### Abstract

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.

Language | English |
---|---|

Publication status | Published - 2008 |

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

### Other

Other | Fifth World Congress of Nonlinear Analysts |
---|---|

Period | 31/03/11 → … |

### Fingerprint

### Keywords

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

### Cite this

*A delay differential equation mathematical model for the control of the hormonal system of the hypothalamus, the pituitary and the testis in man*. Fifth World Congress of Nonlinear Analysts, .

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**A delay differential equation mathematical model for the control of the hormonal system of the hypothalamus, the pituitary and the testis in man.** / Greenhalgh, David; Khan, Qamar.

Research output: Contribution to conference › Speech

TY - CONF

T1 - A delay differential equation mathematical model for the control of the hormonal system of the hypothalamus, the pituitary and the testis in man

AU - Greenhalgh, David

AU - Khan, Qamar

PY - 2008

Y1 - 2008

N2 - In this paper we develop previously studied mathematical models ofthe regulation of testosterone by luteinizing hormone andluteinizing hormone release hormone in the human body. We propose adelay differential equation mathematical model which improves onearlier simpler models by taking into account observed experimentalfacts. We show that our model has four possible equilibria, but onlyone unique equilibrium where all three hormones are present. Weperform stability and Hopf bifurcation analyses on the equilibriumwhere all three hormones are present. With no time delay thisequilibrium is unstable, but as the time delay increases through aninfinite sequence of positive values Hopf bifurcation occursrepeatedly. This is of practical interest as biological evidenceshows that the levels of these hormones in the body oscillateperiodically. 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.

AB - In this paper we develop previously studied mathematical models ofthe regulation of testosterone by luteinizing hormone andluteinizing hormone release hormone in the human body. We propose adelay differential equation mathematical model which improves onearlier simpler models by taking into account observed experimentalfacts. We show that our model has four possible equilibria, but onlyone unique equilibrium where all three hormones are present. Weperform stability and Hopf bifurcation analyses on the equilibriumwhere all three hormones are present. With no time delay thisequilibrium is unstable, but as the time delay increases through aninfinite sequence of positive values Hopf bifurcation occursrepeatedly. This is of practical interest as biological evidenceshows that the levels of these hormones in the body oscillateperiodically. 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.

KW - human hormonal systems

KW - testosterone

KW - luteinizing hormone

KW - luteinizing hormone release hormone

KW - time delay

KW - differential equations

KW - equilibrium and stability analysis

KW - limit cycles

KW - Hopf bifurcation

UR - http://ifnaworld.org/

UR - http://dx.doi.org/10.1016/j.na.2009.01.031

M3 - Speech

ER -