Periodic solution for a two-species nonautonomous competition Lotka-Votterra patch system with time delay

By using the continuation theorem of coincidence degree theory, the existence of a positive periodic solution for a two-species nonautonomous competition Lotka-Volterra patch system with time delay, x'(1)(t) = x(1)(t)[r(1)(t) - a(1)(t)x(1)(t) - b(1)(t)y(t)] + D-1(t)[x(2)(t) - x(1)(t)], x'(2)(t) = x(2)(t)[r(2)(t) - a(2)(t)x(2)(t)] + D-2(t)[x(1)(t) - x(2)(t)], y'(t) = y(t)[r(3)(t) - a(3)(t)x(1)(t) - b(3)(t)y(t) - beta(t)f(tau)(0) K(s)y(t + s)ds], is established, where r(i)(t), a(i)(t) (i = 1, 2, 3), D-i(t) (i = 1, 2), b(i)(t) (i = 1, 3), and beta(t) are all positive periodic continuous functions with period to w > 0, tau is a nonnegative constant, and K(s) is a continuous nonnegative function on [- tau, 0]. (C) 2002 Elsevier Science.