A discrete-time delayed diffusion model governed by backward difference equations is investigated. By using the coincidence degree and the related continuation theorem as well as some priori estimates, easily verifiable sufficient criteria are established for the existence of positive periodic solutions. where x i (t) represents the prey population in the ith patch (i = 1,2), and x 3 (t) represents the predator population, τ > 0 is a constant delay due to gestation, and D i (t) denotes the

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... rsal rate of the prey in the ith patch (i = 1,2). D i (t) (i = 1,2), a i (t) (i = 1,2,3), a 11 (t), a 13 (t), a 22 (t), a 31 (t), and m(t) are strictly positive continuous ω-periodic functions. They proved that system (1.1) has at least one positive ω-periodic solution if the conditions a 31 (t) > a 3 (t) and m(t)a 1 (t) > a 13 (t) are satisfied.