一类捕食—食饵模型解的性质
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摘要
当今，数学生物学已成为一个受到广泛关注的热门学科，人们对许多生命现象建立了数学模型，并应用现代数学理论不断地对其加以研究，取得了许多有价值的研究成果．由于种群间捕食关系的普遍存在性及重要性，捕食-食饵模型更加受到国内外学者的广泛关注．研究具有捕食-食饵关系的种群的共存性，稳定性或周期持续生存，对于保持生态平衡，保护生态环境甚至挽救濒临灭绝的珍稀生物等具有非常重要的实际意义．
二阶捕食-食饵系统的典型模型是Lotka-Volterra模型，它在种群动力学的理论研究中具有非常重要的地位，已被广泛的研究．针对具体的数学模型，一个关键的因素即所谓的“功能反应函数”，它表示食饵的种群密度关于时间的变化率．它不仅受食饵密度大小的影响，而且受捕食者本身密度的影响．Holling反应函数较为合理地反映了捕食者与食饵的相互作用关系，将其进行推广，得到一类捕食-食饵模型
本文运用非线性分析和非线性偏微分方程的知识，特别是抛物型方程(组)和对应椭圆型方程(组)的理论和方法，研究了以上捕食-食饵模型的动力学行为，包括正平衡解存在的充分条件、不同参数下平衡态系统正分歧解的结构和局部分歧解的稳定性以及解的渐近性和稳定性．所涉及的数学理论包括：上下解方法、比较原理、全局分歧理论、稳定性理论、拓扑度理论等．本文主要有三章内容：
第一章是绪论，主要介绍了问题背景、相关工作和本文的内容框架．
第二章研究了该模型正平衡解的性质，可分为两部分：第一部分运用极值原理、上下解方法和锥映射不动点指标理论得到正平衡解存在的充分条件；第二部分利用分歧理论给出了平衡态系统在不同参数下，正分歧解的结构，并讨论了局部分歧解的稳定性．
第三章研究了该模型解的渐近性和稳定性．运用上下解方法和稳定性理论，考察了平凡解、半平凡解处的渐近性情况和稳定性情况．
Today, the so called Mathematical Biology becomes an active branch of modern science. A lot of mathematical models are established successfully, and significant achievements are obtained. The dynamic interaction between predators and their prey has long been one of the dominant themes in mathematical biology due to its universal existence and importance. Studies on the coexistence, stability and persistence for predator-prey systems have very important practical significance to equilibrium of ecology, ecological environments preservation and even saving the rare and precious creature on the brink of extinction.
One classical model of predator-prey systems in ecology is Lotka-Volterra model, it is a kind of the most significant models in mathematical biology and has been widely studied. A crucial element of all models is the so-called "functional response " , which is the function representing the prey consumption per unit time. Holling functional response produces richer dynamics.
Modified the Holling functional response, we get a class of predator-prey systems as below
Mainly using the theories of nonlinear analysis and nonlinear partial differential equations, especially those of parabolic equations and corresponding elliptic equations, we have systematically studied the dynamical behavior of the above predator-prey model, such as several sufficient conditions for coexistence solutions of the steady-state, the global structure of the coexistence solutions, stability of positive steady states and asymptotic behavior and stability of positive solutions. The tools used here include super-sub solutions method, comparison principle, global bifurcation theory, linear stability theory, and fixed-point theory of topology.
The main contents and results in this paper are as follows:
section 1 is introduction, mainly introduces the background of the question, interrelated study and the main contents and results in this paper.
In section 2, we state our main analytical results on equilibria of the above system. First, several sufficient conditions for coexistence of the steady-state are given by the maximum principle, methods of the super-sub solution and the standard fixed-point index theory in cone. Second, the global structure of the coexistence solutions and their local stability are established by using bifurcation theory when the bifurcation parameters are different, stability of positive steady states.
Section 3 contains asymptotic behavior and stability of positive solutions. Asymptotic behavior and stability of positive solutions are given by methods of the supersub solution and linear stability theory.
二阶捕食-食饵系统的典型模型是Lotka-Volterra模型，它在种群动力学的理论研究中具有非常重要的地位，已被广泛的研究．针对具体的数学模型，一个关键的因素即所谓的“功能反应函数”，它表示食饵的种群密度关于时间的变化率．它不仅受食饵密度大小的影响，而且受捕食者本身密度的影响．Holling反应函数较为合理地反映了捕食者与食饵的相互作用关系，将其进行推广，得到一类捕食-食饵模型
本文运用非线性分析和非线性偏微分方程的知识，特别是抛物型方程(组)和对应椭圆型方程(组)的理论和方法，研究了以上捕食-食饵模型的动力学行为，包括正平衡解存在的充分条件、不同参数下平衡态系统正分歧解的结构和局部分歧解的稳定性以及解的渐近性和稳定性．所涉及的数学理论包括：上下解方法、比较原理、全局分歧理论、稳定性理论、拓扑度理论等．本文主要有三章内容：
第一章是绪论，主要介绍了问题背景、相关工作和本文的内容框架．
第二章研究了该模型正平衡解的性质，可分为两部分：第一部分运用极值原理、上下解方法和锥映射不动点指标理论得到正平衡解存在的充分条件；第二部分利用分歧理论给出了平衡态系统在不同参数下，正分歧解的结构，并讨论了局部分歧解的稳定性．
第三章研究了该模型解的渐近性和稳定性．运用上下解方法和稳定性理论，考察了平凡解、半平凡解处的渐近性情况和稳定性情况．
Today, the so called Mathematical Biology becomes an active branch of modern science. A lot of mathematical models are established successfully, and significant achievements are obtained. The dynamic interaction between predators and their prey has long been one of the dominant themes in mathematical biology due to its universal existence and importance. Studies on the coexistence, stability and persistence for predator-prey systems have very important practical significance to equilibrium of ecology, ecological environments preservation and even saving the rare and precious creature on the brink of extinction.
One classical model of predator-prey systems in ecology is Lotka-Volterra model, it is a kind of the most significant models in mathematical biology and has been widely studied. A crucial element of all models is the so-called "functional response " , which is the function representing the prey consumption per unit time. Holling functional response produces richer dynamics.
Modified the Holling functional response, we get a class of predator-prey systems as below
Mainly using the theories of nonlinear analysis and nonlinear partial differential equations, especially those of parabolic equations and corresponding elliptic equations, we have systematically studied the dynamical behavior of the above predator-prey model, such as several sufficient conditions for coexistence solutions of the steady-state, the global structure of the coexistence solutions, stability of positive steady states and asymptotic behavior and stability of positive solutions. The tools used here include super-sub solutions method, comparison principle, global bifurcation theory, linear stability theory, and fixed-point theory of topology.
The main contents and results in this paper are as follows:
section 1 is introduction, mainly introduces the background of the question, interrelated study and the main contents and results in this paper.
In section 2, we state our main analytical results on equilibria of the above system. First, several sufficient conditions for coexistence of the steady-state are given by the maximum principle, methods of the super-sub solution and the standard fixed-point index theory in cone. Second, the global structure of the coexistence solutions and their local stability are established by using bifurcation theory when the bifurcation parameters are different, stability of positive steady states.
Section 3 contains asymptotic behavior and stability of positive solutions. Asymptotic behavior and stability of positive solutions are given by methods of the supersub solution and linear stability theory.
引文
[1] J.Blat, K.J. Brown. Bifurcation of Steady-state Solutions in Predator-Prey and Competition Systems[J]. Proc. R. Soc. Edinburgh Sect. A. 1984, 97: 21-34.
[2] L. Li. Coexistence Theorems of Steady States for Predator-Prey Interacting Sys-tems[J]. Trans. Amer. Math. Soc. 1988, 305(1): 143-166.
[3] C. F. Gui, Y. Lou. Uniqueness and Nonuniqueness of Coexistence States in the Lotka-Volterra Competiiton Model [J].Comm Pure Appl.Math.l994,47(12):1571-1594.
[4] E. N. Dancer, Z. T. Zhang. Dynamics of Lotka-Volterra Competition System with Large Interaction [J].J.Differential Equations.2002,82:470-489.
[5] C. S. Holling. The Functional Response of Predator to prey Density and Its Role in Mimicry and Population Regulation[J].Mem Ent Soc Canda. 1965,45:1-60.
[6] J. F. Andrews. A Mathematical Model for the Continuous Culture of Microorganisms Utilizing Inhibitory Substrates[J].Biotechnol.Bioeng.1968,10:707-723.
[7] J. B. Collings. The Effects of the Functional Response on the Bifurcation Behavior of a Mite Predator-Prey Interaction Model[J].J.Math.Biol. 1997,36:149-168.
[8] W. Sokol, J. A. Howell.Kinetics of Phenol Oxidation by Washed Cells[J].Biotechn ol. Bioeng. 1981,23(9) :2039-2049.
[9] S. Ruan , D. Xiao. Global Analysis in a Predator-Prey System with Nonmonnotonic Functional Response[J].SIAM J.APPL.2001,61:1445-1472.
[10]黄继才．具有HolingType-Ⅳ功能反应函数的捕食与被捕食系统的定性分析[D]． 武汉，华中师范大学，2002．
[11]马知恩．种群生态学的数学建模与研究[M]．合肥：安徽教育出版社，1996．
[12] J. Blat, K. J. Brown. Global Bifurcation of Positive Solutions in Some Systems ofElliptic Equations[J]. SIAM J. Math. Anal.1986, 17(6): 1339-1353.
[13] WONLYUL KO, KIMUN RYU. Qualitative Analysis of a Predator-Prey Modelwith Holling Type 2 Functional Response Incorporating a Prey Refuge [J].J.Diff.Equ.2006,231:534-550.
[14] K. S. Cheng. Uniqueness of Limit Cycle for a Predator - Prey System[J]. SIAMJ.Math. Anal.l981,12(4):541-548.
[15] A. Gasull, A. Guillamon. Non - Existence of Limit Cycle for Some Predator -Prey Systems[J]. Proceedings of Equad2 iff. 91 World Scientific,Singapore.1993, 91 :538-543.
[16] J. Sugie, R. Kohno, R. Miyazaki. On a Predator - Prey System of Holling Type[J].Proc. Soc. Amer. 1997 ,125(7):2041- 2050.
[17] A. Casal, J. C. Eilbeck, J. LOPEZ-GOMEZ. Existence and Uniqueness of Coexistence States for a Predator-Prey Model with Diffusion[J]. Differential Integral Equations . 1994, 7(2): 411-439.
[18] Y. H. Du, Y. Lou. Some Uniqueness and Exact Multiplicity Results for a Predator-Prey Model[J]. Trans. Amer. Math. Soc. 1997, 349(6): 2443-2475.
[19] Y. H. Du, Y. Lou. S-Shaped Global Bifurcation Curve and Hopf Bifurcation ofPositive Solutions to a Predator-Prey Model[J]. J. Differential Equations. 1998,144(2): 390-440.
[20] Y. H. Du, Y. Lou. Qualitative Behaviour of Positive Solutions of a Predator-PreyModel: Effects of Saturation [J]. Proc. R. Soc. Edinburgh Sect. A. 2001, 131(2):321-349.
[21] J. H. Wu. Maximal Attractor, Stability, and Persistence for Prey-Predator Modelwith Saturation[J]. Mathematical and Computer Modeüing. 1999, 30(11-12): 7-16.
[22]叶其孝，李正元．反应扩散方程引论[M]．北京：科学出版社，1994，118-134．
[23] C. S. Cassanova. Existence and Structure of the Set of Positive Solutions of a General Class of Sublinear Elliptic Non-Classical Mixed Boundary Value Problems[J]. Nonlinear Anal. Ser. A: Theory Methods. 2002, 49(3): 361-430.
[24] C. V. Pao. Nonlinear Parabolic and Elliptic Equations[M]. Plenum Press, New York, 1992.
[25] M. G. Crandall, P. H. Rabinowitz. Bifurcation from Simple Eigenvalues [J]. J. Functional Analysis. 1971, 8(2): 321-340.
[26] M. G. Crandall, P. H. Rabinowitz. Bifurcation, Perturbation of Simple Eigenvalues and Linearized Stability[J]. Arch. Rational Mech. Anal. 1973, 52(2): 161-180.
[27] J. Smoller. Shock Waves and Reaction-Diffusion Equations[M]. New York: Spring-Verlag, 1983: 93-122, 126-156, 167-181.
[28] P. H. Rabinowitz. Some Global Results for Nonlinear Problems[J]. J. Functional Analysis. 1971, 7(3): 487-513.
[29] E. N. Dancer. On the Indices of Fixed Points of Mappings in Cones and Applica-tions[J]. J. Math. Anal. Appl. 1983, 91(1): 131-151.
[30] W. H. Ruan, W. Feng. On the Fixed Point Index and Multiple Steady-State Solutions of Reaction-Diffusion Systems[J]. Differential Integral Equations. 1995, 8(2): 371-392.
[31] J. H. Wu. Global Bifurcation of Coexistence State for the Competition Model in the Chemostat[J]. Nonlinear Analysis. 2000, 39(7): 817-835.
[32] J. H. Wu, G. S. Wei. Coexistence State for Cooperative Model with Diffusion[J]. Computers & Mathematics with Applications. 2002, 43(10-11): 1277-1290.
[33]聂华．两类生物模型的共存态和渐近行为[D]．西安：陕西师范大学，2006．
[34]郭改慧．带B-D反应项的捕食-食饵模型的共存态及渐近行为[D]．西安：陕西师范 大学，2007．
[2] L. Li. Coexistence Theorems of Steady States for Predator-Prey Interacting Sys-tems[J]. Trans. Amer. Math. Soc. 1988, 305(1): 143-166.
[3] C. F. Gui, Y. Lou. Uniqueness and Nonuniqueness of Coexistence States in the Lotka-Volterra Competiiton Model [J].Comm Pure Appl.Math.l994,47(12):1571-1594.
[4] E. N. Dancer, Z. T. Zhang. Dynamics of Lotka-Volterra Competition System with Large Interaction [J].J.Differential Equations.2002,82:470-489.
[5] C. S. Holling. The Functional Response of Predator to prey Density and Its Role in Mimicry and Population Regulation[J].Mem Ent Soc Canda. 1965,45:1-60.
[6] J. F. Andrews. A Mathematical Model for the Continuous Culture of Microorganisms Utilizing Inhibitory Substrates[J].Biotechnol.Bioeng.1968,10:707-723.
[7] J. B. Collings. The Effects of the Functional Response on the Bifurcation Behavior of a Mite Predator-Prey Interaction Model[J].J.Math.Biol. 1997,36:149-168.
[8] W. Sokol, J. A. Howell.Kinetics of Phenol Oxidation by Washed Cells[J].Biotechn ol. Bioeng. 1981,23(9) :2039-2049.
[9] S. Ruan , D. Xiao. Global Analysis in a Predator-Prey System with Nonmonnotonic Functional Response[J].SIAM J.APPL.2001,61:1445-1472.
[10]黄继才．具有HolingType-Ⅳ功能反应函数的捕食与被捕食系统的定性分析[D]． 武汉，华中师范大学，2002．
[11]马知恩．种群生态学的数学建模与研究[M]．合肥：安徽教育出版社，1996．
[12] J. Blat, K. J. Brown. Global Bifurcation of Positive Solutions in Some Systems ofElliptic Equations[J]. SIAM J. Math. Anal.1986, 17(6): 1339-1353.
[13] WONLYUL KO, KIMUN RYU. Qualitative Analysis of a Predator-Prey Modelwith Holling Type 2 Functional Response Incorporating a Prey Refuge [J].J.Diff.Equ.2006,231:534-550.
[14] K. S. Cheng. Uniqueness of Limit Cycle for a Predator - Prey System[J]. SIAMJ.Math. Anal.l981,12(4):541-548.
[15] A. Gasull, A. Guillamon. Non - Existence of Limit Cycle for Some Predator -Prey Systems[J]. Proceedings of Equad2 iff. 91 World Scientific,Singapore.1993, 91 :538-543.
[16] J. Sugie, R. Kohno, R. Miyazaki. On a Predator - Prey System of Holling Type[J].Proc. Soc. Amer. 1997 ,125(7):2041- 2050.
[17] A. Casal, J. C. Eilbeck, J. LOPEZ-GOMEZ. Existence and Uniqueness of Coexistence States for a Predator-Prey Model with Diffusion[J]. Differential Integral Equations . 1994, 7(2): 411-439.
[18] Y. H. Du, Y. Lou. Some Uniqueness and Exact Multiplicity Results for a Predator-Prey Model[J]. Trans. Amer. Math. Soc. 1997, 349(6): 2443-2475.
[19] Y. H. Du, Y. Lou. S-Shaped Global Bifurcation Curve and Hopf Bifurcation ofPositive Solutions to a Predator-Prey Model[J]. J. Differential Equations. 1998,144(2): 390-440.
[20] Y. H. Du, Y. Lou. Qualitative Behaviour of Positive Solutions of a Predator-PreyModel: Effects of Saturation [J]. Proc. R. Soc. Edinburgh Sect. A. 2001, 131(2):321-349.
[21] J. H. Wu. Maximal Attractor, Stability, and Persistence for Prey-Predator Modelwith Saturation[J]. Mathematical and Computer Modeüing. 1999, 30(11-12): 7-16.
[22]叶其孝，李正元．反应扩散方程引论[M]．北京：科学出版社，1994，118-134．
[23] C. S. Cassanova. Existence and Structure of the Set of Positive Solutions of a General Class of Sublinear Elliptic Non-Classical Mixed Boundary Value Problems[J]. Nonlinear Anal. Ser. A: Theory Methods. 2002, 49(3): 361-430.
[24] C. V. Pao. Nonlinear Parabolic and Elliptic Equations[M]. Plenum Press, New York, 1992.
[25] M. G. Crandall, P. H. Rabinowitz. Bifurcation from Simple Eigenvalues [J]. J. Functional Analysis. 1971, 8(2): 321-340.
[26] M. G. Crandall, P. H. Rabinowitz. Bifurcation, Perturbation of Simple Eigenvalues and Linearized Stability[J]. Arch. Rational Mech. Anal. 1973, 52(2): 161-180.
[27] J. Smoller. Shock Waves and Reaction-Diffusion Equations[M]. New York: Spring-Verlag, 1983: 93-122, 126-156, 167-181.
[28] P. H. Rabinowitz. Some Global Results for Nonlinear Problems[J]. J. Functional Analysis. 1971, 7(3): 487-513.
[29] E. N. Dancer. On the Indices of Fixed Points of Mappings in Cones and Applica-tions[J]. J. Math. Anal. Appl. 1983, 91(1): 131-151.
[30] W. H. Ruan, W. Feng. On the Fixed Point Index and Multiple Steady-State Solutions of Reaction-Diffusion Systems[J]. Differential Integral Equations. 1995, 8(2): 371-392.
[31] J. H. Wu. Global Bifurcation of Coexistence State for the Competition Model in the Chemostat[J]. Nonlinear Analysis. 2000, 39(7): 817-835.
[32] J. H. Wu, G. S. Wei. Coexistence State for Cooperative Model with Diffusion[J]. Computers & Mathematics with Applications. 2002, 43(10-11): 1277-1290.
[33]聂华．两类生物模型的共存态和渐近行为[D]．西安：陕西师范大学，2006．
[34]郭改慧．带B-D反应项的捕食-食饵模型的共存态及渐近行为[D]．西安：陕西师范 大学，2007．