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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

Zhicheng Wang 1, and  Jun Wu 1,

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.

Keywords: Positive periodic solution, ratio-dependent system, stocking rate, continuation theorem., predator-prey system.
Mathematics Subject Classification: Primary: 34K15, 34K60; Secondary: 45M2.
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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