# 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 of a discrete $N$-species cooperation system with time delays and feedback controls. (English) Zbl 1113.93063
Summary: A discrete $N$-species cooperation system with time delays and feedback controls is proposed. By applying the comparison theorem of difference equation, sufficient conditions are obtained for the permanence of the system.

##### MSC:
 93C30 Control systems governed by other functional relations 93B52 Feedback control 93D99 Stability of control systems
Full Text:
##### References:
 [1] Fan, M.; Wong, Patrica J. Y.; Agarwal, Ravi P.: Periodicity and stability in periodic n-species Lotka -- Volterra competition system with feedback controls and deviating arguments. Acta math. Sinica 19, No. 4, 801-822 (2003) · Zbl 1047.34080 [2] Huo, H.; Li, W.: Positive periodic solutions of a class of delay differential system with feedback control. Appl. math. Comput. 148, No. 1, 35-46 (2004) · Zbl 1057.34093 [3] May, R. M.: Theoretical ecology, principles and applications. (1976) [4] Cui, J. A.; Chen, L. S.: Global asymptotic stability in a nonautonomous cooperative system. Syst. sci. Math. sci. 6, No. 1, 44-51 (1993) · Zbl 0788.34051 [5] Yang, P.; Xu, R.: Global asymptotic stability of periodic solution in n-species cooperative system with time delays. J. biomath. 13, No. 6, 841-846 (1998) [6] Zhang, X.; Wang, K.: Almost periodic solution for n-species cooperative system with time delay. J. northeast normal univ. 34, No. 3, 9-13 (2002) [7] Wei, F. Y.; Wang, K.: Asymptotically periodic solution of N-species cooperation system with time delay. Nonlinear anal.: real world appl. 7, No. 4, 591-596 (2006) · Zbl 1114.34340 [8] Bai, L.; Fan, M.; Wang, K.: Existence of positive solution for difference equation of the cooperative system. J. biomath. 19, No. 3, 271-279 (2004) [9] Weng, P. X.: Global attractivity in a periodic competition system with feedback controls. Acta appl. Math. 12, 11-21 (1996) · Zbl 0859.34061 [10] Weng, P. X.: Existence and global stability of positive periodic solution of periodic integro-differential systems with feedback controls. Comput. math. Appl. 40, 747-759 (2000) · Zbl 0962.45003 [11] Xiao, Y. N.; Tang, S. Y.; Chen, J. F.: Permanence and periodic solution in competition system with feedback controls. Math. comput. Model. 27, No. 6, 33-37 (1998) · Zbl 0896.92032 [12] Gopalsamy, K.; Weng, P. X.: Feedback regulation of logistic growth. Int. J. Math. sci. 16, No. 1, 177-192 (1993) · Zbl 0765.34058 [13] Yang, F.; Jiang, D. Q.: Existence and global attractivity of positive periodic solution of a logistic growth system with feedback control and deviating arguments. Ann. diff. Eqs. 17, No. 4, 337-384 (2001) · Zbl 1004.34030 [14] Li, X. Y.; Fan, M.; Wang, K.: Positive periodic solution of single species model with feedback regulation and infinite delay. Appl. math. J. chin. Univ. ser. A 17, No. 1, 13-21 (2002) · Zbl 1005.34039 [15] Zhang, H. Y.; Chen, F. D.: Permanence and positive periodic solution of n-species nonautonomous T-periodic lokta -- Volterra competition system with feedback controls. J. ningxia univ. (Natural science edition) 25, No. 2, 114-120 (2004) [16] Chen, F. D.: Global asymptotic stability in n-species non-autonomous Lotka -- Volterra competitive systems with infinite delays and feedback control. Appl. math. Comput. 170, No. 2, 1452-1468 (2005) · Zbl 1081.92038 [17] Yin, F. Q.; Li, Y. K.: Positive periodic solutions of a single species model with feedback regulation and distributed time delay. Appl. math. Comput. 153, 475-484 (2004) · Zbl 1087.34051 [18] Fan, G. H.; Li, Y. K.; Qin, M. C.: The existence of positive periodic solutions for periodic feedback control systems with delays. ZAMM Z. Angew. math. Mech. 84, No. 6, 425-430 (2004) · Zbl 1118.34328 [19] Chen, F. D.; Lin, F. X.; Chen, X. X.: Sufficient conditions for the existence of positive periodic solutions of a class of neutral delay models with feedback control. Appl. math. Comput. 158, No. 1, 45-68 (2004) · Zbl 1096.93017 [20] Chen, F. D.: Positive periodic solutions of neutral Lotka -- Volterra system with feedback control. Appl. math. Comput. 162, No. 3, 1279-1302 (2005) · Zbl 1125.93031 [21] Chen, F. D.: Positive periodic solutions of a class of non-autonomous single species population model with delays and feedback control. Acta math. Sinica 21, No. 6, 1319-1336 (2005) · Zbl 1110.34049 [22] Huang, Z. K.; Chen, F. D.: Almost periodic solution of two species model with feedback regulation and infinite delay. J. eng. Math. 20, No. 5, 33-40 (2004) · Zbl 1138.34344 [23] Xia, Y. H.; Cao, J. D.; Zhang, H. Y.; Chen, F. D.: Almost periodic solutions in n-species competitive system with feedback controls. J. math. Anal. appl. 294, 504-522 (2004) · Zbl 1053.34040 [24] Chen, X. X.; Chen, F. D.: Almost periodic solutions of a delay population equation with feedback control. Nonlinear anal.: real world appl. 7, No. 4, 559-571 (2006) · Zbl 1128.34045 [25] X.X. Chen, Almost periodic solutions of nonlinear delay population equation with feedback control, Nonlinear Anal.: Real World Appl., in press. [26] Chen, F. D.: The permanence and global attractivity of Lotka -- Volterra competition system with feedback controls. Nonlinear anal.: real world appl. 7, No. 1, 133-143 (2006) · Zbl 1103.34038 [27] Chen, F. D.: Permanence in nonautonomous multi-species predator -- prey system with feedback controls. Appl. math. Comput. 173, No. 2, 694-709 (2006) · Zbl 1087.92059 [28] Liu, Q. M.; Xu, R.: Persistence and global stability for a delayed nonautonomous single-species model with dispersal and feedback control. Diff. equat. Dyn. syst. 11, No. 3 -- 4, 353-367 (2003) · Zbl 1231.34131 [29] X.X. Chen, F.D. Chen, Stable periodic solution of a discrete periodic Lotka -- Volterra competition system with a feedback control, Appl. Math. Comput., in press, doi:10.1016/j.amc.2006.02.039. · Zbl 1106.39003 [30] F.D. Chen, Permanence and global attractivity of a discrete multispecies Lotka -- Volterra competition predator -- prey systems, Appl. Math. Comput., in press, doi:10.1016/j.amc.2006.03.026. · Zbl 1113.92061 [31] Chen, F. D.: On a nonlinear nonautonomous predator -- prey model with diffusion and distributed delay. J. comput. Appl. math. 180, No. 1, 33-49 (2005) · Zbl 1061.92058 [32] Chen, F. D.: Global stability of a single species model with feedback control and distributed time delay. Appl. math. Comput. 178, No. 2, 474-479 (2006) · Zbl 1101.92035 [33] Wang, L.; Wang, M. Q.: Ordinary difference equation. (1991) · Zbl 0734.34024 [34] Zhou, Z.; Zou, X.: Stable periodic solutions in a discrete periodic logistic equation. Appl. math. Lett. 16, No. 2, 165-171 (2003) · Zbl 1049.39017