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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Existence of Periodic Solutions for a Delayed Ratio-Dependent Three-Species Predator-Prey Diffusion System on Time Scales | Hindawi Publishing Corporation Advances in Difference Equations Volume 2009 Article ID 141589 15 pages doi 10.1155 2009 141589 Research Article Existence of Periodic Solutions for a Delayed Ratio-Dependent Three-Species Predator-Prey Diffusion System on Time Scales Zhenjie Liu School of Mathematics and Computer Harbin University Harbin Heilongjiang 150086 China Correspondence should be addressed to Zhenjie Liu liouj2008@126.com Received 3 September 2008 Accepted 21 January 2009 Recommended by Binggen Zhang This paper investigates the existence of periodic solutions of a ratio-dependent predator-prey diffusion system with Michaelis-Menten functional responses and time delays in a two-patch environment on time scales. By using a continuation theorem based on coincidence degree theory we obtain suffcient criteria for the existence of periodic solutions for the system. Moreover when the time scale T is chosen as R or Z the existence of the periodic solutions of the corresponding continuous and discrete models follows. Therefore the methods are unified to provide the existence of the desired solutions for the continuous differential equations and discrete difference equations. Copyright 2009 Zhenjie Liu. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction The traditional predator-prey model has received great attention from both theoretical and mathematical biologists and has been studied extensively e.g. see 1-4 and references therein . Based on growing biological and physiological evidences some biologists have argued that in many situations especially when predators have to search for food and therefore have to share or compete for food the functional response in a prey-predator model should be ratio-dependent which can be roughly stated as that the per capita predator growth rate should be a .
