Global Positive Periodic Solutions of Generalized n-Species Gilpin-Ayala Delayed Competition Systems with Impulses.

We consider the following generalized n-species Lotka-Volterra type and Gilpin-Ayala type competition systems with multiple delays and impulses: x'i(t) = xi(t) [ai(t) - bi(t)xi(t) - Σnj=1 Ci j(t)xαijj (t-pi j(t)) - Σnj=1 dij(t)xᵦij j (t - τi j(t)) - Σnj=1 ei j(t) ∫0-952;ij kij(s)x ϒijj (t + s)ds - Σn j=1 fij (t) ∫0-ϑij Kij(ξ)xiδij (t +ξ)xjσij (t + ξ)dξ], a.e,t > 0, t ≠ tk; xi(tk+) - xi(tk-) = hikxi(tk), i = 1,2,...,n,k ∈ Z+. By applying the Krasnoselskii fixed-point theorem in a cone of Banach space, we derive some verifiable necessary and sufficient conditions for the existence of positive periodic solutions of the previously mentioned. As applications, some special cases of the previous system are examined and some earlier results are extended and improved. [ABSTRACT FROM AUTHOR]