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Oscillations and global attractivity in models of hematopoiesis. (English) Zbl 0694.34057

Summary: Let P(t) denote the density of mature cells in blood circulation. M. C. Mackey and L. Glass [Science 197, 287-289 (1977)] have proposed the following equations: \[ \dot P(t)=\frac{\beta_ 0\theta^ n}{\theta^ n+[P(t-\tau)]^ n}-\gamma P(t) \] and \[ \dot P(t)=\frac{\beta_ 0\theta^ nP(t-\tau)}{\theta^ n+[P(t-\tau)]^ n}- \gamma P(t) \] 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.

MSC:

34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
92D25 Population dynamics (general)
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
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