Lengyel-Epstein方程与反应—扩散—迁徙方程
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摘要
Lengyel-Epstein方程的提出来自于对CIMA反应实验的建模,而反应-扩散-迁徙模型则来自于生物数学中对种群竞争的建模。本文主要研究了这两类方程中不同的参数取值对方程解的稳定性的影响。具体的说,对于Lengyel-Epstein方程,本文将给出一个关于a的取值范围,使得方程在满足这个条件时,方程的解具有全局渐近稳定性；对于反应-扩散-迁徙模型,本文主要通过对竞争模型的分析,得到了方程中含有的参数在取值不同时,对模型稳定性的影响。
本论文的内容安排如下：第一章是对Lengyel-Epstein方程和反应-扩散-迁徙模型做一个简单的叙述；第二章则介绍了与本论文相关的一些预备知识；第三章讨论Lengyel-Epstein方程中注入催化剂的浓度对常数解稳定性的影响,通过构造适当的Lyapunov函数,证明了当注入催化剂的浓度不是很大时,Lengyel-Epstein方程的常数解是全局渐进稳定的,对任意的初值,方程解最终一致收敛到这个常数解。第四章则主要研究了在空间资源分布不均匀的环境下。含有两个竞争种群的反应-扩散-迁徙模型,竞争结果对模型中各个参数的依赖性.通过对各个参数的扰动分析,得到了一些结果.
Lengyel-Epstein equation is Proposed from the Modeling experiments on the CIMA reaction, while the Reaction-Diffusion-Advection model is derived from bio-mathematics to the Population competitive model. The main work of this paper is studies the stability of two types of equations with different values of the parameters. Specifically, for Lengyel-Epstein equation, we get a rage of a, when a satisfy the conditions, the solutions of the Lengyel-Epstein equation are all global asymptotic stability; For the Reaction-Diffusion-Advection model, through the analysis of com-petition model, we get the Stability of the model when the parameters are different.
This paper is organized as follows:In the first chapter, we make a simple introduction to "Lengyel-Epstein equation" and "reaction-diffusion-advection model". Some notations and preliminaries relate to the paper are described in the second chapter. we consider the in-fluence of the feed concentration of activator in the Lengyel-Epstein reaction-diffusion system. By constructing a proper Lyapunov function, we show that when the feed concentration is small enough, the constant equilibrium solution of the Lengyel-Epstein reaction-diffusion system is globally asymptotically stable. We also show that all solutions converge uniformly to the con-stant equilibrium solution. Finally, in the fourth chapter, we study a competing species model for tow organisms with different feed inhabiting a spatially heterogeneous environment. by using the perturbation analysis, we get some results.
本论文的内容安排如下：第一章是对Lengyel-Epstein方程和反应-扩散-迁徙模型做一个简单的叙述；第二章则介绍了与本论文相关的一些预备知识；第三章讨论Lengyel-Epstein方程中注入催化剂的浓度对常数解稳定性的影响,通过构造适当的Lyapunov函数,证明了当注入催化剂的浓度不是很大时,Lengyel-Epstein方程的常数解是全局渐进稳定的,对任意的初值,方程解最终一致收敛到这个常数解。第四章则主要研究了在空间资源分布不均匀的环境下。含有两个竞争种群的反应-扩散-迁徙模型,竞争结果对模型中各个参数的依赖性.通过对各个参数的扰动分析,得到了一些结果.
Lengyel-Epstein equation is Proposed from the Modeling experiments on the CIMA reaction, while the Reaction-Diffusion-Advection model is derived from bio-mathematics to the Population competitive model. The main work of this paper is studies the stability of two types of equations with different values of the parameters. Specifically, for Lengyel-Epstein equation, we get a rage of a, when a satisfy the conditions, the solutions of the Lengyel-Epstein equation are all global asymptotic stability; For the Reaction-Diffusion-Advection model, through the analysis of com-petition model, we get the Stability of the model when the parameters are different.
This paper is organized as follows:In the first chapter, we make a simple introduction to "Lengyel-Epstein equation" and "reaction-diffusion-advection model". Some notations and preliminaries relate to the paper are described in the second chapter. we consider the in-fluence of the feed concentration of activator in the Lengyel-Epstein reaction-diffusion system. By constructing a proper Lyapunov function, we show that when the feed concentration is small enough, the constant equilibrium solution of the Lengyel-Epstein reaction-diffusion system is globally asymptotically stable. We also show that all solutions converge uniformly to the con-stant equilibrium solution. Finally, in the fourth chapter, we study a competing species model for tow organisms with different feed inhabiting a spatially heterogeneous environment. by using the perturbation analysis, we get some results.
引文
[1]TURNING A. The Chemical Basis of Morphogenesis. Phil Trans Roy Soc London,1952,237:37-72.
[2]KONDO S, ASAI R, A Reaction-Diffusion Wave on The Skin of The Marine Angelfish Pomacanthus. Nature, 1995,376:765-768.
[3]LENGYEL, I, EPSTEIN, I R. Modeling of Turing structure in the chlorite-malonic-acid-starch reaction system. Science,1991,251:650-652.
[4]NI W M, TANG M. Tuing patterns in Lengel-Epstien system for the CIMA reaction. Tran Amer Math Soc,2005, 357:3935-3969.
[5]JANG J, NI W M, TANG M. Global bifurcation and structure of Turing patterns in the 1-D Lengel-Epstein model. J Dynam Differential Equations,2005,16(2):297-320.
[6]YI F, WEI J, SHI J. Globle asymptotical behavior of the Lengyel-Epstein reaction-diffusion system. Appl Math Lett,2009,22(1):52-55.
[7]YI F, WEI J, SHI J. Diffusion-driyen instability and bifurcation in the Lengel-Epstein system. Nonlinear Anal, Real World Appl,2008,9:1038-1051.
[8]Hsu S B. A survey of constructing Laypunov function for mathematical models in populaion biology. Taiwanese J Math,2005,9(2):151-173.
[9]张锦炎,冯贝叶,常微分方程几何理论与分支问题(第三版),北京大学出版社,2000.
[10]A. Okubo, S.A. Levin, Diffusion and Ecological Problems:Modern Perspectives, second ed., Springer-Verlag, New York,2001.
[11]R.S. Cantrell, C. Cosner, Y. Lou, Movement towards better environments and the evolution of rapid diffusion, Math. Biosci.204 (2006) 199-214.
[12]R.S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations. Wiley, Chichester, West Sussex, U.K.,2003.
[13]J. Dockery, V. Hutson, K. Mischaikow, and M. Pernarowski, The evolution of slow dispersal rates:a reaction-diffusion model. J. Math. Biol.37 (1998),61.
[14]S-B.HSU, A survey of constructing Laypunov function for mathematical models in populaion biology, Taiwanese J.math,2005,9(2):151-173.
[15]R.S. Cantrell, C. Cosner, Y. Lou, Advection-mediated coexistence of competing species, Proc. R. Soc. Edin., to appear.
[16]F. Belgacem, C. Cosner, The effects of dispersal along environmental gradients on the dynamics of populations in heterogeneous environments, Can. Appl. Math. Quar.3 (1995) 379.
[17]P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics 247, Longman, Harlow, Essex, UK,1991.
[18]R.S. Cantrell, C. Cosner, Y. Lou, Approximating the ideal free distribution via reaction-diffusion-advection equations, J. Differential Equations 2008,245(3687-3703).
[19]T. Hillen, Transport equations with resting phases, Eur. J. Appl. Math.14 (2003) 613.
[20]P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics 247, Longman, Harlow, Essex, UK,1991.
[21]H. Smith, Monotone Dynamical Systems. Mathematical Surveys and Monographs 41. American Mathematical Society, Providence, Rhode Island, USA,1995.
[22]S. Senn, On a nonlinear elliptic eigenvalue problem with Neumann boundary conditions with an application to population genetics, Comm. PDE 8 (1983) 1199.
[23]S. Senn, P. Hess, On positive solutions of a linear elliptic boundary value problem with Neumann boundary conditions, Math. Ann.258 (1982) 459.
[24]F. Belgacem, Boundary value problems with indefinite weights:variational formulations of the principal eigen-value and applications, Pitman Research Notes in Mathematics 368, Longman, Harlow, Essex, UK,1997.
[25]R.S. Cantrell, C. Cosner, On the steady-state problem for the Volterra-Lotka competition model with diffusion, Houston J. Math.13 (1987) 337.
[26]R.S. Cantrell, C. Cosner, Y. Lou, Multiple reversals of competitive dominance in ecological reserves via external habitat degradation, J. Dynamics Differential Equations 16 (2004) 973.
[27]R.G. Casten, C. Holland, Instability results for reaction-diffusion equations with Neumann boundary condi-tions, J. Differential Equations 27 (1978) 266.
[28]H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci.15(1979)401.
[29]D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equation of Second Order,2nd ed., Springer-Verlag/Berlin/Heidelberg/New York/Tokyo,1983.11.
[30]J.K. Hale, G. Raugel, Reaction-diffusion equation on thin domains, J. Math. Pures Appl.71 (1992) 33-95.
[31]J.K. Hale, G. Raugel, A reaction-diffusion equation on a thin L-shaped domain, Proc. Roy. Soc. Edinburgh Sect. A 125(1995)283-327.
[32]E. Yanagida, Existence of stable stationary solutions of scalar reaction-diffusion equations in thin tubular domains, Appl. Anal.36 (1990) 171-188.
[33]Y. Kan-on, E. Yanagida, Existence of non-constant stable equilibria in competition-diffusion equations, Hi-roshima Math. J.23 (1993) 193-221.
[34]H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci.15 (1979) 401-454.
[35]Y. Du, Realization of prescribed patterns in the competition model, J. Differential Equations 193 (2003) 147-179.
[36]A. Okubo, Diffusion and Ecological Problems:Mathematical Models, in:Biomathematics Texts, Vol.10, Springer, Berlin,1980.
[2]KONDO S, ASAI R, A Reaction-Diffusion Wave on The Skin of The Marine Angelfish Pomacanthus. Nature, 1995,376:765-768.
[3]LENGYEL, I, EPSTEIN, I R. Modeling of Turing structure in the chlorite-malonic-acid-starch reaction system. Science,1991,251:650-652.
[4]NI W M, TANG M. Tuing patterns in Lengel-Epstien system for the CIMA reaction. Tran Amer Math Soc,2005, 357:3935-3969.
[5]JANG J, NI W M, TANG M. Global bifurcation and structure of Turing patterns in the 1-D Lengel-Epstein model. J Dynam Differential Equations,2005,16(2):297-320.
[6]YI F, WEI J, SHI J. Globle asymptotical behavior of the Lengyel-Epstein reaction-diffusion system. Appl Math Lett,2009,22(1):52-55.
[7]YI F, WEI J, SHI J. Diffusion-driyen instability and bifurcation in the Lengel-Epstein system. Nonlinear Anal, Real World Appl,2008,9:1038-1051.
[8]Hsu S B. A survey of constructing Laypunov function for mathematical models in populaion biology. Taiwanese J Math,2005,9(2):151-173.
[9]张锦炎,冯贝叶,常微分方程几何理论与分支问题(第三版),北京大学出版社,2000.
[10]A. Okubo, S.A. Levin, Diffusion and Ecological Problems:Modern Perspectives, second ed., Springer-Verlag, New York,2001.
[11]R.S. Cantrell, C. Cosner, Y. Lou, Movement towards better environments and the evolution of rapid diffusion, Math. Biosci.204 (2006) 199-214.
[12]R.S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations. Wiley, Chichester, West Sussex, U.K.,2003.
[13]J. Dockery, V. Hutson, K. Mischaikow, and M. Pernarowski, The evolution of slow dispersal rates:a reaction-diffusion model. J. Math. Biol.37 (1998),61.
[14]S-B.HSU, A survey of constructing Laypunov function for mathematical models in populaion biology, Taiwanese J.math,2005,9(2):151-173.
[15]R.S. Cantrell, C. Cosner, Y. Lou, Advection-mediated coexistence of competing species, Proc. R. Soc. Edin., to appear.
[16]F. Belgacem, C. Cosner, The effects of dispersal along environmental gradients on the dynamics of populations in heterogeneous environments, Can. Appl. Math. Quar.3 (1995) 379.
[17]P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics 247, Longman, Harlow, Essex, UK,1991.
[18]R.S. Cantrell, C. Cosner, Y. Lou, Approximating the ideal free distribution via reaction-diffusion-advection equations, J. Differential Equations 2008,245(3687-3703).
[19]T. Hillen, Transport equations with resting phases, Eur. J. Appl. Math.14 (2003) 613.
[20]P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics 247, Longman, Harlow, Essex, UK,1991.
[21]H. Smith, Monotone Dynamical Systems. Mathematical Surveys and Monographs 41. American Mathematical Society, Providence, Rhode Island, USA,1995.
[22]S. Senn, On a nonlinear elliptic eigenvalue problem with Neumann boundary conditions with an application to population genetics, Comm. PDE 8 (1983) 1199.
[23]S. Senn, P. Hess, On positive solutions of a linear elliptic boundary value problem with Neumann boundary conditions, Math. Ann.258 (1982) 459.
[24]F. Belgacem, Boundary value problems with indefinite weights:variational formulations of the principal eigen-value and applications, Pitman Research Notes in Mathematics 368, Longman, Harlow, Essex, UK,1997.
[25]R.S. Cantrell, C. Cosner, On the steady-state problem for the Volterra-Lotka competition model with diffusion, Houston J. Math.13 (1987) 337.
[26]R.S. Cantrell, C. Cosner, Y. Lou, Multiple reversals of competitive dominance in ecological reserves via external habitat degradation, J. Dynamics Differential Equations 16 (2004) 973.
[27]R.G. Casten, C. Holland, Instability results for reaction-diffusion equations with Neumann boundary condi-tions, J. Differential Equations 27 (1978) 266.
[28]H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci.15(1979)401.
[29]D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equation of Second Order,2nd ed., Springer-Verlag/Berlin/Heidelberg/New York/Tokyo,1983.11.
[30]J.K. Hale, G. Raugel, Reaction-diffusion equation on thin domains, J. Math. Pures Appl.71 (1992) 33-95.
[31]J.K. Hale, G. Raugel, A reaction-diffusion equation on a thin L-shaped domain, Proc. Roy. Soc. Edinburgh Sect. A 125(1995)283-327.
[32]E. Yanagida, Existence of stable stationary solutions of scalar reaction-diffusion equations in thin tubular domains, Appl. Anal.36 (1990) 171-188.
[33]Y. Kan-on, E. Yanagida, Existence of non-constant stable equilibria in competition-diffusion equations, Hi-roshima Math. J.23 (1993) 193-221.
[34]H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci.15 (1979) 401-454.
[35]Y. Du, Realization of prescribed patterns in the competition model, J. Differential Equations 193 (2003) 147-179.
[36]A. Okubo, Diffusion and Ecological Problems:Mathematical Models, in:Biomathematics Texts, Vol.10, Springer, Berlin,1980.
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