Mutual Inhibition in a Two Species Chemostat-like System G. E. Kimathi * , A. A. M. Wasike † , and G. P. Pokhariyal ‡ Abstract A general model of the chemostat with mutual inhibition is provided in this paper. Global behavior of the solutions of this model are discussed and we show that for two species competing for a single nutrient available in limiting supply, at most one species will survive. Key words: Mutual Inhibition, Chemostat, Competitive exlusion. AMS (MOS) subject classifications: 94A15, 92A17, 34C15, 34C35. 1 Introduction Finding criteria for the long term coexistence of species is an important problem in population biology. This long term coexistence, also known to as permanence or uniform persistence, requires non-extinction as well as non-explosion of species, but allows otherwise arbitrary asymptotic behaviour. Most studies dealing with permanence are centered on models gov- erned by autonomous systems of differential equations. The models predict that when species compete for n resources available in limiting supply, at most n species can survive. In na- ture, only a few resources are usually in limiting supply, but the number of species surviving on those few resources is abundant. Specifically, for a single limiting resource, the models predict that at most one species can survive. This competition for a single resource is called exploitative competition [4]. A good place where the predictions of these mathematical models may be tested experi- mentally is in a laboratory device called a chemostat. Basically, the chemostat contains three chambers where the first chamber contains the nutrient, the second contains the competing species and the third is for collecting the overflow. The nutrient is pumped into the second * School of Mathematics, University of Nairobi, P.O. Box 30197 - 00100, Nairobi. gkimathi@daystar.ac.ke † Department of Mathematics, Masinde Muliro University of Science and Technology, P.O. Box 190 - 50100 Kakamega. awasike@mmust.ac.ke ‡ School of Mathematics, University of Nairobi, P.O. Box 30197 - 00100, Nairobi. pokhariyal@uonbi.ac.ke 1