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On a mutualism model with feedback controls. (English) Zbl 1194.93069
Summary: We propose and study a mutualism model with feedback controls. By applying a new differential inequality, we show that the conditions which ensure the permanence of the system are the same as that of the model without feedback controls, which means that the feedback control variables have no influence on the persistent property of the system. Our results not only improve but also complement some existing ones.

MSC:
93B52 Feedback control
93C15 Control/observation systems governed by ordinary differential equations
34H05 Control problems involving ordinary differential equations
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[1] Fan, M.; Wang, K.; Patricia, J.Y.W.; Agarwal, R.P., Periodicity and stability in periodic n-species lotka – volterra competition system with feedback controls and deviating arguments, Acta math. sinica, 19, 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, 35-46, (2004) · Zbl 1057.34093
[3] Hale, J., Theory of functional differential equations, (1977), Springer-Verlag Heidelberg, pp. 30-59
[4] Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Publishers, pp. 319-327 · Zbl 0752.34039
[5] Li, Y.K., On a periodic mutualism model, Anziam j., 42, 569-580, (2001) · Zbl 0996.34059
[6] Huang, A.M., Permanence of a delayed n-species mutualism system, J. fuzhou univ. nat. sci. ed., 45, 499-501, (2007) · Zbl 1150.92342
[7] Cui, J.A.; Chen, L.S., Global asymptotic stability in a nonautonomous cooperative system, Systems sci. math. sci., 6, 44-51, (1993) · Zbl 0788.34051
[8] Cui, J.A., Global asymptotic stability in n-species cooperative system with time delays, Systems sci. math. sci., 7, 45-48, (1994) · Zbl 0807.34088
[9] Yang, P.; Xu, R., Global asymptotic stability of periodic solution in n-species cooperative system with time delays, J. biomath., 13, 841-846, (1998)
[10] Zhang, X.; Wang, K., Almost periodic solution for n-species cooperative system with time delay, J. northeast univ. nat. sci., 34, 9-13, (2002)
[11] Wei, F.Y.; Wang, K., Asymptotically periodic solution of N-species cooperation system with time delay, Nonlinear anal. real world appl., 7, 591-596, (2006) · Zbl 1114.34340
[12] Wei, F.Y.; Wang, K., Global stability and asymptotically periodic solution for nonautonomous cooperative lotka – volterra diffusion system, Appl. math. comput., 182, 161-165, (2006) · Zbl 1113.92062
[13] Chen, F.D.; Liao, X.Y.; Huang, Z.K., The dynamic behavior of N-species cooperation system with continuous time delays and feedback controls, Appl. math. comput., 181, 803-815, (2006) · Zbl 1102.93021
[14] Chen, F.D., Permanence of a discrete N-species cooperation system with time delays and feedback controls, Appl. math. comput., 186, 23-29, (2007) · Zbl 1113.93063
[15] 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
[16] Xiao, Y.N.; Tang, S.Y.; Chen, J.F., Permanence and periodic solution in competition system with feedback controls, Math. comput. model., 27, 33-37, (1998) · Zbl 0896.92032
[17] Gopalsamy, K.; Weng, P.X., Feedback regulation of logistic growth, Int. J. math. sci., 16, 177-192, (1993) · Zbl 0765.34058
[18] 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. differ. equations, 17, 337-384, (2001) · Zbl 1004.34030
[19] Li, X.Y.; Fan, M.; Wang, K., Positive periodic solution of single species model with feedback regulation and infinite delay, Appl. math. J. Chinese univ. ser. A, 17, 13-21, (2002) · Zbl 1005.34039
[20] 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, 1452-1468, (2005) · Zbl 1081.92038
[21] 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
[22] 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, 425-430, (2004) · Zbl 1118.34328
[23] 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, 45-68, (2004) · Zbl 1096.93017
[24] Chen, F.D., Positive periodic solutions of neutral lotka – volterra system with feedback control, Appl. math. comput., 162, 1279-1302, (2005) · Zbl 1125.93031
[25] Huang, Z.K.; Chen, F.D., Almost periodic solution of two species model with feedback regulation and infinite delay, J. eng. math., 20, 33-40, (2004) · Zbl 1138.34344
[26] 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)
[27] Chen, X.X.; Chen, F.D., Almost periodic solutions of a delay population equation with feedback control, Nonlinear anal. real world appl., 7, 559-571, (2006) · Zbl 1128.34045
[28] Chen, X.X., Almost periodic solutions of nonlinear delay population equation with feedback control, Nonlinear anal. real world appl., 8, 62-72, (2007) · Zbl 1120.34054
[29] Chen, F.D., The permanence and global attractivity of lotka – volterra competition system with feedback controls, Nonlinear anal. real world appl., 7, 133-143, (2006) · Zbl 1103.34038
[30] Chen, F.D., Permanence in nonautonomous multi-species predator – prey system with feedback controls, Appl. math. comput., 173, 694-709, (2006) · Zbl 1087.92059
[31] Liu, Q.M.; Xu, R., Persistence and global stability for a delayed nonautonomous single-species model with dispersal and feedback control, Differ. eqs. dyn. syst., 11, 353-367, (2003) · Zbl 1231.34131
[32] Fan, Y.H.; Wang, L.L., Permanence for a discrete model with feedback control and delay, Discrete dyn. nat. soc., 2008, (2008), Article ID 945109, 8 p.
[33] Nie, L.F.; Peng, J.G.; Teng, Z.D., Permanence and stability in multi-species non-autonomous lotka – volterra competitive systems with delays and feedback controls, Math. comput. model., 49, 295-306, (2009) · Zbl 1165.34373
[34] Hu, H.X.; Teng, Z.D.; Jiang, H.J., Permanence of the nonautonomous competitive systems with infinite delay and feedback controls, Nonlinear anal. real world appl., 10, 2420-2433, (2009) · Zbl 1163.45302
[35] Liao, X.Y., Permanence and global stability in a discrete n-species competition system with feedback controls, Nonlinear anal. real world appl., 9, 1661-1671, (2008) · Zbl 1154.34352
[36] Wang, Q.; Dai, B.X., Almost periodic solution for n-species lotka – volterra competitive system with delay and feedback controls, Appl. math. comput., 200, 133-146, (2008) · Zbl 1146.93021
[37] F.D. Chen, J.H. Yang, L.J. Chen, Note on the persistent property of a feedback control system with delays, Nonlinear Anal. Real World Appl., in press.
[38] Teng, Z.; Chen, L., The positive periodic solutions in periodic Kolmogorov type systems with delays, Acta math. appl. sinica, 22, 446-456, (1999), (in Chinese) · Zbl 0976.34063
[39] Chen, F.D.; Li, Z.; Huang, Y.J., Note on the permanence of a competitive system with infinite delay and feedback controls, Nonlinear anal. real world appl., 8, 680-687, (2007) · Zbl 1152.34366
[40] Murray, J.D., Mathematical biology, (1998), Springer-Verlag, p. 83
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