Periodic Solution for a Two-Species Nonautonomous Competition Lotka–Volterra Patch System with Time Delay☆☆☆

Abstract 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) − β(t) ∫ − τ 0 K(s)y(t + s)ds , is established, where ri(t), ai(t)(i = 1, 2, 3), Di(t)(i = 1, 2), bi(t)(i = 1, 3), and β(t) are all positive periodic continuous functions with period w > 0, τ is a nonnegative constant, and K(s) is a continuous nonnegative function on [− τ, 0].