Periodic solutions for delay Lotka-Volterra competition systems

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(t)(t) - Sigma(j=1j not equal i)(n)a(ij)(t) integral(-Tij)(0)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) - Sigma(j=1j not equal i)(n)a(ij)(t)u(j)(t-tau(j)(t,u(1)(t),..., u(n)(t)))], i = 1, 2,...,n, where r(i), a(ii) > 0, a(ij) greater than or equal to 0 (j not equal i, i, j = 1, 2,..., n) are continuous omega-periodic functions, T-ij epsilon (0, infinity)(j not equal i, i, j = 1, 2,..., n), K-ij epsilon C([- T-ij, 0], (0, infinity)), integral(-Tij)(0)K(ij)(s) ds = 1 (j not equal i, i, j = 1, 2,..., n), tau(i) epsilon C(Rn+1, R), and tau(i) (i = 1, 2,..., n) are omega-periodic with respect to their first arguments, respectively. (C) 2000 Academic Press.