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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. {\it M. C. Mackey} and {\it L. Glass} [Science 197, 287-289 (1977)] have proposed the following equations: $$ \dot P(t)=\frac{\beta\sb 0\theta\sp n}{\theta\sp n+[P(t-\tau)]\sp n}-\gamma P(t) $$ and $$ \dot P(t)=\frac{\beta\sb 0\theta\sp nP(t-\tau)}{\theta\sp n+[P(t-\tau)]\sp n}- \gamma P(t) $$ 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.

34K99Functional-differential equations
92D25Population dynamics (general)
34C15Nonlinear oscillations, coupled oscillators (ODE)
Full Text: DOI
[1] Glass, L., and Mackey, M. C. (1979). Pathological conditions resulting from instabilities in physiological control systems.Ann. N.Y. Acad. Sci. 316, 214--235. · Zbl 0427.92004 · doi:10.1111/j.1749-6632.1979.tb29471.x
[2] Gopalsamy, K., Kulenovic, M. R. S., and Ladas, G. (1989a). Time lags in a ”food-limited” population modelAppl. Anal.31 (1988), 225--237. · Zbl 0639.34070 · doi:10.1080/00036818808839826
[3] Gopalsamy, K., Kulenovic, M. R. S., and Ladas, G. (1989b). Oscillations and global attractivity in respiratory dynamics.Dynamics and Stability of Systems (in press). · Zbl 0683.92009
[4] Kulenovic, M. R. S., Ladas, G., and Meimaridou, A. (1987). On oscillations of nonlinear delay equations.Q. Appl. Math. XLV 155--164. · Zbl 0627.34076
[5] Mackey, M. C. (1978a). Dynamic haematological disorders in stem cell origin. InCellular Mechanisms of Reproduction and Aging, J. Vassileva-Popova (ed.), Plenum Press, New York, pp. 373--409.
[6] Mackey, M. C. (1978b). A unified hypothesis for the origin of aplastic anemia and periodic haematopoiesis.Blood 51, 941--956.
[7] Mackey, M. C. (1979). Periodic auto-immune hemolytic anemia; An induced dynamical disease.Bull. Math. Biol. 41, 829--834. · Zbl 0414.92012
[8] Mackey, M. C. (1981). Some models in hemopoiesis: Predictions and problems. InBiomathematics and Cell Kinetics, M. Rotenberg (ed.), Elsevier/North-Holland, Amsterdam, pp. 23--38.
[9] Mackey, M. C., and an der Heiden, U. (1982). Dynamical diseases and bifurcations: Understanding functional disorders in physiological systems.Funk. Biol. Med. 156, 156--164.
[10] Mackey, M. C., and Glass, L. (1977). Oscillations and chaos in physiological control systems.Science 197, 287--289. · doi:10.1126/science.267326
[11] Mackey, M. C., and Milton, J. G. (1979). Dynamical diseases.Ann. N.Y. Acad. Sci. 316, 214--235. · doi:10.1111/j.1749-6632.1979.tb29471.x
[12] Mallet-Paret, J., and Nussbaum, R. D. (1986). Global continuation and asymptotic behavior for periodic solutions of a differential-delay equation.Ann. Mat. Pura Appl. 145, 33--128. · Zbl 0617.34071 · doi:10.1007/BF01790539