Global Positive Periodic Solutions of Generalized n-Species Gilpin-Ayala Delayed Competition Systems with Impulses. (26th November 2013) - Luo, Zhenguo; Luo, Liping; Huang, Jianhua; Dai, Binxiang
Abstract : We consider the following generalizedn -species Lotka-Volterra type and Gilpin-Ayala type competition systems with multiple delays and impulses:x i ′ ( t ) = x i ( t ) [ a i ( t ) - b i ( t ) x i ( t ) - ∑ j = 1 n ‍ c i j ( t ) x j α i j ( t - ρ i j ( t ) ) - ∑ j = 1 n ‍ d i j ( t ) x j β i j ( t - τ i j ( t ) ) - ∑ j = 1 n ‍ e i j ( t ) ∫ - η i j 0 ‍ k i j ( s ) x j γ i j ( t + s ) d s - ∑ j = 1 n ‍ f i j ( t ) ∫ - θ i j 0 ‍ K i j ( ξ ) x i δ i j ( t + ξ ) x j σ i j ( t + ξ ) d ξ ], a . e, t > 0, t ≠ t k ;x i ( t k + ) - x i ( t k - ) = h i k x i ( t k ), 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.