tailieunhanh - Báo cáo sinh học: " Research Article Asymptotical Behaviors of Nonautonomous Discrete Kolmogorov System with Time Lags"

Tuyển tập các báo cáo nghiên cứu về sinh học được đăng trên tạp chí sinh học Journal of Biology đề tài: Research Article Asymptotical Behaviors of Nonautonomous Discrete Kolmogorov System with Time Lags | Hindawi Publishing Corporation Advances in Difference Equations Volume 2010 Article ID 517378 19 pages doi 2010 517378 Research Article Asymptotical Behaviors of Nonautonomous Discrete Kolmogorov System with Time Lags Shengqiang Liu Natural Science Research Center The Academy of Fundamental and Interdisciplinary Science Harbin Institute of Technology 3041 2 Yi-Kuang Street Nan-Gang District Harbin 150080 China Correspondence should be addressed to Shengqiang Liu sqliu@ Received 23 November 2009 Accepted 24 April 2010 Academic Editor Elena Braverman Copyright 2010 Shengqiang Liu. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. We discuss a general n-species discrete Kolmogorov system with time lags. We build some new results about the sufficient conditions for permanence extinction and balancing survival. When applying these results to some Lotka-Volterra systems we obtain the criteria on harmless delay for the permanence as well as profitless delay for balancing survival. 1. Introduction Difference equations are frequently used in modelling the interactions of populations with nonoverlapping generations see . May and Oster 1 for the one-species difference equations 2 for how populations regulate Hassel 3 Basson and Fogarty 4 and Beddington et al. 5 for predator-prey models . Since one of the most important ecological problems associated with the populations dynamical system is to study the long-term coexistence of all the involved species such problem in the nondelayed discrete systems had already attracted much attention and thereby many excellent results had appeared 6-11 . However recent studies of the natural populations indicated that the interactions of populations for example the density-dependent population regulation sometimes takes place over many generations see 12-29 and .

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