Applicable Analysis, Vol. 81, pp. 575–587 Necessary^Sufficient Conditions for Permanence and Extinction in Lotka^ Volterra System with Discrete Delays SHENGQIANG LIU * and LANSUN CHEN y Academy of Mathematics and System Sciences, Academia Sinica, Beijing, 100080, P.R. China Communicated by R. Agarwal (Received 26 September 2001; In final form 15 November 2001) In this article, we give the necessary and sufficient conditions for permanence and extinc- tion of a two-species Lotka–Volterra system with different discrete delays. We improve some previous results. Moreover, we show the permanence and extinction for this system is equivalent to that of its corresponding nondelayed system. Keywords: Necessary and sufficient conditions; Lotka–Volterra system; Discrete delays; Extinction; Permanence AMS Subject Classification: 92D25 1. INTRODUCTION AND MAIN RESULTS Consider the following autonomous competitive system with delays. _ x 1 ðtÞ¼ x 1 ðtÞ b 1  P 2 j ¼1 P n k¼1 a ðkÞ 1j x j t  ( ðkÞ 1j    ! , _ x 2 ðtÞ¼ x 2 ðtÞ b 2  P 2 j ¼1 P n k¼1 a ðkÞ 2j x j t  ( ðkÞ 2j    ! : 8 > > > > > < > > > > > : ð1Þ *Corresponding author. E-mail: Isq@math03.math.ac.cn y E-mail: Ischen@math08.math.ac.cn ISSN 0003-6811 print: ISSN 1563-504X online ß 2002 Taylor & Francis Ltd DOI: 10.1080/0003681021000004302