Analysis of the dynamical behavior of solutions for a class of hybrid generalized Lotka–Volterra models

Abstract

In the paper, the generalized Lotka–Volterra systems with switching are investigated. It is assumed that functions included in the right parts of the considered systems satisfy nonlinear constraints of the power type. Conditions are found for the switching law that provide the desired features of the dynamical behavior of solutions of such systems. In particular, the problems to guarantee the asymptotic stability of a given equilibrium position with the required region of attraction, the ultimate boundedness of solutions, the global asymptotic stability are studied. To solve these problems, the method of splitting the phase space of the studied systems into several parts is used. In each of these parts, some partial estimates for the chosen Lyapunov function are constructed and conditions are established to ensure the transition of solutions from one part to another. Moreover, different Lyapunov functions can be used in different parts of the phase space.