周期非单调反应扩散模型的空间动力学
发布时间：2018-09-02 08:30
【摘要】：作为化学、生态学、流行病学中重要研究对象,反应扩散方程得到了广泛的关注与研究.通常,描述种群的增长以及多种群之间的相互作用的许多非线性反应扩散系统都不是单调的,典型的如捕食者和食饵系统、疾病在易感者与染病者之间的传播模型等.由于此类系统比较原理和单调性的缺失,使得研究其空间动力学变得困难.此外,在研究种群增长、疾病传播过程中,昼夜更替、季节变迁等周期变化因素也不容忽视.因此,研究非自治反应扩散方程具有重要的意义.本文主要致力于几类时间周期的非单调反应扩散系统空间动力学的探究.主要内容如下:首先,我们研究一类具有年龄阶段结构和非单调出生函数的周期反应扩散单种群模型的渐近传播速度和周期行波解.由于出生函数非单调,标准的单调性方法不再适用.此外,自治发展方程的行波解相关方法很难直接应用到周期非单调方程,所以我们试图寻求新的方法证明渐近传播速度和周期行波解.本文通过将给定的出生函数夹于两个非降函数之间从而构造原方程的两个控制方程,进而利用比较方法结合单调方程的渐近传播速度的相关结论得到了原方程渐近传播速度的存在性.然后通过构造闭凸集上的一个适当的非线性非单调算子,结合Schauder’s不动点定理证明了周期行波解的存在性.此处非线性算子的构造方法与证明自治系统行波解时构造的非线性算子非常不同.利用已经得到的该模型的渐近传播速度相关结论,我们证明了周期行波解的渐近行为以及周期行波解的非存在性.其次,我们研究一类具有标准发生率的周期反应扩散SIR模型的周期行波解.关于周期行波解的存在性证明,基本思想与周期单种群模型类似.此时,证明周期行波解的渐近边界条件成为困难,我们主要使用Laudau型不等式、合作抛物系统的Harnack不等式以及标量方程的比较方法来证明周期行波解满足的边界条件.此外,利用标量周期反应扩散方程的渐近传播速度以及标量抛物方程的比较方法,我们对两种情形证明了周期行波解的非存在性.再次,我们研究一类具有固定潜伏期的周期反应扩散SIR模型的动力学.通过考虑季节变迁、扩散以及潜伏期等因素,我们导出一个有界区域上周期非局部时滞反应扩散系统.与自治的时滞微分方程不同,线性周期时滞微分方程的稳定性与相应的无时滞周期微分方程的稳定性不再一致.这对发展周期时滞模型的基本再生数?0理论带来了极大的困难.我们首先利用次代算子方法引进?0,然后结合线性算子特征值理论进一步得到了?0与相应的线性方程的Poincar′e映射的谱半径之间的关系.最后,应用比较方法和持久性理论证明了阈值动力学.最后,我们研究一类具有固定传染期的反应扩散SIR模型,具体地,由一个周期非局部时滞反应扩散系统描述.在该模型中,时滞项是负的且初值满足一个非线性约束条件,这与以往的反应扩散传染病模型有着本质的不同,对分析模型的动力学行为带来新的数学上的困难.我们首先利用次代算子方法引入?0,然后通过一个线性积分方程结合扰动技术来克服负的时滞项带来的困难,从而给出了疾病灭绝和持久的充分条件.需要强调的是,以前的工作通常基于相应的线性微分方程的主特征值讨论,我们的方法与此明显不同.
[Abstract]:As an important research object in chemistry, ecology and epidemiology, reaction-diffusion equations have attracted extensive attention and research. Generally, many nonlinear reaction-diffusion systems describing population growth and interactions among populations are not monotonic. Typical systems such as predator-prey systems, diseases in susceptible and infected persons are not monotonic. Because of the lack of the comparison principle and monotonicity of these systems, it is difficult to study their spatial dynamics. In addition, the periodic factors such as population growth, disease transmission, day-night change, seasonal change and so on can not be ignored. Therefore, it is of great significance to study the non-autonomous reaction-diffusion equation. The main contents are as follows: Firstly, we study the asymptotic propagation velocity and periodic traveling wave solutions of a class of periodic reaction-diffusion monopopulation model with age-stage structure and nonmonotonic birth function. In addition, the traveling wave solution correlation method for autonomous evolution equations is difficult to be directly applied to periodic nonmonotone equations, so we try to find a new method to prove the asymptotic propagation velocity and periodic traveling wave solutions. The existence of the asymptotic propagation velocity of the original equation is obtained by using the comparison method and the related conclusion of the asymptotic propagation velocity of the monotone equation. Then the existence of periodic traveling wave solutions is proved by constructing an appropriate nonlinear nonmonotone operator on a closed convex set and combining Schauder's fixed point theorem. The construction method is very different from that of the nonlinear operators used to prove the traveling wave solutions of autonomous systems. By using the asymptotic propagation velocity dependence of the model, we prove the asymptotic behavior of the periodic traveling wave solutions and the nonexistence of the periodic traveling wave solutions. Secondly, we study a class of periodic reaction-diffusion SIR with standard incidence. On the existence of periodic traveling wave solutions, the basic idea is similar to that of periodic single-species model. At this point, it is difficult to prove the asymptotic boundary conditions of periodic traveling wave solutions. We mainly use Laudau-type inequality, Harnack inequality of cooperative parabolic systems and scalar equation comparison methods to prove periodic traveling wave solutions. Furthermore, by using the asymptotic propagation velocity of the scalar periodic reaction-diffusion equation and the comparison method of the scalar parabolic equation, we prove the nonexistence of the periodic traveling wave solution for two cases. Thirdly, we study the dynamics of a periodic reaction-diffusion SIR model with a fixed latency. Unlike autonomous delay differential equations, the stability of linear periodic delay differential equations is no longer consistent with that of corresponding delay-free periodic differential equations. Number? 0 theory brings great difficulties. We first introduce? 0 by using the method of subgeneration operator, and then combine the eigenvalue theory of linear operator to obtain the relation between? 0 and the spectral radius of Poincar'e mapping of the corresponding linear equation. Finally, we prove the threshold dynamics by using the comparison method and the persistence theory. A reaction-diffusion SIR model with a fixed infectious period is described in detail by a periodic nonlocal reaction-diffusion system with delays. In this model, the delay term is negative and the initial value satisfies a nonlinear constraint condition, which is essentially different from the previous reaction-diffusion epidemic model and brings about the dynamic behavior of the analytical model. A new mathematical difficulty is presented. We first introduce? 0 by using the method of subordinate operators, and then overcome the difficulty caused by negative delay terms by a linear integral equation combined with perturbation technique. Sufficient conditions for the extinction and persistence of the disease are given. Our method is obviously different from the discussion of the sign value.
【学位授予单位】：兰州大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175
，
本文编号：2218816
[Abstract]:As an important research object in chemistry, ecology and epidemiology, reaction-diffusion equations have attracted extensive attention and research. Generally, many nonlinear reaction-diffusion systems describing population growth and interactions among populations are not monotonic. Typical systems such as predator-prey systems, diseases in susceptible and infected persons are not monotonic. Because of the lack of the comparison principle and monotonicity of these systems, it is difficult to study their spatial dynamics. In addition, the periodic factors such as population growth, disease transmission, day-night change, seasonal change and so on can not be ignored. Therefore, it is of great significance to study the non-autonomous reaction-diffusion equation. The main contents are as follows: Firstly, we study the asymptotic propagation velocity and periodic traveling wave solutions of a class of periodic reaction-diffusion monopopulation model with age-stage structure and nonmonotonic birth function. In addition, the traveling wave solution correlation method for autonomous evolution equations is difficult to be directly applied to periodic nonmonotone equations, so we try to find a new method to prove the asymptotic propagation velocity and periodic traveling wave solutions. The existence of the asymptotic propagation velocity of the original equation is obtained by using the comparison method and the related conclusion of the asymptotic propagation velocity of the monotone equation. Then the existence of periodic traveling wave solutions is proved by constructing an appropriate nonlinear nonmonotone operator on a closed convex set and combining Schauder's fixed point theorem. The construction method is very different from that of the nonlinear operators used to prove the traveling wave solutions of autonomous systems. By using the asymptotic propagation velocity dependence of the model, we prove the asymptotic behavior of the periodic traveling wave solutions and the nonexistence of the periodic traveling wave solutions. Secondly, we study a class of periodic reaction-diffusion SIR with standard incidence. On the existence of periodic traveling wave solutions, the basic idea is similar to that of periodic single-species model. At this point, it is difficult to prove the asymptotic boundary conditions of periodic traveling wave solutions. We mainly use Laudau-type inequality, Harnack inequality of cooperative parabolic systems and scalar equation comparison methods to prove periodic traveling wave solutions. Furthermore, by using the asymptotic propagation velocity of the scalar periodic reaction-diffusion equation and the comparison method of the scalar parabolic equation, we prove the nonexistence of the periodic traveling wave solution for two cases. Thirdly, we study the dynamics of a periodic reaction-diffusion SIR model with a fixed latency. Unlike autonomous delay differential equations, the stability of linear periodic delay differential equations is no longer consistent with that of corresponding delay-free periodic differential equations. Number? 0 theory brings great difficulties. We first introduce? 0 by using the method of subgeneration operator, and then combine the eigenvalue theory of linear operator to obtain the relation between? 0 and the spectral radius of Poincar'e mapping of the corresponding linear equation. Finally, we prove the threshold dynamics by using the comparison method and the persistence theory. A reaction-diffusion SIR model with a fixed infectious period is described in detail by a periodic nonlocal reaction-diffusion system with delays. In this model, the delay term is negative and the initial value satisfies a nonlinear constraint condition, which is essentially different from the previous reaction-diffusion epidemic model and brings about the dynamic behavior of the analytical model. A new mathematical difficulty is presented. We first introduce? 0 by using the method of subordinate operators, and then overcome the difficulty caused by negative delay terms by a linear integral equation combined with perturbation technique. Sufficient conditions for the extinction and persistence of the disease are given. Our method is obviously different from the discussion of the sign value.
【学位授予单位】：兰州大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175
，
本文编号：2218816
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