×

Positive periodic solutions of neutral Lotka-Volterra system with feedback control. (English) Zbl 1125.93031

Summary: With the help of a continuation theorem based on Gaines and Mawhin’s coincidence degree, easily verifiable criteria are established for the global existence of positive periodic solutions of neutral Lotka-Volterra system with periodic delays and feedback control. Our results extend and improve existing results, and have further applications in population dynamics.

MSC:

93C23 Control/observation systems governed by functional-differential equations
93B52 Feedback control
34K13 Periodic solutions to functional-differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Li, Y. K., Periodic solutions of a periodic delay predator-prey system, Proc. Am. Math. Soc., 127, 1331-1335 (1999) · Zbl 0917.34057
[2] Freedman, H. I.; Wu, J., Periodic solutions of single-species models with periodic delay, SIAM J. Math. Anal., 23, 689-701 (1992) · Zbl 0764.92016
[3] Tang, B. R.; Kuang, Y., Existence, uniqueness and asymptotic stability of periodic solutions of periodic functional differential system, Tohoku Math., 49, 217-239 (1997) · Zbl 0883.34074
[4] Li, Y. K., Periodic solutions for delay Lotka-Volterra competition systems, J. Math. Anal. Appl., 246, 230-244 (2000) · Zbl 0972.34057
[5] Li, Y. K.; Kuang, Y., Periodic solutions of periodic delay Lotka-Volterra equations and systems, J. Math. Anal. Appl., 255, 265-280 (2001) · Zbl 1024.34062
[6] Li, Y. K.; Kuang, Y., Periodic solutions in periodic state-dependent delay equations and population models, J. Math. Anal. Appl., 255, 265-280 (2001)
[7] Wan, L. L.; Li, W. T., Existence and global stability of positive periodic solutions of a predator-prey system with delays, Appl. Math. Comput., 146, 167-185 (2003) · Zbl 1029.92025
[8] Fan, M.; Wang, K., Positive periodic solutions of a periodic integro-differential competition system with infinite delays, ZAMM. Z. Angew. Math. Mech., 81, 3, 197-203 (2001) · Zbl 0977.45005
[9] Fan, M.; Wang, K., Periodicity in a delayed ratio-dependent pedator-prey system, J. Math. Anal. Appl., 262, 179-190 (2001) · Zbl 0994.34058
[10] Fan, M.; Wang, K., Global existence of positive periodic solutions of periodic predator-prey system with infinite delay, J. Math. Anal. Appl., 262, 1-11 (2001) · Zbl 0995.34063
[11] Gopalsamy, K.; Zhang, B. G., On a neutral delay logistic equation, Dyn. Stab. Syst., 2, 183-195 (1998) · Zbl 0665.34066
[12] Freedman, H. I.; Kuang, Y., Stability switches in linear scalar neutral delay equations, Funkc. Ekvac., 34, 187-209 (1991) · Zbl 0749.34045
[13] Gopalsamy, K.; He, X.; Wen, L., On a periodic neutral logistic equation, Glasgow Math. J., 33, 281-286 (1991) · Zbl 0737.34050
[14] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press: Academic Press Boston · Zbl 0777.34002
[15] Li, Y. K., Positive periodic solution for neutral delay model, Acta Math. Sin., 39, 6, 789-795 (1996), (in Chinese)
[16] Li, Q.; Cao, J. D., On positive periodic solutions for neutral delay model, J. Math. Exposition, 20, 4, 562-565 (1999)
[17] Fang, H.; Li, J., On the existence of periodic solutions of a neutral delay model of single-species population growth, J. Math. Anal. Appl., 259, 8-17 (2001) · Zbl 0995.34073
[18] Lu, S. P.; Ge, W. G., Existence of positive periodic solutions for neutral functional differential equations with deviating arguments, Appl. Math. J. Chinese Univ. Ser. B, 17, 4, 382-390 (2002) · Zbl 1025.34073
[19] Yang, Z. H.; Cao, J. D., Sufficient conditions for the existence of positive periodic solutions of a class of neutral delays models, Appl. Math. Computat., 142, 123-142 (2003) · Zbl 1037.34066
[20] Yang, Z. H.; Cao, J. D., Positive periodic solutions of neutral Lotka-Volterra system with periodic delays, Appl. Math. Computat., 149, 661-687 (2004) · Zbl 1045.92037
[21] Li, Y. K., On a periodic neutral delay Lotka-Volterra system, Nonlinear Anal., 39, 767-778 (2000) · Zbl 0943.34058
[22] Weng, P. X., Global attractivity in a periodic competition system with feedback controls, Acta Appl. Math., 12, 11-21 (1996) · Zbl 0859.34061
[23] 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
[24] Weng, P. X.; Jiang, D. Q., Existence and global stability of positive periodic solution of \(n\)-species periodic Lotka-Volterra competition systems with feedback controls and deviating arguments, Far East J. Math. Sci., 7, 1, 45-65 (2002) · Zbl 1043.34075
[25] Xiao, Y. N.; Tang, S. Y.; Chen, J. F., Permanence and periodic solution in competition system with feedback controls, Math. Comput. Model., 27, 6, 33-37 (1998) · Zbl 0896.92032
[26] Gopalsamy, K.; Weng, P. X., Feedback regulation of logistic growth, Int. J. Math. Sci., 16, 1, 177-192 (1993) · Zbl 0765.34058
[27] 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. Equat., 17, 4, 337-384 (2001) · Zbl 1004.34030
[28] 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, 1, 13-21 (2002), (in Chinese) · Zbl 1005.34039
[29] Fan, M., Periodicity and stability in periodic \(n\)-species Lotka-Volterra competition system with feedback controls and deviating arguments, Acta Math. Sin., 19, 801-822 (2003) · Zbl 1047.34080
[30] Gaines, R. E.; Mawhin, J. L., Coincidence Degree and Nonlinear Differential Equations (1977), Springer-Verlag: Springer-Verlag Berlin · Zbl 0326.34021
[31] Chen, F. D., Persistence and global stability for non-autonomous cooperative system with diffusion and time delay, Acta Sci. Nat. Univer. Pekinen., 39, 1, 22-28 (2003) · Zbl 1042.92036
[32] Chen, F. D.; Chen, X. X., The \(n\)-competing Volterra-Lotka almost periodic systems with grazing rates, J. Biomath., 18, 4, 411-416 (2003), (in Chinese)
[33] Chen, F. D., Periodic solution of nonlinear integral-differential equations with infinite delay, Acta Math. Appl. Sin., 26, 1, 1-8 (2003), (in Chinese)
[34] F.D. Chen, J.L. Shi, Periodic solution of neutral higher order periodic systems, Acta Math. Sin. (in press); F.D. Chen, J.L. Shi, Periodic solution of neutral higher order periodic systems, Acta Math. Sin. (in press)
[35] Chen, F. D., Existence and uniqueness of almost periodic solution for a class of non-autonomous system, Math. Practice Theory, 31, 5, 532-538 (2001), (in Chinese) · Zbl 1493.34138
[36] Gopalsamy, K., Stability and Oscillation in Delay Differential Equations of Population Dynamics. Stability and Oscillation in Delay Differential Equations of Population Dynamics, Mathematics and its Applications, 74 (1992), Kluwer Academic Publishers Group: Kluwer Academic Publishers Group Dordrecht · Zbl 0752.34039
[37] Li, Y. K., Periodic solution of a periodic neutral delay equation, J. Math. Anal. Appl., 214, 11-21 (1997) · Zbl 0894.34075
[38] Lu, S. P., On the existence of positive periodic solutions for neutral functional differential equation with multiple deviating arguments, J. Math. Anal. Appl., 280, 321-333 (2003) · Zbl 1034.34084
[39] F.D. Chen, F. Lin, X. Chen, Sufficient conditions for the existence of positive periodic solutions of a class of neutral delay models with feedback control, Appl. Math. Computat. (accepted); F.D. Chen, F. Lin, X. Chen, Sufficient conditions for the existence of positive periodic solutions of a class of neutral delay models with feedback control, Appl. Math. Computat. (accepted) · Zbl 1096.93017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.