一类具自由扩散和交错扩散的传染病模型分析
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摘要
传统看来,流行病学和生态学是两个相互独立的研究领域,但是它们之间有着很大的联系.在自然界中我们经常会发现流行病学已经渗透到生态学领域中,从而使其动力系统发生很大改变.如稳定性的改变,解的存在性以及震荡问题改变等等.另外,人为的捕捞量在寄主-宿主系统中也起着至关重要的作用,合理的捕捞量可以消灭捕食系统宿主中的寄主.
基于此,本文中我们研究了疾病在捕食系统中的传播情况,并讨论相应的具齐次Neumann边界条件反应扩散方程组的平衡解的渐近性质.
第一章简要介绍相关工作的背景和发展概况,并阐述本文的研究内容.
第二章给出了该抛物问题解的先验估计及常微分系统稳态解存在的充分条件和渐近状态.
第三章用特征子空间的分解和线性化方法讨论具有自由扩散偏微分系统各平衡点解的局部稳定性,进一步地,我们给出了偏微分系统在平衡点的全局稳定性.我们还讨论了自由扩散系数对系统平衡解稳定性的影响,我们的结果表明:当扩散系数满足一定条件时,对于正平衡解来说,在常微分系统局部渐近稳定的条件下,自由扩散系数的引入导致了系统不稳定.
第四章我们引入了交错扩散系数来讨论系统在平衡点处的稳定性变化,给出因交错扩散系数而引起系统不稳定的相关条件.我们的结果表明,引入交错扩散系数,可以使在原自扩散动力学系统中稳定的平衡解变得不稳定.
第五章通过数值模拟和一些讨论来说明我们的结论.
Epidemiology and ecology are traditionally treated as independent research areas, but there are a lot of commonalities between these two fields. It is usually observed in nature that epidemiology has an encroachment into the later. Wherefore, it changes the system dynamics significantly. The predator-prey interaction in presence of parasites can produce more complex dynamics, such as switching of stability, extinction and oscillations and so on. On the other hand, another variable—the harvesting practices may play an important role in the host-parasite system. Reasonable harvesting can remove a parasite, in principle, from their host.
In this paper, we are concerned with the disease transmission in the predator-prey system, and consider the asymptotic behavior of the corresponding reaction -diffusion systems with homogeneous Neumann boundary conditions.
In Chapter 1, the background and history about the related work are first introduced and the major work of this paper is presented.
Chapter 2 is devoted to the prior estimates for the solutions of parabolic system and the sufficient conditions for the existence of each steady solution of the ODE system.
In Chapter 3, the local asymptotic stabilities around each of the equilibrium are discussed by using the characteristic decomposition and linearization. Furthermore, some sufficient conditions for the global asymptotic stabilities around some equilibriums are given.
The cross-diffusion is introduced in Chapter 4 to discuss the stability of the system at the equilibrium point, and the conditions for the instability caused by cross-diffusion coefficient are given. Our results show that cross-diffusion can induce the instability of an equilibrium which is stable for the kinetic system and for the self-diffusion reaction system.
Chapter 5 deals with numerical simulations and some discussions are presented to illustrate the main result.
基于此,本文中我们研究了疾病在捕食系统中的传播情况,并讨论相应的具齐次Neumann边界条件反应扩散方程组的平衡解的渐近性质.
第一章简要介绍相关工作的背景和发展概况,并阐述本文的研究内容.
第二章给出了该抛物问题解的先验估计及常微分系统稳态解存在的充分条件和渐近状态.
第三章用特征子空间的分解和线性化方法讨论具有自由扩散偏微分系统各平衡点解的局部稳定性,进一步地,我们给出了偏微分系统在平衡点的全局稳定性.我们还讨论了自由扩散系数对系统平衡解稳定性的影响,我们的结果表明:当扩散系数满足一定条件时,对于正平衡解来说,在常微分系统局部渐近稳定的条件下,自由扩散系数的引入导致了系统不稳定.
第四章我们引入了交错扩散系数来讨论系统在平衡点处的稳定性变化,给出因交错扩散系数而引起系统不稳定的相关条件.我们的结果表明,引入交错扩散系数,可以使在原自扩散动力学系统中稳定的平衡解变得不稳定.
第五章通过数值模拟和一些讨论来说明我们的结论.
Epidemiology and ecology are traditionally treated as independent research areas, but there are a lot of commonalities between these two fields. It is usually observed in nature that epidemiology has an encroachment into the later. Wherefore, it changes the system dynamics significantly. The predator-prey interaction in presence of parasites can produce more complex dynamics, such as switching of stability, extinction and oscillations and so on. On the other hand, another variable—the harvesting practices may play an important role in the host-parasite system. Reasonable harvesting can remove a parasite, in principle, from their host.
In this paper, we are concerned with the disease transmission in the predator-prey system, and consider the asymptotic behavior of the corresponding reaction -diffusion systems with homogeneous Neumann boundary conditions.
In Chapter 1, the background and history about the related work are first introduced and the major work of this paper is presented.
Chapter 2 is devoted to the prior estimates for the solutions of parabolic system and the sufficient conditions for the existence of each steady solution of the ODE system.
In Chapter 3, the local asymptotic stabilities around each of the equilibrium are discussed by using the characteristic decomposition and linearization. Furthermore, some sufficient conditions for the global asymptotic stabilities around some equilibriums are given.
The cross-diffusion is introduced in Chapter 4 to discuss the stability of the system at the equilibrium point, and the conditions for the instability caused by cross-diffusion coefficient are given. Our results show that cross-diffusion can induce the instability of an equilibrium which is stable for the kinetic system and for the self-diffusion reaction system.
Chapter 5 deals with numerical simulations and some discussions are presented to illustrate the main result.
引文
[1] P. A. Amundsen, R. Kristoffersen, Infection of white fish: a case study in parasite control, Can. J. Zoo. , 68 (1990) 1187.
[2] R. M. Anderson, R. M. May, The invasion, persistence and spread of infectious diseases within animal and plant communities, Phil. Trans. R. Soc. London,B314 (1986), 533-570.
[3] E. Beltrami, T. O. Carroll, Modeling the role of viral disease in recurrent phytoplankton blooms, J. Math. Biol.,32 (1994), 857-863.
[4] N. Bairagi, S. Chaudhuri, J. Chattopadhyay, Harvesting as a disease control measure in an eco-epidemiological system-A theoretical study, Math. Bio.,217 (2009) 134-144.
[5] N. Bairagi, P. K. Roy, J. Chattopadhyay, Role of infection on the stability of a predator-prey system with several response functions-A comparative study, Journal of Theoretical Biology 248 (2007) 10–25.
[6] J. Chattopadhyay, N. Bairagi, Pelicans at risk in Salton Sea- an eco-epidemiological model,Ecol. Model. , 136 (2001) 103–112.
[7] J. Chattopadhyay, O. Arino, A Predator-prey model with disease in the prey, Nonlinear Anal., 36 (2) (1999), 749-766.
[8] J. Chattopadhyay, S. Pal, A. E. Abdllaoui, Classical Predator-Prey system with infection of prey population-a mathematical model, Math. Appl. Sci. 26 (2003), 1211-1222.
[9] M. E. Gilpin, Spiral chaos in a Predator-Prey model, Am. Nat. 113(1979),306-308.
[10] K. P. Hadeler, H. I. Freedman, Predator-Prey Population with Parasite infection, J. Math. Biol. 27(1989), 609-631.
[11] L. T. Han, Z. E. Ma, S. Tan, An SIRS epidemic model of two competitive species, Math. Computation,37 (2003), 87-108.
[12] J. Hofbauer, K. Sigmund K, The theory of Evolution and Dynamical Systems, Cambridge University Press, Cambridge, 1988.
[13] W. O. Kermack, A. G. Mckendrick, Contributions to the mathematical theory of epidemics,Proceedings of the Royal Society of London,Series A, 115 (1927), 700-721.
[14] A. J. Lotka, Elements of Physical biology, Williams and Wilkins Co.,Inc. Baltimore, 1924.
[15] Z. G. Lin, M. Pedersen, Stability in a diffusive food-chain model with Michaelis-Menten functional response, Nonlinear Analysis., 57 (2004), 421-433.
[16] C. V. Pao, Convergence of solutions of reaction-diffusion systems with time delays, Nonlinear Anal., 48(2002), 349-362.
[17] C. V. Pao, Dynamics of nonlinear parabolic systems with time delays, J. Math. Anal. Appl., 198 (1996), 751-779.
[18] C. V. Pao, Parabolic system in unbounded domains I: existence and dynamics, J. Math. Anal. Appl., 205 (1997), 157-185.
[19] C. V. Pao,Nonlinear Parabolic and Elliptic Equations, Plenum,New York,1992, 201.
[20] R. O. Peterson, R.E. Page, Wolf density as a predictor of predation rate, Swedish Wildlife Research, 1 (Suppl.), (1987), 771-773.
[21] G. Slack, Salton Sea Sickness, Pacific Discovery, Winter, 1997.
[22] P. K. Upadhya, N. Bairagi, J. Chattopadhyay, Chaos in eco-epidemiological problem of the Salton Sea and its possible control, Mathematics and Computation, 196 (2008) 392-401.
[23] V. Volterra, Variations efluttauazionidel numbered individual in specie animals conviventi, Mem. Acad. Lincei,2 (1926),31-33.(Translation in a appendix to Chap main’s, Animal Ecology New York,1931).
[24] E. Venturino, Epidemics in predator-prey models: disease in the prey, InMathematical Population Dynamics: Analysis of Heterogeneity, Arino. O, Axelrod, Kimmel M, Langlais M (eds), 1 (1995), 381-393.
[25] E. Venturino, The influence of diseases on Lotka-Volterra systems, J. Math., 24 (1) (1994), 389-402.
[26] Y. N. Xiao, L. S. Chen, Modeling and analysis of a Predator-Prey model with disease in the prey, Math. Bio.,171 (2001), 59-82,
[27] Y. N. Xiao, L. S. Chen, A ratio-dependent Predator-Prey model with disease in the Prey, Applied Mathematics and Computation, 131 (2002), 397-414.
[28] Y. N. Xiao, L. S. Chen, Analysis of a three Species Ecoepidem-iological model, Math Appl., 258 (2) (2001), 733-754.
[29]陈兰荪,生物数学引论,北京:科学出版社,1983.
[30]陈兰荪,宋新宇,陆征一,数学生态学模型与研究方法,成都:四川科学技术出版社,2003.
[31]马知恩,种群生态学的数学建模与研究,合肥:安徽教育出版社,1996
[32]孙树林,原存孙,捕食者具有传染病的捕食——被捕食(SI)模型分析,生物数学学报,21(1)(2006),97-104.
[33]马之恩,周义仓,王稳地,靳祯,传染病动力学的数学建模与研究,北京:科学出版社,2004.
[34]王明新,非线性抛物方程,科学出版社, 1993.
[35]叶其孝,李正元,反应扩散方程引论,科学出版社, 1995.
[2] R. M. Anderson, R. M. May, The invasion, persistence and spread of infectious diseases within animal and plant communities, Phil. Trans. R. Soc. London,B314 (1986), 533-570.
[3] E. Beltrami, T. O. Carroll, Modeling the role of viral disease in recurrent phytoplankton blooms, J. Math. Biol.,32 (1994), 857-863.
[4] N. Bairagi, S. Chaudhuri, J. Chattopadhyay, Harvesting as a disease control measure in an eco-epidemiological system-A theoretical study, Math. Bio.,217 (2009) 134-144.
[5] N. Bairagi, P. K. Roy, J. Chattopadhyay, Role of infection on the stability of a predator-prey system with several response functions-A comparative study, Journal of Theoretical Biology 248 (2007) 10–25.
[6] J. Chattopadhyay, N. Bairagi, Pelicans at risk in Salton Sea- an eco-epidemiological model,Ecol. Model. , 136 (2001) 103–112.
[7] J. Chattopadhyay, O. Arino, A Predator-prey model with disease in the prey, Nonlinear Anal., 36 (2) (1999), 749-766.
[8] J. Chattopadhyay, S. Pal, A. E. Abdllaoui, Classical Predator-Prey system with infection of prey population-a mathematical model, Math. Appl. Sci. 26 (2003), 1211-1222.
[9] M. E. Gilpin, Spiral chaos in a Predator-Prey model, Am. Nat. 113(1979),306-308.
[10] K. P. Hadeler, H. I. Freedman, Predator-Prey Population with Parasite infection, J. Math. Biol. 27(1989), 609-631.
[11] L. T. Han, Z. E. Ma, S. Tan, An SIRS epidemic model of two competitive species, Math. Computation,37 (2003), 87-108.
[12] J. Hofbauer, K. Sigmund K, The theory of Evolution and Dynamical Systems, Cambridge University Press, Cambridge, 1988.
[13] W. O. Kermack, A. G. Mckendrick, Contributions to the mathematical theory of epidemics,Proceedings of the Royal Society of London,Series A, 115 (1927), 700-721.
[14] A. J. Lotka, Elements of Physical biology, Williams and Wilkins Co.,Inc. Baltimore, 1924.
[15] Z. G. Lin, M. Pedersen, Stability in a diffusive food-chain model with Michaelis-Menten functional response, Nonlinear Analysis., 57 (2004), 421-433.
[16] C. V. Pao, Convergence of solutions of reaction-diffusion systems with time delays, Nonlinear Anal., 48(2002), 349-362.
[17] C. V. Pao, Dynamics of nonlinear parabolic systems with time delays, J. Math. Anal. Appl., 198 (1996), 751-779.
[18] C. V. Pao, Parabolic system in unbounded domains I: existence and dynamics, J. Math. Anal. Appl., 205 (1997), 157-185.
[19] C. V. Pao,Nonlinear Parabolic and Elliptic Equations, Plenum,New York,1992, 201.
[20] R. O. Peterson, R.E. Page, Wolf density as a predictor of predation rate, Swedish Wildlife Research, 1 (Suppl.), (1987), 771-773.
[21] G. Slack, Salton Sea Sickness, Pacific Discovery, Winter, 1997.
[22] P. K. Upadhya, N. Bairagi, J. Chattopadhyay, Chaos in eco-epidemiological problem of the Salton Sea and its possible control, Mathematics and Computation, 196 (2008) 392-401.
[23] V. Volterra, Variations efluttauazionidel numbered individual in specie animals conviventi, Mem. Acad. Lincei,2 (1926),31-33.(Translation in a appendix to Chap main’s, Animal Ecology New York,1931).
[24] E. Venturino, Epidemics in predator-prey models: disease in the prey, InMathematical Population Dynamics: Analysis of Heterogeneity, Arino. O, Axelrod, Kimmel M, Langlais M (eds), 1 (1995), 381-393.
[25] E. Venturino, The influence of diseases on Lotka-Volterra systems, J. Math., 24 (1) (1994), 389-402.
[26] Y. N. Xiao, L. S. Chen, Modeling and analysis of a Predator-Prey model with disease in the prey, Math. Bio.,171 (2001), 59-82,
[27] Y. N. Xiao, L. S. Chen, A ratio-dependent Predator-Prey model with disease in the Prey, Applied Mathematics and Computation, 131 (2002), 397-414.
[28] Y. N. Xiao, L. S. Chen, Analysis of a three Species Ecoepidem-iological model, Math Appl., 258 (2) (2001), 733-754.
[29]陈兰荪,生物数学引论,北京:科学出版社,1983.
[30]陈兰荪,宋新宇,陆征一,数学生态学模型与研究方法,成都:四川科学技术出版社,2003.
[31]马知恩,种群生态学的数学建模与研究,合肥:安徽教育出版社,1996
[32]孙树林,原存孙,捕食者具有传染病的捕食——被捕食(SI)模型分析,生物数学学报,21(1)(2006),97-104.
[33]马之恩,周义仓,王稳地,靳祯,传染病动力学的数学建模与研究,北京:科学出版社,2004.
[34]王明新,非线性抛物方程,科学出版社, 1993.
[35]叶其孝,李正元,反应扩散方程引论,科学出版社, 1995.
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