## Abstract

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.

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.

Original language | English |
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Publication status | Published - 2008 |

Event | Fifth World Congress of Nonlinear Analysts - Duration: 31 Mar 2011 → … |

### Other

Other | Fifth World Congress of Nonlinear Analysts |
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Period | 31/03/11 → … |

## Keywords

- human hormonal systems
- testosterone
- luteinizing hormone
- luteinizing hormone release hormone
- time delay
- differential equations
- equilibrium and stability analysis
- limit cycles
- Hopf bifurcation