A delay differential equation mathematical model for the control of the hormonal system of the hypothalamus, the pituitary and the testis in man

In this paper we develop previously studied mathematical models of
the regulation of testosterone by luteinizing hormone and
luteinizing hormone release hormone in the human body. We propose a
delay differential equation mathematical model which improves on
earlier simpler models by taking into account observed experimental
facts. We show that our model has four possible equilibria, but only
one unique equilibrium where all three hormones are present. We
perform stability and Hopf bifurcation analyses on the equilibrium
where all three hormones are present. With no time delay this
equilibrium is unstable, but as the time delay increases through an
infinite sequence of positive values Hopf bifurcation occurs
repeatedly. This is of practical interest as biological evidence
shows that the levels of these hormones in the body oscillate
periodically. We next discuss stability of the other equilibria heuristically using analytical methods. Then we describe simulations with realistic parameter values and show that our model can mimic the regular fluctuations of the three hormones in the body and explore numerically some of our heuristic conjectures. A brief discussion concludes the paper.