Existence and global attractivity of positive periodic solutions of periodic n-species Lotka-Volterra competition systems with several deviating arguments.

In this paper, we study the existence and global attractivity of positive periodic solutions of periodic n-species Lotka-Volterra competition systems. By using the method of coincidence degree and Lyapunov functional, a set of easily verifiable sufficient conditions are derived for the existence of at least one strictly positive (componentwise) periodic solution of periodic n-species Lotka-Volterra competition systems with several deviating arguments and the existence of a unique globally asymptotically stable periodic solution with strictly positive components of periodic n-species Lotka-Volterra competition system with several delays. Some new results are obtained. As an application, we also examine some special cases of the system we considered, which have been studied extensively in the literature. Some known results are improved and generalized.

[1]  Lawrence Stark.  Stability and Oscillations , 1968 .

[2]  Antonio Tineo,et al.  A, Different Consideraton about the Globally Asymptotically Stable Solution of the Periodic n-Competing Species Problem* , 1991 .

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

[4]  F. Zanolin,et al.  Coexistence States for Periodic Competitive Kolmogorov Systems , 1998 .

[5]  Hal L. Smith.  Periodic solutions of periodic competitive and cooperative systems , 1986 .

[6]  F. Zanolin.  Permanence and Positive Periodic Solutions for Kolmogorov Competing Species Systems , 1992 .

[7]  P. de Mottoni,et al.  Competition systems with periodic coefficients: A geometric approach , 1981 .

[8]  G. Ladas,et al.  Oscillation Theory of Delay Differential Equations: With Applications , 1992 .

[9]  Kondalsamy Gopalsamy,et al.  Global attractivity and oscillations in a periodic delay-logistic equation , 1990 .

[10]  H. I. Freedman,et al.  Persistence in a model of three competitive populations , 1985 .

[11]  Jim M Cushing,et al.  Two species competition in a periodic environment , 1980 .

[12]  Shair Ahmad,et al.  On the nonautonomous Volterra-Lotka competition equations , 1993 .

[13]  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.

[14]  R. May,et al.  Nonlinear Aspects of Competition Between Three Species , 1975 .

[15]  J. K. Hale,et al.  Competition for fluctuating nutrient , 1983 .

[16]  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.

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

[18]  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.

[19]  Philip Korman.  Some new results on the periodic competition model , 1992 .

[20]  Hal L. Smith.  Periodic competitive differential equations and the discrete dynamics of competitive maps , 1986 .

[21]  Nobuhiko Saitô,et al.  Time delays and chaos in two competing species , 1980 .