Asymptotic behavior of nonlinear compartmental systems: Nonoscillation and stability

This paper discusses properties related to the stability of a class of nonlinear compartmental systems. Specifically, mathematical conditions which guarantee the same qualitative behavior inherent in linear compartmental systems are considered. We first consider the nonoscillatory property of solutions and show that the system has no periodic oscillation under a mild condition. The result is then used to derive a necessary and sufficient condition for every solution to converge to a set of equlibrium points which may depend both on the input and the initial state. A sufficient condition is also given for an equilibrium state to depend only on the input. The asymptotic behavior of the free systems is also considered, and a sufficient condition is given for the origin to be globally asymptotically stable. Furthermore, for a closed compartmental system it is shown that for each given initial state, unique equilibrium state, if it exists, depends only on the total sum of the components of the initial state. Finally a sufficient condition is given for solutions to converge to the unique point.