Globally asymptotic stability in two periodic delayed competitive systems

Based on two types of well known periodic single-species population growth models with time delay, we propose two corresponding periodic competitive systems with multiple delays. By using some new analysis techniques, we derive the same criteria for the existence and globally asymptotic stability of positive periodic solutions of the above two competitive systems. The easily verifiable criteria show that the existence and globally asymptotic stability of positive periodic solutions of two delayed nonautonomous competitive systems correspond to the existence and global stability of positive equilibrium of corresponding undelayed autonomous system. As an application, some special cases of the above systems are examined and some earlier results are extended and improved. Biological interpretations on the main results are also given.

[1]  Zhijun Liu,et al.  Persistence and periodic solutions of a nonautonomous predator-prey diffusion with Holling III functional response and continuous delay , 2004 .

[2]  M. Zhien,et al.  Harmless delays for uniform persistence , 1991 .

[3]  Yang Kuang,et al.  Existence, uniqueness and asymptotic stability of periodic solutions of periodic functional-differential systems , 1997 .

[4]  G. Seifert,et al.  On a delay-differential equation for single specie population variations , 1987 .

[5]  Udo Halbach,et al.  Life Table Data and Population Dynamics of the Rotifer Brachionus calyciflorus Pallas as Influenced by Periodically Oscillating Temperature , 1973 .

[6]  X. Zou,et al.  Global attractivity of positive periodic solution to periodic Lotka-Volterra competition systems with pure delay , 2006 .

[7]  D. Jiang,et al.  Existence and global attractivity of positive periodic solutions of periodic n-species Lotka-Volterra competition systems with several deviating arguments. , 1999, Mathematical biosciences.

[8]  S. Yau.  Mathematics and its applications , 2002 .

[9]  Lansun Chen,et al.  Global stability in a periodic delayed predator-prey system , 2007, Appl. Math. Comput..

[10]  R. K. Miller.  On Volterra’s Population Equation , 1966 .

[11]  Yang Kuang,et al.  Global stability for infinite delay Lotka-Volterra type systems , 1993 .

[12]  Shair Ahmad.  Convergence and ultimate bounds of solutions of the nonautonomous Volterra-Lotka competition equations , 1987 .

[13]  J. Maynard Smith,et al.  Mathematical Ideas in Biology , 1968 .

[14]  Zhidong Teng,et al.  Nonautonomous Lotka–Volterra Systems with Delays , 2002 .

[15]  Chen Lansun,et al.  Periodic solutions of single-species nonautonomous diffusion models with continuous time delays , 1996 .

[16]  Jianhong Wu,et al.  Periodic solutions of single-species models with periodic delay , 1992 .

[17]  T. A. Burton,et al.  Stability and Periodic Solutions of Ordinary and Functional Differential Equations , 1986 .

[18]  WANG WENDI,et al.  GLOBAL STABILITY OF A COMPETITION MODEL WITH PERIODIC COEFFICIENTS AND TIME DELAYS , 2005 .

[19]  Fan Meng,et al.  Periodic Solutions of Single Population Model with Hereditary Effects , 2000 .

[20]  K. Gopalsamy,et al.  Global asymptotic stability in a periodic Lotka-Volterra system , 1985, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[21]  Hisashi Fujimoto,et al.  Dynamical behaviors for population growth equations with delays , 1998 .

[22]  Ravi P. Agarwal,et al.  Periodicity and Stability in Periodic n-Species Lotka-Volterra Competition System with Feedback Controls and Deviating Arguments , 2003 .

[23]  Yongkun Li,et al.  Periodic Solutions for Delay Lotka–Volterra Competition Systems , 2000 .

[24]  K. Gopalsamy,et al.  Exchange of equilibria in two species Lotka-Volterra competition models , 1982, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[25]  R. Gaines,et al.  Coincidence Degree and Nonlinear Differential Equations , 1977 .

[26]  K. Gopalsamy.  Stability and Oscillations in Delay Differential Equations of Population Dynamics , 1992 .

[27]  Yi Xie,et al.  Almost periodic solutions of single population model with hereditary effects , 2008, Appl. Math. Comput..

[28]  Alan C. Lazer,et al.  An application of topological degree to the periodic competing species problem , 1986, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.