TY - JOUR

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 J.A.

N1 - Funding Information:
This work was made possible by the award of two research exchange travel grants from Sultan Qaboos University, Oman, and we would like to express our gratitude for this financial support. This work was completed with the support of two research grants from Sultan Qaboos University, Oman.

PY - 2009/12/15

Y1 - 2009/12/15

N2 - 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.

AB - 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.

KW - Differential equations

KW - Equilibrium and stability analysis

KW - Hopf bifurcation

KW - Human hormonal systems

KW - Limit cycles

KW - Luteinizing hormone

KW - Luteinizing hormone release hormone

KW - Testosterone

KW - Time delay

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U2 - 10.1016/j.na.2009.01.031

DO - 10.1016/j.na.2009.01.031

M3 - Article

AN - SCOPUS:72449156725

SN - 0362-546X

VL - 71

SP - e925-e935

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

IS - 12

ER -