————— * Research supported by the Natural Sciences and Engineering Research Council of Canada. ** Research supported by the Alexander von Humboldt Foundation of Germany. Mathematics Subject Classification (1991): 34D20, 34K20, 45M10, 92D25 J. Math. Biol. (1999) 38: 285 —316 Global dynamics of a chemostat competition model with distributed delay Gail S. K. Wolkowicz *, Huaxing Xia, Jianhong Wu * **  Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4K1. e-mail: wolkowic@mcmail.cis.mcmaster.ca  Department of Mathematics and Statistics, York University, North York, Ontario, Canada M3J 1P3 Received: 9 August 1997 / Revised version: 2 July 1998 Abstract. We study the global dynamics of n-species competition in a chemostat with distributed delay describing the time-lag involved in the conversion of nutrient to viable biomass. The delay phenomenon is modelled by the gamma distribution. The linear chain trick and a fluctu- ation lemma are applied to obtain the global limiting behavior of the model. When each population can survive if it is cultured alone, we prove that at most one competitor survives. The winner is the popula- tion that has the smallest delayed break-even concentration, provided that the orders of the delay kernels are large and the mean delays modified to include the washout rate (which we call the virtual mean delays) are bounded and close to each other, or the delay kernels modified to include the washout factor (which we call the virtual delay kernels) are close in ¸-norm. Also, when the virtual mean delays are relatively small, it is shown that the predictions of the distributed delay model are identical with the predictions of the corresponding ODEs model without delay. However, since the delayed break-even concentra- tions are functions of the parameters appearing in the delay kernels, if the delays are sufficiently large, the prediction of which competitor survives, given by the ODEs model, can differ from that given by the delay model. Key words: Distributed delay — Chemostat — Competitive exclusion — Global dynamics