Estimates for solutions in a predator-prey model with delay

In the paper we consider a system of delay differential equations describing the interaction between predator and prey populations living on the same territory. The system consists of three equations, herewith the components of solutions characterize the number of individuals of prey population, the number of adult predators, and the number of juvenile predators. It is assumed that only adult predators can attack the individuals of prey population and reproduce. The delay parameter is assumed to be constant and denotes the time that the predators need to become adult. For the system we consider the initial value problem, for which it is discussed the existence, uniqueness, nonnegativity, and boundedness of solutions. It is also discussed the stability of stationary solutions (equilibrium points) corresponding to complete extinction of populations, extinction of only predator populations and coexistence of predator and prey populations. The main attention in the paper is paid to obtaining estimates of solutions characterizing the rate of convergence to the equilibrium point corresponding to the coexistence of populations and the establishment of estimates for the attraction set, i.e. the admissible conditions for the initial data under which the convergence takes place. When obtaining the results, we use the method of Lyapunov–Krasovskii functionals, which is an analogue of the method of Lyapunov functions for ordinary differential equations. Herewith in the paper it is significantly used the modified Lyapunov–Krasovskii functional proposed by G.V. Demidenko and I.I. Matveeva. It is important to note that this functional allows to obtain estimates of solutions to delay systems, which are analogues of the Krein's estimate for ordinary differential equations, and the construction of such functional is reduced to solving well-conditioned problems.