## On Approach to Equilibrium in Nonlinear Compartmental Systems

##### Abstract

A closed compartmental system is a set of nonnegative interdependent functions,
[see pdf for notation]
such that their sum is constant. The functions can represent populations, masses or concentrations, depending on the particular application. It
is convenient to normalize so that
[see pdf for notation]
in which case the functions are proportions. It is assumed that the (nonnegative) flow rate from j to i has the form fjjxj. Thus, the rate of change,
[see pdf for notation]
The first term is the inflow to i from the other "compartments" and the second term is the outflow from i to the other compartments. Setting
[see pdf for notation]
we obtain the system in vector form,
[see pdf for notation]
In classical compartmental analysis [1]-[4], which deals mainly with tracer and drug studies, each xi represents the amount of tracer or drug in an organ or a compartment of the human body, hence the term "compartment". Moreover, in classical work, the fij are treated as constants, however, in more recent work [5]-[12], they are functions,
[see pdf for notation]
Let us consider a classical tracer study.