Estimates of solutions to differential equations with distributed delay describing the competition of several types of microorganisms

We consider a model of competition of n species in a chemostat. This model is a system of $n+1$ differential equations with infinite distributed delay. One equation is responsible for the change in nutrient concentration, and the other n are responsible for the change in the number of species. The transformation of a nutrient into viable cells does not occur instantly, and requires some time, which is taken into account by the presence of a delay. Under the condition when the concentration of the introduced nutrient is below a certain level, we have constructed Lyapunov–Krasovskii functionals, with the help of which we obtain estimates for all components of solutions. The estimates characterize the extinction rates of all species in the chemostat and the stabilization rate of the nutrient concentration to a constant value.