With the help of a continuation theorem based on Gaines and Mawhinscoincidence degree, easily verifiable criteria are established for the global existence of positiveperiodic solutions of the following nonlinear state dependent delays predator-prey system{dN_1(t)/dt = N_1(t)[b_1(t) - ∑ from i=1 to n of ai(t)(N_1(t-τ_i(t,N_1(t), N_2(t))))^(α_i) - ∑from j=1 to m of c_j(t)(N_2(t - σ_j(t,N1(t),N_2(t))))^(β_j)] dN_2(t)/dt = N_2(t)[-b_2(t) + ∑ fromi=1 to n of d_i(5)(N_1(t - ρ_i(t,N_1(t),N_2(t))))^(γ_i)], where a_i(t), c_j(t), d_i(t) arecontinuous positive periodic functions with periodic > 0, b_1(t), b_2(t) are continuous periodicfunctions with periodic ω and ∫_0~ωb_i(t)dt > 0. τ_i, σ_j, ρ_i (i = 1,2,…,m) are continuousand ω-periodic with respect to their first arguments, respectively. α_i, β_j, γ_i (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.
With the help of a continuation theorem based on Gaines and Mawhin＇s coincidence degree, several verifiable criteria are established for the global existence of positive periodic solutions of a class of non-autonomous single species population model with delays （both state-dependent delays and continuous delays） and feedback control. After that, by constructing a suitable Lyapunov functional, sufficient conditions which guarantee the existence of a unique globally asymptotic stable positive periodic solution of a kind of nonlinear feedback control ecosystem are obtained. Our results extend and improve the existing results, and have further applications in population dynamics.
