210 Communications in Nonlinear Science & Numerical Simulation December 2001 Profitless delays for extinction in nonautonomous Lotka-Volterra system Shengqiang LIU and Lansun CHEN Academy of Mathematics and System Sciences, Chinese Academy sf Sciences, Beijing 100080, China; e-mail: ZsqOmath03.math.ac.cn (Received July 22, 2001) Abstract: We study the delayed periodic n-species Lotka-Voll;erra systems where the growth rate of each species is not always positive. The sufficient conditions for the extinction that are independent of the delays are obtained. Some known results are improved and generalized. Our results suggest that under some conditions, the introduction and the variance of the time delays can be both harmless and profitless. Discussion about the effect of time delays on the extinction of the system is also advanced. Keywords: competitive, Lo&-Volterra system, periodic, profitless delays, extinction Introduction Consider the following system of differential equations: This system models competition among n species. Here xi(t) denotes the concentration of the ith species at time t, and hi(t) is its growth rate at this time, aij(t) denote the competitive coefficient between the ith species and the jth species, and when i = j, q(t) denotes the density-dependent rate of the ith species; Tj(t) is the time deviating arguments at time t (i,j = 1,2,... ,n). As Tj(t) E 0 for j = l;.. ,n, system (1) is known as the famous Lotka- Volterra system: &(t) = xi(t) hi(t) - 2 a,ct)z,(t)) , i = 1,2,.-e ,n j=l (2) There has been much work devoted to studying (2): de IOca and ZeemanI’], Ahmad and de Oca12] and Ahmad13] considered its extinction, while de Oca and ZeemanL4] constructed the conditions for its balancing survival. Others[5y6] examined the globally asymptotic stability of system (2). Less work was done on (1). Lu and Takeuchi17] studied the permanence for the two species autonomous system of (1) with constant delays. Hisashi[*] extended the model of [7] into the nonautonomous case - still with constant delays, though he extended the result in [7]. The results of [7,8] implied that the time delays had no effect on the permanence of the system, that is, these delays are “harmless” for the permanence. Fan, Wang and JiangLg] considered the existence of the positive periodic solution of system (1). They showed that the existence of the positive periodic solution might keep independent of the time delays under proper conditions, which indicated delays can also be harmless to the existence of periodic solution. Then what is the effect of time delays on the extinction of the periodic (1) with nonconstant delays? As far as we know, no author had studied this problem. Then it is significant for us to study the extinction of (1). We obtain the sufficient conditions for the extinction of all species