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We study the two state model which describes the balance equation for carbondioxide and oxygen. These are nonlinear parameter dependent and because of thetransport delay in the respiratory control system, they are modeled with delaydifferential equation. So, the dynamics of a two state one delay model areinvestigated. By choosing the delay as a parameter, the stability and Hopfbifurcation conditions are obtained. We notice that as the delay passes throughits critical value, the positive equilibrium loses its stability and Hopfbifurcation occurs. The stable region of the system with delay against theother parameters and bifurcation diagrams are also plotted. The threedimensional stability chart of the two state model is constructed. We find thatthe delay parameter has effect on the stability but not on the equilibriumstate. The explicit derivation of the direction of Hopf bifurcation and thestability of the bifurcation periodic solutions are determined with the help ofnormal form theory and center manifold theorem to delay differential equations.Finally, some numerical example and simulations are carried out to confirm theanalytical findings. The numerical simulations verify the theoretical results.

Stability and Hopf bifurcation analysis of a two state delay differential equation modeling the human respiratory system