Rocky Mountain Journal of Mathematics

Permanence in Multi-Species Competitive Systems with Delays And Feedback Controls

Linfei Nie, Jigen Peng, and Zhidong Teng

Full-text: Open access

Article information

Rocky Mountain J. Math., Volume 38, Number 5 (2008), 1609-1631.

First available in Project Euclid: 22 September 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Kolomogorov Lotka-Volterra system competitive system delay feedback control Ultimate boundedness permanence


Nie, Linfei; Peng, Jigen; Teng, Zhidong. Permanence in Multi-Species Competitive Systems with Delays And Feedback Controls. Rocky Mountain J. Math. 38 (2008), no. 5, 1609--1631. doi:10.1216/RMJ-2008-38-5-1609.

Export citation


  • S. Ahmad and A.C. Lazer, On a property of nonautonomous Lotka-Volterra competition model, Nonlinear Anal. 37 (1999), 603-611.
  • F. Chen, The permanence and global attractivity of Lotka-Volterra competition system with feedback controls, Nonlinear Analysis: Real World Appl. 7 (2006), 133-143.
  • --------, On the periodic solutions of periodic multi-species Kolmogorov type competitive system with delays and feedback cont, Appl. Math. Comp. 180 (2006), 366-373.
  • X. Chen, Almost periodic solutions of nonlinear delay population equation with feedback control, Nonlinear Analysis: Real World Appl. 8 (2007), 62-72.
  • X. Chen and F. Chen, Almost-periodic solutions of a delay population equation with feedback control, Nonlinear Analysis: Real World Appl. 7 (2006), 559-571.
  • K. Gopalsamy, Stability and oscillations in delay different equations of population dynamics, Kluwer Academic, Dordrecht, MA, 1992.
  • K. Gopalsamy and P. Wen, Feedback regulation of logistic growth, Internat. J. Math. Math. Sci. 16 (1993), 177-192.
  • J. Hale, Theory of functional differential equations, Springer-Verlag, Heidelberg, 1977.
  • H. Huo and W. Li, Positive periodic solutions of a class of delay differential system with feedback control, Appl. Math. Comp. 148 (2004), 35-46.
  • Y. Kuang, Delay differential equations, with applications in population dynamics, Academic Press, New York, 1993.
  • J. Lasalle and S. Lefschetz, Stability by Lyapunov's direct method, Academic Press, New York, 1961.
  • S. Lefschetz, Stability of nonlinear control system, Academic Press, New York, 1965.
  • Y. Li and P. Liu, Positive periodic solutions of a class of functional differential systems with feedback controls, Nonlinear Anal. 57 (2004), 655-666.
  • B. Lisena, Global attractivity in nonautonomous logistic equations with delay, Nonlinear Analysis: Real World Appl., in press.
  • Q. Liu and R. Xu, Persistence and global stability for a delayed nonautonomous single-species model with dispersal and feedback control, Diff. Equat. Dynam. Syst. 11 (2003), 353-367.
  • Y. Muroya, Permanence of nonautonomous Lotka-Volterra delay differential systems, Appl. Math. Letters 19 (2006), 445-450.
  • L. Nie, J. Peng and Z. Teng, Permanence and stability in non-autonomous predator-prey Lotka-Volterra systems with feedback controls, preprint.
  • --------, Harmless feedback control for permanence and global asymptotic stability in nonlinear delay population equation, Stud. Appl. Math. 120 (2008), 247-263.
  • G. Seifert, On a delay-differential equation for single specie population variation, Nonlinear Analysis: TMA 11 (1987), 1051-1059.
  • Z. Teng, Permanence and stability in general nonautonomous single-species Kolmogorov systems with delays, Nonlinear Analysis: Real World Appl. 8 (2007), 230-248.
  • Z. Teng and L. Chen, The positive periodic solutions of periodic Kolmogorov type systems with delays, Acta Math. Appl. Sinica 22 (1999), 456-464.
  • Z. Teng and Z. Li, Permanence and asymptotic behavior of the $N$-Species nonautonomous Lotka-Volterra competitive systems, Comput. Math. Appl. 39 (2000), 107-116.
  • Z. Teng, Z. Li and H. Jiang, Pemanence criteria in non-autonomous predator-prey Kolmogorov systems and its applications, Dynam. Systems 19 (2004), 171-194.
  • Z. Teng and M. Rehim, Permanence in nonautonomous predator-prey systems with infinite delays, J. Appl. Math. Comput. 197 (2006), 302-321.
  • R.R. Vance and E.A. Coddington, A nonautonomous model of population growth, J. Math. Biol. 27 (1989), 491-506.
  • W. Wang and Z. Ma, Harmless delays for uniform permanence, J. Math. Anal. Appl. 158 (1991), 256-268.
  • P. Weng, Existence and global stability of positive periodic solution of periodic integrodifferential systems with feedback controls, Comput. Math. Appl. 40 (2000), 747-759.
  • Y. Xia and J. Cao, Almost periodic solutions of $n$-species competitive system with feedback controls, J. Math. Anal. Appl. 294 (2004), 503-522.
  • Y. Xiao and S. Tang, Permanence and periodic solution in competition system with feedback controls, Math. Comput. Model. 27 (1998), 33-37.