Periodic Solutions for Delay Lotka–Volterra Competition Systems

Abstract By using the continuation theorem of coincidence degree theory, sufficient and realistic conditions are obtained for the existence of positive periodic solutions for the periodic distributed delay Lotka–Volterra competition system du i t dt = u i t r i t − a ii t u i t − ∑ j = 1 j ≠ i n a ij t ∫ 0 − T ij K ij s u j t + s ds , i = 1, 2,…,n, and the periodic state dependent delay Lotka–Volterra competition system du i t dt = u i t r i t − a ii t u i t − ∑ j = 1 j ≠ i n a ij t u j t − τ j t , u 1 t ,…, u n t , i = 1, 2,…,n, where ri, aii > 0, aij ≥ 0 (j ≠ i, i, j = 1, 2,…,n) are continuous ω-periodic functions, Tij ∈ (0, ∞)(j ≠ i, i, j = 1, 2,…,n), Kij ∈ C([−Tij, 0], (0, ∞)), ∫0 − TijKij(s) ds = 1 (j ≠ i, i, j = 1, 2,…,n), τi ∈ C(Rn + 1, R), and τi (i = 1, 2,…,n) are ω-periodic with respect to their first arguments, respectively.