Global stability and estimates for solutions in a model of population dynamics with delay

We consider a model of the isolated population dynamics described by a delay differential equation. We study the case when the model has no more than two equilibrium points corresponding to the complete extinction of the population and to the constant positive population size. We indicate conditions for the right side of the equation, under which solutions are stabilized to equilibrium points for arbitrary non-negative initial data. We obtain estimates for the stabilization rate depending on the coefficients of the equation, the nonlinear function from the right side of the equation, and the function at the initial time interval. The established estimates characterize the rate of population extinction and the rate of stabilization of the population to a constant value. The results are obtained using Lyapunov–Krasovskii functionals.