Periodic solutions and stability for a delayed discrete ratio-dependent predator-prey system with Holling-type functional response

The existence of positive periodic solutions for a delayed discrete predator-prey model with Holling-type-III functional response N 1 ( k + 1 ) = N 1 ( k ) exp { b 1 ( k ) − a 1 ( k ) N 1 ( k − [ τ 1 ] ) − α 1 ( k ) N 1 ( k ) N 2 ( k ) / ( N 1 2 ( k ) + m 2 N 2 2 ( k ) ) } , N 2 ( k + 1 ) = N 2 ( k ) exp { − b 2 ( k ) + α 2 ( k ) N 1 2 ( k − [ τ 2 ] ) / ( N 1 2 ( k − [ τ 2 ] ) + m 2 N 2 2 ( k − [ τ 2 ] ) ) } is established by using the coincidence degree theory. We also present sufficient conditions for the globally asymptotical stability of this system when all the delays are zero. Our investigation gives an affirmative exemplum for the claim that the ratio-dependent predator-prey theory is more reasonable than the traditional prey-dependent predator-prey theory.