# American Institute of Mathematical Sciences

September  2006, 5(3): 423-433. doi: 10.3934/cpaa.2006.5.423

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

 1 College of Mathematics and Econometrics, Hunan University, Hunan 410082, China, China

Received  August 2005 Revised  January 2006 Published  June 2006

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.

Citation: Zhicheng Wang, Jun Wu. Existence of positive periodic solutions for delayed ratio-dependent predator-prey system with stocking. Communications on Pure & Applied Analysis, 2006, 5 (3) : 423-433. doi: 10.3934/cpaa.2006.5.423
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