Global attractivity of non-autonomous Lotka-Volterra competition system without instantaneous negative feedback

Abstract We consider a non-autonomous Lotka–Volterra competition system with distributed delays but without instantaneous negative feedbacks (i.e., pure delay systems). We establish some 3/2-type and M -matrix-type criteria for global attractivity of the positive equilibrium of the system, which generalise and improve the existing ones.

[1]  Daniel I. Barnea.  A Method and New Results for Stability and Instability of Autonomous Functional Differential Equations , 1969 .

[2]  Hal L. Smith,et al.  Monotone Dynamical Systems: An Introduction To The Theory Of Competitive And Cooperative Systems (Mathematical Surveys And Monographs) By Hal L. Smith , 1995 .

[3]  Xue-Zhong He.  The Lyapunov Functionals for Delay Lotka-Volterra-Type Models , 1998, SIAM J. Appl. Math..

[4]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[5]  Xianhua Tang,et al.  Stability in a Linear Delay System without Instantaneous Negative Feedback , 2002, SIAM J. Math. Anal..

[6]  Yang Kuang,et al.  Global Stability in Delay Differential Systems without Dominating Instantaneous Negative Feedbacks , 1995 .

[7]  Tu Jian.  Global Attractivity for a Population Model with Time Delay , 2000 .

[8]  Joseph W.-H. So,et al.  On the uniform stability for a ‘food-limited’ population model with time delay , 1995, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[9]  Josef Hofbauer,et al.  Diagonal dominance and harmless off-diagonal delays , 2000 .

[10]  Yang Kuang,et al.  Convergence in Lotka-Volterra type diffusive delay systems without dominating instantaneous negative feedbacks , 1993, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[11]  Peter N. Brown,et al.  Decay to Uniform States in Ecological Interactions , 1980 .

[12]  M. Fiedler.  Special matrices and their applications in numerical mathematics , 1986 .

[13]  Kondalsamy Gopalsamy.  Stability criteria for the linear system [Xdot](t) + A(t) X(t—τ) = 0 and an application to a non-linear system , 1990 .

[14]  J. Yorke.  Asymptotic stability for one dimensional differential-delay equations☆ , 1970 .

[15]  Hal L. Smith,et al.  Convergence in Lotka–Volterra-type delay systems without instantaneous feedbacks , 1993, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[16]  Toshiaki Yoneyama.  The 32 stability theorem for one-dimensional delay-differential equations with unbounded delay , 1992 .

[17]  E. M. Wright.  A non-linear difference-differential equation. , 1946 .

[18]  Xue-Zhong He,et al.  Global stability in nonautonomous Lotka-Volterra systems of "pure-delay type" , 1998, Differential and Integral Equations.

[19]  Xianhua Tang,et al.  3/2-type criteria for global attractivity of Lotka-Volterra competition system without instantaneous negative feedbacks $ , 2002 .

[20]  T. Yoneyama.  On the 32 stability theorem for one-dimensional delay-differential equations , 1987 .