Periodicities of open linear systems with positive steady states

Abstract

Systems of two, three, and four linear non-homogeneous differential equations are examined with a view toward determining whether they can possibly serve as mathematical models to describe periodicities in the concentrations of substances which enhance or inhibit each other's rate of production (or dissipation). The nature of the model demands that the solutions of the differential equations be non-negative at all times, i.e., that all the steady states be positive. Conditions for periodicity and for positive steady states are derived, and it is shown that these conditions are not always compatible with each other. In particular it is shown that certain three- and four-hormone models proposed to account for the periodicities observed in the menstrual cycle cannot satisfy the above conditions for any values of the parameters and hence are inadequate.