2004 Conference on Diff. Eqns. and Appl. in Math. Biology, Nanaimo, BC, Canada.
Electron. J. Diff. Eqns., Conference 12, 2005, pp. 65-78.

Numerical stability analysis in respiratory control system models

Laszlo E. Kollar, Janos Turi

Stability of the unique equilibrium in two mathematical models (based on chemical balance dynamics) of human respiration is examined using numerical methods. Due to the transport delays in the respiratory control system these models are governed by delay differential equations. First, a simplified two-state model with one delay is considered, then a five-state model with four delays (where the application of numerical methods is essential) is investigated. In particular, software is developed to perform linearized stability analysis and simulations of the model equations. Furthermore, the Matlab package DDE-BIFTOOL v. 2.00 is employed to carry out numerical bifurcation analysis. Our main goal is to study the effects of transport delays on the stability of the model equations. Critical values of the transport delays (i.e., where Hopf bifurcations occur) are determined, and stable periodic solutions are found as the delays pass their critical values. The numerical findings are in good agreement with analytic results obtained earlier for the two-state model.

Published April 20, 2005.
Math Subject Classifications: 92C30, 93C23.
Key Words: Delay differential equations; human respiratory system; transport delay; numerical analysis.

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Laszlo E. Kollar
Department of Applied Sciences
The University of Quebec at Chicoutimi
555 Boul. de l'Universite, Chicoutimi, Quebec G7H 2B1, Canada
email: laszlo_kollar@uqac.ca
  Janos Turi
Department of Mathematical Sciences
The University of Texas at Dallas
P. O. Box 830688, MS EC 35, Richardson, Texas 75083-0688, USA
email: turi@utdallas.edu

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