Oscillations and global attractivity in models of hematopoiesis

Abstract

LetP(t) denote the density of mature cells in blood circulation. Mackey and Glass (1977) have proposed the following equations:

$$\dot P(t) = \frac{{\beta _0 \theta ^n }}{{\theta ^n + [P(t - \tau )]^n }} - \gamma P(t)$$

and

$$\dot P(t) = \frac{{\beta _0 \theta ^n P(t - \tau )}}{{\theta ^n + [P(t - \tau )]^n }} - \gamma P(t)$$

as models of hematopoiesis. We obtain sufficient and also necessary and sufficient conditions for all positive solutions to oscillate about their respective positive steady states. We also obtain sufficient conditions for the positive equilibrium to be a global attractor.