-bearing and plasmid-free organisms, and the permanence of the system are carried out. Furthermore, sufficient conditions ensuring the existence and global stability of the positive periodic solution are established. Numerical simulations illustrate the theoretical results. Finally, we present a procedure by which one can control the parameters of the model to keep the plasmid-bearing organism stay eventually in a desired set. [Copyright &y& Elsevier]

Abstract: The asymptotic behavior of solutions of a chemostat-type model in which two species compete for a limiting nutrient supplied at a constant rate is considered. The model incorporates a general nutrient uptake function and two distributed delays. The first delay models the fact that the nutrient is partially recycled after the death of the biomass by bacterial decomposition and the second indicates that the growth of the species depends on the past concentration of the nutrient. Furthermore, it is assumed that there is interspecific competition between the two species as well as intraspecific competition within each species. Conditions for boundedness of solutions and existence of nonnegative equilibria are given. By constructing appropriate Liapunov-like functionals, some sufficient conditions for global attractivity of the positive equilibrium is obtained. The combined effects of the two different delays are studied. The main results of Freedman and Xu [H.I. Freedman, Y. Xu, Models of competition in the chemostat with instantaneous and delayed nutrient recycling, J. Math. Biol. 31 (1993) 513–527] and Ruan and He [S. Ruan, X.-Z. He, Global stability in chemostat-type competition models with nutrient recycling, SIAM J. Appl. Math. 58 (1) (1998) 170–192] are improved and extended. [Copyright &y& Elsevier]