# 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)
Permanence, oscillation and attractivity of the discrete hematopoiesis model with variable coefficients. (English) Zbl 1125.39002
The authors study a periodic discrete Mackey-Glass equation $$p_{n+1}-p_n= -\delta_np_n+\frac{\beta_n}{1+p_{n-\omega}^m},$$ where $\delta_n\in(0,1)$, $\beta_n>0$ are $\omega$-periodic sequences and $m>1$. For positive initial values it is shown that every solution is positive, permanent and entering a bounded set. Moreover, every nonoscillatory solution tends to a $\omega$-periodic positive solution $\bar p_n$, and conditions for $\bar p_n$ to be the global attractor are given. Finally, some sufficient condition for the oscillation of every positive solution about $\bar p_n$ are established.

##### MSC:
 39A14 Partial difference equations 92D25 Population dynamics (general) 39A12 Discrete version of topics in analysis 39A20 Generalized difference equations
Full Text:
##### References:
 [1] Chen, M. P.; Yu, J. S.: Oscillation of delay difference equations with variable coefficients. Proceeding of the first international conference on difference equations, 105-114 (1994) [2] El-Morshedy, H. A.; Liz, E.: Convergence to equlibria in discrete population models. J. difference equ. Appl. 11, 117-131 (2005) · Zbl 1070.39022 [3] Erbe, L.; Xia, H.; Yu, J. S.: Global stability of a linear nonautonomous delay difference equation. J. difference equ. Appl. 1, 151-161 (1995) · Zbl 0855.39007 [4] Erbe, L. H.; Zhang, B. G.: Oscillation of discrete analogue of delay equations. Differential integral equations 2, No. 3, 300-309 (1989) · Zbl 0723.39004 [5] Györi, I.; Ladas, G.: Oscillation theory of delay differential equations with applications. (1991) · Zbl 0780.34048 [6] Györi, I.; Pituk, M.: Asymptotic stability in a linear delay difference equation. Advances in difference equations, 295-299 (1997) · Zbl 0892.39007 [7] Ivanov, A. F.: On global stability in a nonlinear discrete model. Nonlinear anal. 23, 1383-1389 (1994) · Zbl 0842.39005 [8] Karakostas, G.; Philos, Ch.G.; Sficas, Y. G.: The dynamics of some discrete population models. Nonlinear anal. 17, 1064-1084 (1991) · Zbl 0760.92019 [9] Kocic, V. L.; Ladas, G.: Oscillation and global attractivity in a discrete model of Nicholson’s blowflies. Appl. anal. 38, 21-31 (1990) · Zbl 0715.39003 [10] Ladas, G.; Qian, C.; Vlahos, P. N.; Yan, J.: Stability of solutions of linear nonautonomous difference equations. Appl. anal. 41, 183-191 (1991) · Zbl 0701.39001 [11] Ladas, G.; Philos, C. H.; Sficas, Y.: Sharp condition for the oscillation of delay difference equations. J. appl. Math. simul. 2, 101-111 (1989) · Zbl 0685.39004 [12] Mackey, M. C.; Glass, L.: Oscillation and chaos in physiological control system. Science 197, 287-289 (1977) [13] Saker, S. H.: Oscillation and global attractivity of hematopoiesis model with delay time. Appl. math. Comput. 136, No. 2-3, 27-36 (2003) · Zbl 1026.34082 [14] Tang, X. H.; Yu, J. S.: A further result on the oscillation of delay difference equations. Comput. math. Appl. 38, 229-237 (1999) · Zbl 0976.39005 [15] Tang, X. H.; Yu, J. S.: Oscillation of delay difference equations. Hokkaido math. J. 29, 213-228 (2000) · Zbl 0958.39015 [16] Stavroulakis, I. P.: Oscillation of delay difference equations. Comput. math. Appl. 29, 83-88 (1995) · Zbl 0832.39002 [17] Zaghrout, A.; Ammar, A.; El-Sheikh, M. M. A.: Oscillation and global attractivity in delay equation of population dynamics. Appl. math. Comput. 77, 195-204 (1996) · Zbl 0848.92018 [18] Zhang, B. G.; Tian, C. J.: Nonexistence and existence of positive solutions for difference equations with unbounded delay. Comput. math. Appl. 36, No. 1, 1-8 (1998) · Zbl 0932.39007 [19] Zhou, Y.; Zhang, B. G.: Oscillation for difference equations with variable delay. Dynam. systems appl. 10, 133-144 (2001) · Zbl 1007.39004