Asymptotic properties of solutions in a model of interaction of populations with several delays

We consider a model describing the interaction of $n$ species of microorganisms. The model is presented as a system of nonlinear differential equations with several constant delays. The components of solutions to the system are responsible for the biomass of the $i$-th species and for the concentration of the nutrient. The delay parameters denote the time required for the conversion of the nutrient to viable biomass for the $i$-th species. In the paper, we study asymptotic properties of the solutions to the considered system. Under certain conditions on the system parameters, we obtain estimates for all components of the solutions. The established estimates characterize the rate of decrease in the biomass of microorganisms and the rate of stabilization of the nutrient concentration to the initial value of the concentration. The estimates are constructive and all values characterizing the stabilization rate are indicated explicitly. The modified Lyapunov–Krasovskii functionals were used to obtain the estimates.