Abstract
By using the method of coincidence degree and Lyapunov functional, a set of easily applicable criteria are established for the global existence and global asymptotic stability of strictly positive (componentwise) periodic solution of a periodic n-species Lotka-Volterra competition system with feedback controls and several deviating arguments. The problem considered in this paper is in many aspects more general and incorporate as special cases various problems which have been studied extensively in the literature. Moreover, our new criteria, which improve and generalize some well known results, can be easily checked.
Similar content being viewed by others
References
Ahmad, S.: On the non autonomous Volterra-Lotka competition equations. Proc. Amer. Math. Soc., 117, 199–204 (1993)
Fan, M., Wang, K., Jiang, D. Q.: Existence and global attractivity of positive periodic solutions of periodic n-species Lotka-Volterra competition systems with several deviating arguments. Mathematical Biosciences, 160, 47–61 (1999)
Fan, M., Wang, K.: Existence and global attractivity of positive periodic solution of multi species ecological competition system. Acta Math. Sinica, 43(1), 77–82 (2000)
Fan, M., Wang, K.: Positive periodic solution of a periodic integro-differential competition systems with infinite delays. ZAMM Z. Angew. Math. Mech., 81, 197–203 (2001)
Golpalsamy, K.: Globally asymptotic stability in a periodic Lotka-Volterra system. J. Austral. Math. Soc. Ser. B., 24, 160–170 (1982)
Golpalsamy, K.: Globally asymptotic stability in a periodic Lotka-Volterra system. J. Austral. Math. Soc. Ser. B., 29, 66–72 (1985)
Gopalsamy, K.: Global asymptotic stability in a periodic integro differential system. Tôhoku Math. J., 37, 323–332 (1985)
Korman, P.: Some new results on the periodic competition model. J. Math. Anal. Appl., 171, 131–138 (1992)
Kuang, Y.: Delay Differential Equations with Application in Population Dynamics, Academic Press, Boston, 1993
Kuang, Y., Smith H. L.: Global stability for infinite delay Lotka-Volterra type systems. J. Differential Equations, 103, 221–246 (1993)
Kuang, Y.: Global stability in delayed nonautonomous Lotka-Volterra type systems without saturated equilibria. Differential Integral Equations, 9(3), 557–567 (1996)
Leung, A. W., Zhao, Z.: Global stability for a large class of Volterra-Lotka type integro-differential population delay equations. Nonlinear Analysis TMA, 12, 495–505 (1988)
Trieo, A., Alrarez, C.: A different consideration about the globally asymptotically stable solution of the periodic n-competing species problem. J. Math. Anal. Appl., 159, 44–50 (1991)
Zhao, X. Q.: The qualitative analysis of n-species Lotka-Volterra periodic competition systems. Math. Comput. Modelling, 15, 3–8 (1991)
Wang, L., Zhang, Y.: Global stability of Volterra-Lotka systems with delays. Differential Equations Dynamical Systems, 3(2), 205–216 (1995)
Freedman, H. I. and Xia, H. X.: Periodic solution of single species models with delay, differential equations, dynamical systems and control sciences. Lecture Notes in Pure and Appl. Math., 152, 55–74 (1994)
Fujimoto, H.: Dynamical behaviours for population growth equations with delays. Nonlinear Analysis TMA, 31(5), 549–558 (1998)
Gopalsamy, K.: Stability and Oscillation in Delay Differential Equations of Population Dynamics. Mathematicx and its Applications 74, Kluwer Academic Publishers Group, Dordrecht, 1992
Weng, P. X.: Global attractivity in a periodic competition system with feedback controls. Acta Appl. Math., 12, 11–21 (1996)
Xiao, Y. N., Tang, S. Y., Chen J. F.: Permanence and periodic solution in competition system with feedback controls. Mathl. Comput. Modelling, 27(6), 33–37 (1998)
Gopalsamy, K. and Weng, P. X.: Feedback regulation of logistic growth, Internat. J. Math. Math. Sci., 16(1), 177–192 (1993)
Aizerman, M. A., Gantmacher, F. R.: Absolute Stability of Regulator Systems (translated from Russian). Holden Day, San Francisco, 1964
Clark, C. W.: Mathematical Bioeconomics, The Optimal Management of Renewable Resources, Wiley-Interscience, New Work, 1976
Boudjillaba, H., Sari, T.: Stability loss delay in harvesting competing populations. J. Differential Equations, 152(2), 394–408 (1999)
Lefschetz, S.: Stability of Nonlinear Control Systems, Academic Press, New York, 1965
Chen, B. S., Liu, Y. Q.: On the stable periodic solutions of single species model with hereditary effects. Mathematica Applicata, 21(1), 42–46 (1999)
Seifert, G.: On a delay-differential equation for single species population variations. Nonlinear Analysis TMA, 11(9), 1051–1059 (1987)
Zhang, B. G., Gopalsamy, K.: Global attractivity and oscillations in a periodic delay-logistic equation. J. Math. Anal. Appl., 15, 274 (1990)
Gaines, R. E., Mawhin, J. L.: Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977
Barbălat, I.: Systems d’equations differentielle d’oscillations nonlineaires. Rev. Roumaine Math. Pures Appl., 4, 267–270 (1959)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the National Natural Sciences Foundation of P. R. China (No. 10171010 and 10201005), the Key Project on Science and Technology of the Education Ministry of P. R. China (No. Key 01061) and the Science Foundation of Jilin Province of P. R. China for Distinguished Young Scholars
Rights and permissions
About this article
Cite this article
Fan, M., Wang, K., Wong, P.J.Y. et al. Periodicity and Stability in Periodic n-Species Lotka-Volterra Competition System with Feedback Controls and Deviating Arguments. Acta Math Sinica 19, 801–822 (2003). https://doi.org/10.1007/s10114-003-0311-1
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10114-003-0311-1
Keywords
- Strictly positive periodic solutions
- Globally asymptotic stability
- Lotka-Volterra competition system
- Feedback controls
- Deviating arguments
- Coincidence degree
- Lyapunov functional