捕食者食饵均染病的入侵反应扩散捕食系统中扩散的作用
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摘要
研究了捕食者食饵均染病的入侵反应扩散捕食系统.利用特征值方法和构造Lyapunov函数,获得了入侵扩散对正常数平衡解的影响,当入侵扩散系数充分大时,导致平衡态失稳.进一步,利用拓扑度方法,证明了在一定条件下入侵扩散系数很大,自扩散充分小时,有非常数正平衡解存在.
An invasion-diffusion prey-predator epidemic system with disease infection in both populations was studied. The influence of invasion diffusion on the equilibrium solutions of positive constants was obtained through analysis of the eigenvalue and construction of the Lyapunov function. Furthermore, with the topological method, it was proved that the coefficient of invasion diffusion will be big enough while the self-diffusion coefficient is sufficiently small, then there exists a positive non-constant equilibrium solution.
An invasion-diffusion prey-predator epidemic system with disease infection in both populations was studied. The influence of invasion diffusion on the equilibrium solutions of positive constants was obtained through analysis of the eigenvalue and construction of the Lyapunov function. Furthermore, with the topological method, it was proved that the coefficient of invasion diffusion will be big enough while the self-diffusion coefficient is sufficiently small, then there exists a positive non-constant equilibrium solution.
引文
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[8] HEISH Y H, HSIAO C K. Predator-prey model with disease infection in both populations[J]. Mathematical Medicine and Biology: a Journal of the IMA, 2008, 25(3): 247-266.
[9] DAS K P, KUNDU K, CHATTOPADHYAY J. A predator-prey mathematical model with both the populations affected by diseases[J]. Ecological Complexity, 2011, 8: 68-80.
[10] DAS K P, CHATTOPADHYAY J. A mathematical study of a predator-prey model with disease circulating in the both populations[J]. International Journal of Biomathematics, 2015, 8(2): 1-27.
[11] 张丽娜, 鲁引儿. 具有Holling-Ⅲ型功能反应的捕食者-食饵扩散模型中避难所的影响[J]. 应用数学, 2017, 30(2): 359-364.(ZHANG Lina, LU Yiner. Effect of a prey refuge on a predator-prey model with diffusion and Holling type Ⅲ response function[J]. Mathematica Applicata, 2017, 30(2): 359-364.(in Chinese))
[12] OKUBO A. Diffusion and Ecological Problems: Mathematical Models[M]. New York: Springer Verlag, 1980.
[13] 李成林. 捕食者带有疾病的入侵反应扩散捕食系统的空间斑图[J]. 应用数学学报, 2016, 39(6): 832-846.(LI Chenling. Spatiotemporal pattern formation of an invasion-diffusion predator-prey system with disease in the predator[J]. Acta Mathematicae Applicatae Sinica, 2016, 39(6): 832-846.(in Chinese))
[14] 祖力, 黄冬冬, 柳扬. 捕食者和食饵均带有扩散的随机捕食-食饵模型动力学分析[J]. 应用数学和力学, 2017, 38(3): 355-368.(ZU Li, HUANG Dongdong, LIU Yang. Dynamics of dual-dispersal predator-prey systems under stochastic perturbations[J]. Applied Mathematics and Mechanics, 2017, 38(3): 355-368.(in Chinese))
[15] YANG K. Delay Differential Equation Application in Population Dynamics[M]. Boston: Academic Press, 1993.
[16] LIN C S, NIW M, TAKAGI I. Large amplitude stationary solutions to a chemotaxis system[J]. Journal of Differential Equations, 1988, 72(1): 1-27.
[2] VENTURINO E. Epidemics in predator-prey models: disease in the prey[J]. Mathematical Population Dynamics: Analysis of Heterogeneity, 1995, 1: 381-393.
[3] XIAO Y, CHEN L. Modeling and analysis of a predator-prey model with disease in the prey[J]. Mathematical Bioscience, 2001, 171(1): 59-82.
[4] CHATTOPADHYAY J, ARINO O. A predator-prey model with disease in the prey[J]. Nonlinear Analysis: Theory, Methods and Applications, 1999, 36(6): 747-766.
[5] 孙树林, 原存德. 捕食者具有流行疾病的捕食-被捕食模型的分析[J]. 生物数学学报, 2006, 21(1): 97-104.(SUN Shuling, YUAN Cunde. On the analysis of predator-prey model with epidemic in the predator[J]. Journal of Biomathematics, 2006, 21(1): 97-104.(in Chinese))
[6] VENTURINO E. Epidemics in predator-prey models: disease in the predators[J]. Mathematical Medicine and Biology: a Journal of the IMA, 2002, 19(3): 185-205.
[7] DAS K P. A study of chaotic dynamics and its possible control in a predator prey model with disease in the predator[J]. Journal of Dynamical and Control Systems, 2015, 21(4): 605-624.
[8] HEISH Y H, HSIAO C K. Predator-prey model with disease infection in both populations[J]. Mathematical Medicine and Biology: a Journal of the IMA, 2008, 25(3): 247-266.
[9] DAS K P, KUNDU K, CHATTOPADHYAY J. A predator-prey mathematical model with both the populations affected by diseases[J]. Ecological Complexity, 2011, 8: 68-80.
[10] DAS K P, CHATTOPADHYAY J. A mathematical study of a predator-prey model with disease circulating in the both populations[J]. International Journal of Biomathematics, 2015, 8(2): 1-27.
[11] 张丽娜, 鲁引儿. 具有Holling-Ⅲ型功能反应的捕食者-食饵扩散模型中避难所的影响[J]. 应用数学, 2017, 30(2): 359-364.(ZHANG Lina, LU Yiner. Effect of a prey refuge on a predator-prey model with diffusion and Holling type Ⅲ response function[J]. Mathematica Applicata, 2017, 30(2): 359-364.(in Chinese))
[12] OKUBO A. Diffusion and Ecological Problems: Mathematical Models[M]. New York: Springer Verlag, 1980.
[13] 李成林. 捕食者带有疾病的入侵反应扩散捕食系统的空间斑图[J]. 应用数学学报, 2016, 39(6): 832-846.(LI Chenling. Spatiotemporal pattern formation of an invasion-diffusion predator-prey system with disease in the predator[J]. Acta Mathematicae Applicatae Sinica, 2016, 39(6): 832-846.(in Chinese))
[14] 祖力, 黄冬冬, 柳扬. 捕食者和食饵均带有扩散的随机捕食-食饵模型动力学分析[J]. 应用数学和力学, 2017, 38(3): 355-368.(ZU Li, HUANG Dongdong, LIU Yang. Dynamics of dual-dispersal predator-prey systems under stochastic perturbations[J]. Applied Mathematics and Mechanics, 2017, 38(3): 355-368.(in Chinese))
[15] YANG K. Delay Differential Equation Application in Population Dynamics[M]. Boston: Academic Press, 1993.
[16] LIN C S, NIW M, TAKAGI I. Large amplitude stationary solutions to a chemotaxis system[J]. Journal of Differential Equations, 1988, 72(1): 1-27.