| Positive solutions for a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-Ⅱ functional response |
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| 作者姓名： | ZHOU Jun KIM Chan-Gyun |
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| 作者单位： | School of Mathematics and Statistics,Southwest University;Department of Mathematics,College of William and Mary |
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| 基金项目： | supported by National Natural Science Foundation of China(Grant No.11201380);the Fundamental Research Funds for the Central Universities(Grant No.XDJK2012B007);Doctor Fund of Southwest University(Grant No.SWU111021);Educational Fund of Southwest University(Grant No.2010JY053);National Research Foundation of Korea Grant funded by the Korean Government(Ministry of Education,Science and Technology)(Grant No.NRF-2011-357-C00006) |
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| 摘 要： | We consider a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-II functional response.The main concern is the existence of positive solutions under the combined effect of cross-diffusion and Holling type-II functional response.Here,a positive solution corresponds to a coexistence state of the model.Firstly,we study the sufficient conditions to ensure the existence of positive solutions by using degree theory and analyze the coexistence region in parameter plane.In addition,we present the uniqueness of positive solutions in one dimension case.Secondly,we study the stability of the trivial and semi-trivial solutions by analyzing the principal eigenvalue of the corresponding linearized system,and then we characterize the stable/unstable regions of semi-trivial solutions in parameter plane.
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| 关 键 词： | Lotka-Volterra prey-predator model Holling type-II functional response cross-diffusion positive solutions coexistence uniqueness degree theory |
Positive solutions for a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-II functional response |
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| ZHOU Jun,KIM Chan-Gyun.Positive solutions for a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-II functional response[J].中国科学 数学(英文版),2014,57(5):991-1010. |
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| Authors: | Jun Zhou Chan-Gyun Kim |
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| Institution: | 1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China
2. Department of Mathematics, College of William and Mary, Williamsburg, VA, 23187-8795, USA
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| Abstract: | We consider a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-II functional response. The main concern is the existence of positive solutions under the combined effect of cross-diffusion and Holling type-II functional response. Here, a positive solution corresponds to a coexistence state of the model. Firstly, we study the sufficient conditions to ensure the existence of positive solutions by using degree theory and analyze the coexistence region in parameter plane. In addition, we present the uniqueness of positive solutions in one dimension case. Secondly, we study the stability of the trivial and semi-trivial solutions by analyzing the principal eigenvalue of the corresponding linearized system, and then we characterize the stable/unstable regions of semi-trivial solutions in parameter plane. |
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| Keywords: | Lotka-Volterra prey-predator model Holling type-II functional response cross-diffusion positive solutions coexistence uniqueness degree theory |
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