On the absence of oscillations in compartmental dynamical systems

Nonnegative and compartmental dynamical system models are widespread in biological, physiological, and ecological sciences and play a key role in understanding these processes. In the specific field of pharmacokinetics involving the study of drug concentrations (in various tissue groups) as a function of time and dose, nonnegative and compartmental models are vital in understanding system wide effects of pharmacological agents. Since drug concentrations are often assumed to monotonically decline after discontinuation of drug administration, standard pharmacokinetic modeling may ignore the possibility of system oscillation. However, nonnegative and compartmental system models may exhibit non-monotonic solutions resulting in differences between model predictions and experimental data. In this paper, we present necessary and sufficient conditions for identifying nonnegative and compartmental systems that only admit nonoscillatory and monotonic solutions.