2004 Conference on Diff. Eqns. and Appl. in Math. Biology, Nanaimo, 
BC, Canada.
Electronic Journal of Differential Equations,
Conference 12, 2005, pp. 65-78. 

Title: Numerical stability analysis in respiratory
       control system models

Authors: Laszlo E. Kollar (Univ. of Quebec at Chicoutimi, Canada)
         Janos Turi (Univ. of Texas at Dallas, TX, USA)
 
Abstract: 
 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.