On monotonicity of solutions of discrete-time nonnegative and compartmental dynamical systems
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Nonnegative and compartmental dynamical system models are widespread in biological, physiological, and pharmacological sciences. Since the state variables of these systems are typically masses or concentrations of a physical process, it is of interest to determine necessary and sufficient conditions under which the system states possess monotonic solutions. In this paper, we present necessary and sufficient conditions for identifying discrete-time nonnegative and compartmental dynamical systems that only admit monotonic solutions. 1. Introduction Nonnegative dynamical systems are of paramount importance in analyzing dynamical systems involving dynamic states whose values are nonnegative [2, 9, 16, 17]. An important subclass of nonnegative systems is compartmental systems [1, 4, 6, 8, 11, 12, 13, 14, 15, 18]. These systems involve dynamical models derived from mass and energy balance considerations of macroscopic subsystems or compartments which exchange material via intercompartmental flow laws. The range of applications of nonnegative and compartmental systems is widespread in models of biological and physiological processes such as metabolic pathways, tracer kinetics, pharmacokinetics, pharmacodynamics, and epidemic dynamics. Since the state variables of nonnegative and compartmental dynamical systems typically represent masses and concentrations of a physical process, it is of interest to determine necessary and sufficient conditions under which the system states possess monotonic solutions. This is especially relevant in the specific field of pharmacokinetics [7, 19] wherein drug concentrations should monotonically decline after discontinuation of drug administration. In a recent paper [5], necessary and sufficient conditions were developed for identifying continuous-time nonnegative and compartmental dynamical systems that only admit nonoscillatory and monotonic solutions. In this paper, we present analogous results for discrete-time nonnegative and compartmental systems. Copyright © 2004 Hindawi Publishing Corporation Advances in Difference Equations 2004:3 (2004) 261–271 2000 Mathematics Subject Classification: 39A11, 93C55 URL: http://dx.doi.org/10.1155/S1687183904310095
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Monotonicity of solutions
The contents of the paper are as follows. In Section 2, we establish definitions and notation, and review some basic results on nonnegative dynamical systems. In Section 3, we introduce the notion of monotonicity of solutions of nonnegative dynamical systems. Furthermore, we provide necessary and sufficient conditions for monotonicity for linear nonnegative dynamical systems. In Section 4, we generalize the results of Section 3 to nonlinear nonnegative dynamical systems. In addition, we provide sufficient conditions that guarantee the absence of limit cycles in nonlinear compartmental systems. In Section 5, we use the results of Section 3 to characterize the class of all linear, threedimensional compartmental systems that exhibit monotonic solutions. Finally, we draw conclusions in Section 6. 2. Notation and mathematica
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