On the ultimate boundedness and permanence of solutions for a class of discrete-time switched models of population dynamics

A discrete-time Lotka–Volterra type system with switching of parameter values is studied. The system consists of a family of subsystems of nonlinear difference equations and a switching law constantly determining which subsystem is active. It focuses on conditions providing the uniform ultimate boundedness or uniform permanence for the considered system for any switching law. A general approach to the problem is based on the constructing of a common Lyapunov function for the family of subsystems corresponding to the switched system. In the present paper, a new construction of such Lyapunov function for considered equations is suggested. The sufficient conditions of the existence of a common Lyapunov function in the given form satisfying in the positive orthant all the assumptions of the Yoshizawa ultimate boundedness theorem are obtained. These conditions are formulated in terms of solvability of auxiliary systems of algebraic inequalities, and their fulfillment guarantees that the switched system is ultimately bounded or uniformly permanent with respect to switching law. The proposed approach permits to relax some known ultimate boundedness and permanence conditions and to extend them to wider classes of discrete-time models of population dynamics. An example is presented to demonstrate the effectiveness of the obtained results. Bibliogr. 25.