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Global asymptotic stability in \(n\)-species non-autonomous Lotka-Volterra competitive systems with infinite delays and feedback control. (English) Zbl 1081.92038
Summary: A non-autonomous Lotka-Volterra competition system with infinite delays and feedback control and without dominating instantaneous negative feedback is investigated. By means of a suitable Lyapunov functional, sufficient conditions are derived for the global asymptotic stability of the system. Some new results are obtained.

MSC:
92D40 Ecology
34K20 Stability theory of functional-differential equations
93D15 Stabilization of systems by feedback
34K25 Asymptotic theory of functional-differential equations
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[1] Gopalsamy, K., Global asymptotic stability in a periodic integrodifferential system, Tohoku math. J., 37, 323-332, (1985) · Zbl 0587.45013
[2] Murakami, S., Almost periodic solution of a system of integrodifferential equations, Tohoku math. J., 39, 71-79, (1987) · Zbl 0598.45017
[3] Hamaya, Y., Periodic solutions of nonlinear integrodifferential equations, Tohoku math. J., 41, 105-116, (1989) · Zbl 0689.45017
[4] Hamaya, Y.; Yoshizawa, T., Almost periodic solutions in an integrodifferential equation, Proc. roy. soc. edinb. ser. A, 114, 151-159, (1990) · Zbl 0699.45007
[5] Bereketoglu, H.; Gyori, I., Global asymptotic stability in a nonautonomous loka-Volterra type system with infinite delay, J. math. anal. appl., 210, 279-291, (1997) · Zbl 0880.34072
[6] Teng, Z.D.; Yu, Y.H., Some new results of nonautonomous Lotka-Volterra competition system with delays, J. math. anal. appl., 241, 254-275, (2000) · Zbl 0947.34066
[7] Xu, R.; Chaplain, M.A.J.; Chen, L.S., Global asymptotic stability in n-species nonautonomous Lotka-Volterra competitive systems with infinite delays, Appl. math. computat., 130, 295-309, (2002) · Zbl 1029.34060
[8] He, X.Z., Almost periodic solutions of a competition system with dominated infinite delays, Tohoku math. J., 50, 71-89, (1998) · Zbl 0906.34048
[9] Weng, P., Existence and global stability of positive periodic solution of periodic integro-differential systems with feedback controls, Comput. math. appl., 40, 6-7, 747-759, (2000) · Zbl 0962.45003
[10] Fan, M.; Wong, P.J.Y.; Ravi, P., Agarwal, periodicity and stability in periodic n-species Lotka-Volterra competition system with feedback controls and deviating arguments, Acta math. sin., 19, 4, 801-822, (2003) · Zbl 1047.34080
[11] Weng, P.; Jiang, D., Existence and global stability of positive periodic solution of n-species periodic Lotka-Volterra competition system with feedback control and deviating arguments, Far east J. math. sci. (FJMS), 7, 1, 45-65, (2002) · Zbl 1043.34075
[12] Yang, F.; Jiang, D., Existence and global attractivity of positive periodic solution of a logistic growth system with feedback control and deviating arguments, Ann. different. equat., 17, 4, 377-384, (2001) · Zbl 1004.34030
[13] Li, X.; 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, 1, 13-21, (2002), (in Chinese) · Zbl 1005.34039
[14] F.D. Chen et al., Positive periodic solutions of a class of non-autonomous single species population model with delays and feedback control, Acta Math. Sin. (in press).
[15] Liu, Z.J., Persistence and periodic solution in two species competitive system with feedback controls, J. biomath., 17, 2, 251-255, (2002)
[16] Xiao, Y.N.; Tang, S.Y.; Chen, J.F., Permanence and periodic solution in competitive system with feedback controls, Math. comput. model., 27, 6, 33-37, (1998) · Zbl 0896.92032
[17] Chen, F.D., Sufficient conditions for the existence of positive periodic solutions of a class of neutral delay models with feedback control, Appl. math. computat., 158, 1, 45-68, (2004) · Zbl 1096.93017
[18] Chen, F.D., Positive periodic solutions of neutral Lotka-Volterra system with feedback control, Appl. math. computat., 162, 3, 1279-1302, (2005) · Zbl 1125.93031
[19] Chen, F.D.; Lin, S.J., Periodicity in a logistic type system with several delays, Comput. math. applic., 48, 1-2, 35-44, (2004) · Zbl 1061.34050
[20] Chen, F.D., Periodicity in a food-limited population model with toxicants and state dependent delays, J. math. anal. appl., 288, 1, 132-142, (2003)
[21] Huang, Z.K.; Chen, F.D., Almost periodic solution of two species model with feedback regulation and infinite delay, Chin. J. eng. math., 21, 1, 33-40, (2004) · Zbl 1138.34344
[22] Chen, F.D., Periodic solution of nonlinear integral-differential equations with infinite delay, Acta math. applic. sin., 26, 1, 141-148, (2003), (in Chinese) · Zbl 1021.45009
[23] Chen, F.D.; Shi, J.L., On the periodic solution of higher dimensional nonautonomous systems, Acta math. sin., 43, 5, 887-894, (2000), (in Chinese) · Zbl 1024.34033
[24] Chen, F.D., On the existence and uniqueness of periodic solutions of a kind of integro-differential equations, Acta math. sin., 47, 5, 973-984, (2004), (in Chinese) · Zbl 1124.34043
[25] Li, Y.K., Positive periodic solutions of a class of functional differential systems with feedback controls, Nonlinear anal., 57, 5-6, 655-666, (2004) · Zbl 1064.34049
[26] Fan, G.H., The existence of positive periodic solutions for periodic feedback control systems with delays, ZAMM Z. angew. math. mech., 84, 6, 425-430, (2004) · Zbl 1118.34328
[27] Huo, H.F.; Li, W.T., Positive periodic solutions of a class of delay differential system with feedback control, Appl. math. comput., 148, 1, 35-46, (2004) · Zbl 1057.34093
[28] Liu, P.; Li, Y.K., Multiple positive periodic solutions of nonlinear functional differential system with feedback control, J. math. anal. appl., 288, 2, 819-832, (2003) · Zbl 1045.34045
[29] 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, 2, 475-484, (2004) · Zbl 1087.34051
[30] Gopalsamy, K.; Weng, P.X., Global attractivity in a competition system with feedback controls, Comput. math. appl., 45, 4-5, 665-676, (2003) · Zbl 1059.93111
[31] Seifert, G., Almost periodic solutions for delay logistic equations with almost periodic time dependence, Different. integr. equat., 9, 2, 335-342, (1996) · Zbl 0838.34083
[32] Barbălat, I., Systems d’equations differential d’oscillations nonlinearies, Rev. roum. math. pure appl., 4, 2, 267-270, (1959) · Zbl 0090.06601
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