Stability and two-sided estimates of solutions of a linear nonautonomous differential equation with variable delay

The stability conditions of the trivial equilibrium position for a linear nonautonomous differential equation with variable delay arising in the simulation of population dynamics are investigated. The equation under study is supplemented by an auxiliary equation reflecting the dynamics of the number of individuals in the population at an intermediate stage of development. To analyze the stability of the trivial equilibrium position of the basic equation, the Lyapunov-Krasovsky functional and the method of integral inequalities are used. Upper and lower exponential estimates of solutions to the Cauchy problem for the main and auxiliary equations of the system under study are constructed.