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# Existence of positive periodic solutions for delayed ratio-dependent predator-prey system with stocking

• Applying the generalized Mawhin's continuation theorem of coincidence degree, we obtain some easily verifiable conditions for the existence of the positive periodic solutions of the following system

$x_1'(t)=h_1(t,x_1(t))(a_1(t)-a_{1 1}(t)x_1(t)-\frac{a_{1 3}(t)x_3(t)}{m(t)x_3(t)+x_1(t)})+D_1(t)(x_2(t)-x_1(t))+S_1(t),$

$x_2'(t)=h_2(t,x_2(t))(a_2(t)-a_{2 2}(t)x_2(t))+D_2(t)(x_1(t)-x_2(t))+S_2(t),$

$x_3'(t)=h_3(t,x_3(t))(-a_3(t)+\frac{a_{3 1}(t)x_1(t-\tau)}{m(t)x_3(t-\tau)+x_1(t-\tau)})+S_3(t).$

Some corresponding results are generalized or improved.

Mathematics Subject Classification: Primary: 34K15, 34K60; Secondary: 45M20.

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