Abstract
This paper discusses the asymptotic behaviors of a class of nonlinear compartmental systems. The system treated here is a nonlinear reciprocal compartmental system for which the flow of a certain material between compartments depends only on the difference of the amounts of the material in the compartments. The specific problem considered here is the derivation of mathematical conditions which guarantee the following properties of systems: the nonnegativeness, the asymptotic stability, and the nonoscillatory property of solutions.
The compartmental system under consideration is a special class of nonlinear reciprocal RC networks. We introduce a Lyapunov function which corresponds, from the circuit theoretical point of view, to an electric energy stored in nonlinear capacitances and discuss the global asymptotic stability. Necessary and sufficient conditions for the global asymptotic stability are established for open systems. For closed systems, necessary and sufficient conditions with which the solutions approach asymptotically the unique steady state are given.
In this paper we also consider a class of nonreciprocal compartmental systems, which describe the dynamics of pharmacokinetics, and give a mild condition which guarantees the nonoscillatory property of solutions. Under this condition we give several necessary and sufficient conditions for global asymptotic stability, boundedness of solutions, and uniqueness of the steady state of solutions.
