There exist an enormous number of
physiological systems that involve or rely on the periodic behaviour of certain subcomponents of the
system. For example, many
homeostatic processes rely on
negative feedback to control the concentration of substances in the blood;
breathing, for instance, is promoted when the brain detects high CO2 levels in the blood. One way to model such systems mathematically is with the following simple
ordinary differential equation: : y'(t) = k - c y(t) where k is the rate at which a "substance" is produced, and c controls how the current level of the substance
discourages the continuation of its production. The solutions of this equation can be found via an
integrating factor, and have the form: :y(t) = \frac{k}{c} + f(y_0) e^{- c t} where y_0 is any initial condition for the
initial value problem. However, the above model assumes that variations in the substance concentration is detected immediately, which often not the case in physiological systems. In order to ease this problem, proposed changing the production rate to a function k(y(t - \tau)) of the concentration at an earlier point t - \tau in time, in hope that this would better reflect the fact that there is a significant delay before the
bone marrow produces and releases mature cells in the blood, after detecting low cell concentration in the blood. By taking the production rate k as being: : \frac{\beta_0 \theta^n}{\theta^n + P(t - \tau)^n} ~~ \text{ or } ~~ \frac{\beta_0 \theta^n P(t - \tau)}{\theta^n + P(t - \tau)^n} we obtain Equations () and (), respectively. The values used by were \gamma = 0.1, \beta_0 = 0.2 and n = 10, with initial condition P(0) = 0.1. The value of \theta is not relevant for the purpose of analyzing the dynamics of Equation (), since the
change of variable P(t) = \theta \cdot Q(t) reduces the equation to: : Q'(t) = \frac{\beta_0 Q(t - \tau)}{1 + Q(t - \tau)^n} - \gamma Q(t). This is why, in this context, plots often place Q(t) = P(t) / \theta in the y-axis. ==Dynamical behaviour==