zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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:

P ˙(t)=β 0 θ n θ n +[P(t-τ)] n -γP(t)

and

P ˙(t)=β 0 θ n P(t-τ) θ n +[P(t-τ)] n -γP(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:
34K99Functional-differential equations
92D25Population dynamics (general)
34C15Nonlinear oscillations, coupled oscillators (ODE)
References:
[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. · 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).
[4]Kulenovic, M. R. S., Ladas, G., and Meimaridou, A. (1987). On oscillations of nonlinear delay equations.Q. Appl. Math. XLV 155–164.
[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.
[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