On the global asymptotic stability and ultimate boundedness for a class of nonlinear switched systems

In this paper, the problem of global stabilization for one switched model of population dynamics is investigated. It is assumed that population dynamics is determined by the Lotka–Volterra-type differential equations systems. With the aid of multiple Lyapunov functions method, sufficient conditions on switching law are derived under which the given equilibrium position of the system is globally asymptotically stable in the positive orthant. For the switched system without the fixed equilibrium position, conditions of ultimate boundedness and permanence are obtained. The main attention in the paper is focused on the case, where the switching signal depends on the time. Some restrictions on the lengths of intervals between consecutive switching times are imposed for the providing the required properties of solutions of the system considered. Additionally, a switched Lotka–Volterra-type system with conservative subsystems of the second orders is investigated. In this case, some state-dependent switching signals are considered.