Oscillations and global attractivity in delay differential equations of population dynamics. (English) Zbl 0848.92018

Summary: The oscillatory and asymptotic behavior of all positive solutions of \[ x'(t) = \beta_0 \theta^n/(\theta^n + x^n (t - \tau)) - \gamma x(t) \] about the positive steady state \(x^*\) are studied, where \(x(t)\) denotes the density of mature cells in blood circulation, \(\tau\) is the time delay between the production of immature cells in the bone marrow, and \(\beta_0\), \(\theta^n\), \(\gamma\) are positive constants.


92D25 Population dynamics (general)
34K25 Asymptotic theory of functional-differential equations
34K11 Oscillation theory of functional-differential equations
92C30 Physiology (general)
Full Text: DOI


[1] Mackey, M. C.; Glass, L., Oscillation and chaos in physicological control system, Science, 197, 287-289 (1977) · Zbl 1383.92036
[2] Gopalsamy, K.; Kulenovic, M.; Ladas, G., Oscillations and global attractivity in respiratory dynamics, Dynamics and Stability of Systems, 4, 2, 131-139 (1989) · Zbl 0683.92009
[3] Arino, O.; Aexlrod, D.; Kimmel, M., Slow oscillations in a model of cell population dynamics, (Proc. of the 2nd International Conference (1991), Marcel Dekker, Inc: Marcel Dekker, Inc New York)
[4] Arino, O.; Kimmel, M., Aymptotic behaviour of nonlinear semigroup describing a model of selective cell growth regulation, J. Math. Biol., 29, 289-314 (1991) · Zbl 0728.92010
[5] Arinio, O.; Kimmel, M., Comparison of approaches to modeling of cell population dynamics, SIAM J. Appl. Math., 53, 5, 1480-1504 (1993) · Zbl 0796.92019
[6] Glass, L.; Mackey, M., Pathological conditions resulting from instabilities in physiological control systems, Ann. N.Y. Acad. Sci., 316, 214-235 (1979) · Zbl 0427.92004
[7] Mackey, M.; Heiden, U., the dynamics of recurrent inhibition, J. Math Biol., 19, 211-225 (1984) · Zbl 0538.92007
[8] Kuang, Y., Global attractivity and periodic solutions in delay differential equations related to models in physiology and population biology, Japan J. of Idustial & Appl. Math., 9, 2, 205-238 (1992) · Zbl 0758.34065
[9] Ladas, G.; Stravroulakis, I. P., On delay differential inequalities of first order, Funkcialaj Ekvacioj, 25, 105-113 (1982) · Zbl 0492.34060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.