On the mathematical foundations of compartmental analysis in biology, medicine, and ecology

Abstract

The basic equations of compartmental analysis are a system S of differential equations which govern the exchanges of material among various compartuments and an environment. The equations are ordinarily nonlinear. In this paper we show that if certain natural conditions are satisfied S possesses an equilibrium point, and that if additional reasonable conditions are met the equilibrium point is unique and globally stable. These results have natural interpretations and, in particular, provide an analytical basis for the use of the familiar linear tracer-analysis equations. We also consider the case in which S takes into account cyclic variations with a given period <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\tau</tex> (this case often arises in ecological studies) and show that under certain reasonable conditions S has a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\tau</tex> -periodic solution that is approached by every solution of S.