Analysis of the Dynamics of Solutions for Hybrid Difference Lotka–Volterra Systems

A difference system of the Lotka–Volterra type is considered. It is assumed that thissystem can operate both in some program and perturbed modes. The restrictions on the time ofthe system’s stay in these modes, providing the desired dynamical behavior, are investigated. Inparticular, the conditions of the ultimate boundedness of solutions and the permanence of thesystem are obtained. The direct Lyapunov method is used, and different Lyapunov functions areconstructed in different parts of the state space. The sizes of the domain of permissible initialvalues of solutions and the domain of the ultimate bound of solutions corresponding to therequired dynamics of the system are estimated. Constraints are set on the size of the digitizationstep of the system.