Analysis of the Dynamics of Solutions for Hybrid DifferenceLotka–Volterra Systems

A difference system of the Lotka–Volterra type is considered. It is assumed that this system can operate both in some program and perturbed modes. The restrictions on the time of the system’s stay in these modes, providing the desired dynamical behavior, are investigated. In particular, the conditions of the ultimate boundedness of solutions and the permanence of the system are obtained. The direct Lyapunov method is used, and different Lyapunov functions are constructed in different parts of the state space. The sizes of the domain of permissible initial values of solutions and the domain of the ultimate bound of solutions corresponding to the required dynamics of the system are estimated. Constraints are set on the size of the digitization step of the system.