反应扩散方程的行波解与几类方程的多解性
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摘要
本文主要研究了具有非局部影响的扩散Nicholson苍蝇方程的行波解的存在性,离散椭圆型边值问题多个非平凡解的存在性以及微分方程多个周期解的存在性问题.全文由五章组成.
第一章简述了问题产生的历史背景及其研究意义以及本文的主要工作,并提出了可以进一步研究的若干问题.
第二章利用平面相图法、奇异摄动理论、Fredholm理论和不动点定理研究了具有时滞的非单调非线性非局部影响的扩散的Nicholson苍蝇方程的行波解的存在性.本文的方法不仅能处理具有离散时滞和分布时滞的Nicholson模型行波解的存在性,也能够处理具有空间平均时滞Nicholson模型行波解的存在性.本文推广了相关文献的主要结果.数值仿真显示,当时滞τ很大或β很大时,明显的出现一个跳跃,这和理论分析的结果是相吻合的.
第三章利用“山路引理”研究了离散椭圆型方程边值问题多个非平凡解的存在性,得到了离散边值问题至少存在两个非平凡解的若干充分条件,我们的结果改进了现有文献的相关结果.
第四章利用Gains和Mawhin重合度理论研究了一阶泛函微分方程周期解的存在性问题.得到了它至少存在两个周期解的若干充分条件.
第五章同样利用重合度理论研究了一类抽象的生物学模型的周期正解存在性问题.并给出了它至少存在四个周期正解的充分条件.
This dissertation mainly deals with the existence of wavefronts for di?usionNicholson blow?ies equation with non-local e?ects, nontrivial solutions of discreteelliptic boundary and multiple periodic solutions of di?erential equations. Thisdissertation contains five chapters.
Chapter 1 focuses on the brief introduction of the historic background andsignification for all the investigated problems, and gives out some open problemsto research.
Chapter 2 investigates the existence of traveling wavefronts for the di?usiveNicholson blow?ies equations with the delay, non-monotone nonlinearity and non-local e?ects by applying phase-plane technique, singular perturbation theory, Fred-holm theory and fixed-point theorem. The method of this dissertation can dealwith the existence of traveling wavefronts for Nicholson blow?ies models in thediscrete delay case and the distributed delay case, but also in the spatial averagingcase. The chapter improves main results of some relevant literature. Numericalsimulations show that a hump appears when delayτorβis large, which coincideswith the theoretical analysis.
Chapter 3 investigates the existence of multiple nontrivial solutions for dis-crete elliptic boundary value problems by using the”Mountain pass theorem”.Some conditions are obtained for discrete elliptic boundary value problems to haveat least two nontrivial solutions. These improve the related results in some knownliterature.
Chapter 4 is devoted to the existence of periodic solutions for the first orderfunctional di?erential equationsby using Gains and Mawhin’s coincidence degree theory. Some su?cient conditionsare obtained to enuring that the above equation has at least two periodic solutions.
Chapter 5 is devoted to the existence of the positive periodic solutions for aclass of abstract ecology models
The su?cient conditions on the existence of four periodic solutions are obtainedby means of the same theory as that in Chapter 4.
第一章简述了问题产生的历史背景及其研究意义以及本文的主要工作,并提出了可以进一步研究的若干问题.
第二章利用平面相图法、奇异摄动理论、Fredholm理论和不动点定理研究了具有时滞的非单调非线性非局部影响的扩散的Nicholson苍蝇方程的行波解的存在性.本文的方法不仅能处理具有离散时滞和分布时滞的Nicholson模型行波解的存在性,也能够处理具有空间平均时滞Nicholson模型行波解的存在性.本文推广了相关文献的主要结果.数值仿真显示,当时滞τ很大或β很大时,明显的出现一个跳跃,这和理论分析的结果是相吻合的.
第三章利用“山路引理”研究了离散椭圆型方程边值问题多个非平凡解的存在性,得到了离散边值问题至少存在两个非平凡解的若干充分条件,我们的结果改进了现有文献的相关结果.
第四章利用Gains和Mawhin重合度理论研究了一阶泛函微分方程周期解的存在性问题.得到了它至少存在两个周期解的若干充分条件.
第五章同样利用重合度理论研究了一类抽象的生物学模型的周期正解存在性问题.并给出了它至少存在四个周期正解的充分条件.
This dissertation mainly deals with the existence of wavefronts for di?usionNicholson blow?ies equation with non-local e?ects, nontrivial solutions of discreteelliptic boundary and multiple periodic solutions of di?erential equations. Thisdissertation contains five chapters.
Chapter 1 focuses on the brief introduction of the historic background andsignification for all the investigated problems, and gives out some open problemsto research.
Chapter 2 investigates the existence of traveling wavefronts for the di?usiveNicholson blow?ies equations with the delay, non-monotone nonlinearity and non-local e?ects by applying phase-plane technique, singular perturbation theory, Fred-holm theory and fixed-point theorem. The method of this dissertation can dealwith the existence of traveling wavefronts for Nicholson blow?ies models in thediscrete delay case and the distributed delay case, but also in the spatial averagingcase. The chapter improves main results of some relevant literature. Numericalsimulations show that a hump appears when delayτorβis large, which coincideswith the theoretical analysis.
Chapter 3 investigates the existence of multiple nontrivial solutions for dis-crete elliptic boundary value problems by using the”Mountain pass theorem”.Some conditions are obtained for discrete elliptic boundary value problems to haveat least two nontrivial solutions. These improve the related results in some knownliterature.
Chapter 4 is devoted to the existence of periodic solutions for the first orderfunctional di?erential equationsby using Gains and Mawhin’s coincidence degree theory. Some su?cient conditionsare obtained to enuring that the above equation has at least two periodic solutions.
Chapter 5 is devoted to the existence of the positive periodic solutions for aclass of abstract ecology models
The su?cient conditions on the existence of four periodic solutions are obtainedby means of the same theory as that in Chapter 4.
引文
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[3]叶其孝,李正元.反应扩散方程引论.北京:科学出版社, 1999.
[4] Britton N F. Reaction–Di?usion Equations and their Applications to Biology. NewYork: Academic Press, 1986.
[5] Okubo A. Di?usion and Ecological Problems: Mathematical Models. Berlin–Heidelberg–New York: Springer–verlag, 1980.
[6] Briton N F. spatial styuctures and periodic travelling waves in an integro–di?erential reaction–di?usion population model, SIAM J. Appl. Math. 1990, 50(6):1663–1668.
[7]郭大钧.非线性泛函分析.山东:山东科学技术出版社, 2003.
[8] Deimling K. Nonlinear Functional Analysis. New York: Springer–verlag, 1981.
[9] Kerorkian J. and Cole J D. Perturbation Methods in Applied Mathematics. NewYork: Springer–verlag, 1984.
[10] Lagerstrom P A. Matched Asymptotic Expansions. New york: Springer–verlag,1988.
[11] Nicholson A J. The self adjustment of population to change, Cold Spring Harb.Symp. Quant. Biol., 1957, 22: 153–173.
[12] Gurney W S C, Blgthe S P, and Nishet R M. Nicholson’s blow?ies revisited. Nature1980, 287: 17–21.
[13] Yang Y, So J W H. Dynamics for the di?usive Nicholson’s blow?ies equation. Dy-namical systems and di?erential equations. Vol II (Spring field, Mo. 1996). DiscreteContin. Dynam. Systems 1998, Added Volume II, 333–352.
[14] So J W H, Yang Y. Dirichlet problem for the di?usive Nicholson’s blow?ies equation.J. Di?. Eqns. 1998, 15(2): 317–348.
[15] So J W H, Zou X. traveling waves for the di?usive Nicholson’s blow?ies equation.Appl. Math. Comput., 2001, 122(3): 385–392.
[16] Gourley S A. Traveling fronts in the di?usive Nicholson’s blow?ies equation withdistributed delays, Math. Comput. Modelling, 2000, 32(7–8): 843–853.
[17] Fenichel N. Geometric singular perturbation theory for ordinary di?erential equa-tions. J. Di?. Equ. 1979, 31(1): 53–98.
[18] Ruan S, Xiao D. Stability of steady states and existence of traveling waves in aVector–diseas model. Proc. Roy. Soc. Edinburgh Sect. A., 2004, 134(5): 991–1011.
[19] Grouley S A. and Chaplain M A J. Traveling fronts in a food–limited Populationmodel with time delay, Proc. Roy. Soc., Edinburgh Sect. A., 2002, 132(1): 75–89.
[20] Ou C, Wu J. Traveling wavefronts in a delayed food–limited population model.SIAM J. Math Anal., 2007, 39: 103–125.
[21] Ou C, Wu J. Persistence of traveling wavefronts in non–local reaction di?usionequations. J. Di?. Equ., 2007, 235: 219–261.
[22] Faria T, Huang W, and Wu J. Traveling Waves for delayed reaction–di?usions withglobal response. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sc., 2006, 462: 229–261.
[23] Gourley S A. Traveling fronts solutions of a nonlocal Fisher equation. J. Math.Biol., 2000, 41(3): 272–284.
[24] Mischaikow K, Smith H, and Thieme H R. Asympotically autonomous semi?ows:Chain recurrence and Lyapunov functions. Trans. Amer. Math, Soc., 1995. 347:1669–1685.
[25] Akveld M E, Hulshof J. Traveling wave solutions of a fourth–order semilinear dif-fusion equation. Appl. Math. Lett., 1998, 11(3): 115–120.
[26] Chow S N, Hale J K. Methods of Bifurcation Theory, New York: Springer–Verlag,1982.
[27] Dunbar S R. Traves in di?usive perdator–prey equation: periodic orbits and point–periodic heteroclinic orbits. SIAM J. AppL. Math., 1986, 46: 1057–1078.
[28]张恭庆.临界点理论及其应用.上海:上海科学技术出版社, 1986.
[29] Mawhin J, Willem M. Critical point Theory and Hamiltonian systems. New York:Springer–Verlag, 1989.
[30] Ambrosetti A, Rabinowitz P H. Dual variational methods in critical point theoryand applications. J. Funct. Annal., 1973, 14: 349–381.
[31] Yu J, Guo Z, and Zou X. Positive periodic solutions of second order self–adjiontdi?erence equations. J. London Math. Soc., 2003, 68(2): 419–430.
[32] Guo Z, Yu J. The existence of periodic and subharmonic solutions for second ordersuperlinear di?erence equation, Science in China. 2003, 33A: 226–235.
[33] Zhou Z, Yu J, and Guo Z. periodic solutions of higher–dimensional discrete systems.Proc. Royal. Soc. Edinburgh., 2004, 134A: 1–10.
[34] Ma M. Dominant and recessive solutions for second order self–adjoint linear di?er-ence equations. Applied Mathematics Letters, 2005, 18(2): 179–185.
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