Permanence, extinction and balancing survival in nonautonomous Lotka–Volterra system with delays Shengqiang Liu, Lansun Chen * Institute of Mathematics, Academia Sinica, Beijing 100080, People’s Republic of China Abstract In this paper, the periodic n-species Lotka–Volterra competition systems with time delays are considered. The sufficient conditions are derived for the permanence, ex- tinction and balancing survival of the system. Some known results are improved and unified. Further, our results suggest that under some conditions, the introduction and the variance of the time delays will be either harmless or ‘‘useless’’. Conjecture about the effect of time delays on some global behaviors of the system is advanced. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Competitive Lotka–Volterra systems; Periodic; Time delays; Extinction; Permanence; Balancing survival 1. Introduction We consider the following system of differential equations: _ x i ðtÞ¼ x i ðtÞ b i ðtÞ   X n i¼1 a ij ðtÞx j ðt  s ij ðtÞÞ ! ; i ¼ 1; 2; ... ; n: ð1Þ Such a system, models competition among n-species, where x i ðtÞ denotes the concentration of the ith species at time t,and b i ðtÞ is its growth rate at this time, Applied Mathematics and Computation 129 (2002) 481–499 www.elsevier.com/locate/amc * Corresponding author. E-mail address: lschen@math08.math.ac.cn (L. Chen). 0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII:S0096-3003(01)00058-3