Study On The Competitive Chemostat Model With Delays

The chemostat is a device frequently used to perform laboratory experiments for understanding the interactions among species by controlling the input and output. This device plays a very important role in mathematical biology as the parameter can be obtained from relevant experiments, and mathematical results obtained can be easily to conduct. Chemostat experiments also can be performed to verify theoretical predictions arised from mathematical biology.In order to describe the actual ecosystem, n species with different removal rates and general functional response are incorporated into the Chemostat at which n species compete for one finite resource. It is well known that time delay is intrinsic needed for the process of conversion nutrient to new biomass. In present paper, we incorporate two kinds of delays to describe the time lag in the process of conversion of nutrient to new biomass: discrete type and finite-distributed type. By performing mathematical analysis for two kinds of competitive chemostat model, we obtain the sufficient conditions for satisfying competitive exclusion principle. This study is divided into two parts:In the first part, model with discrete delay are considered. By constructing the suitable Lypunov functional, we prove that either all species die out or only the species with the smallest break-even value survives finally, that is, the competitive exclusion principle holds. Furthermore, we incorporate finite-distributed delay to describe the biomass developed at time t is not only related to some fixed time before time t, but also over the whole past time. We show the competitive exclusion principle holds by constructing suitable Lypunov functional.