With the help of a continuation theorem based on Gaines and Mawhin's coincidence degree, easily verifiable criteria are established for the global existence of positive periodic solutions of the following nonlinear state dependent delays predator-prey system {dN1(T)/dt=N1(t)[b1(t)-∑i=1^n αi(t)(N1(t-γi(t,N1(t),N2(5))))^αi;-∑j=1^mcj(t)(N2)(t-σj(t,N1(t),N2(t)))^βj],dN2(t)/dt=N2(t)[-b2(t)+∑i=1^n di(t)(N1(t-ρi(t,N1(t),N2(t))))^γi],where αi(t), cj(t), di(t) are continuous positive periodic functions with periodic ω >0, b1(t), b2(t) are continuous periodic functions with periodic ω and ∫0^ω bi(t)dt >0. γi,σj, ρi (i=1, 2,…,n, j = 1, 2,…,m) are continuous and ω-periodic with respect to their first arguments, respectively, αi,βj, γi (i=1,2,…，n，j=1,2,…, m)are positive constants.
In this paper, we consider a nonautonomous competitive system which is also affected by toxic substances. Some averaged conditions for the permanence of this system are obtained. Our result shows that under some suitable assumption on the coefficients of the system, the toxic has no influence on the permanence of the system. Also, by using a suitable Lyapunov function, sufficient conditions which guarantee the attractivity of any two positive solutions of the system are obtained.
