周期非單調(diào)反應(yīng)擴散模型的空間動力學
發(fā)布時間：2018-09-02 08:30
【摘要】：作為化學、生態(tài)學、流行病學中重要研究對象,反應(yīng)擴散方程得到了廣泛的關(guān)注與研究.通常,描述種群的增長以及多種群之間的相互作用的許多非線性反應(yīng)擴散系統(tǒng)都不是單調(diào)的,典型的如捕食者和食餌系統(tǒng)、疾病在易感者與染病者之間的傳播模型等.由于此類系統(tǒng)比較原理和單調(diào)性的缺失,使得研究其空間動力學變得困難.此外,在研究種群增長、疾病傳播過程中,晝夜更替、季節(jié)變遷等周期變化因素也不容忽視.因此,研究非自治反應(yīng)擴散方程具有重要的意義.本文主要致力于幾類時間周期的非單調(diào)反應(yīng)擴散系統(tǒng)空間動力學的探究.主要內(nèi)容如下:首先,我們研究一類具有年齡階段結(jié)構(gòu)和非單調(diào)出生函數(shù)的周期反應(yīng)擴散單種群模型的漸近傳播速度和周期行波解.由于出生函數(shù)非單調(diào),標準的單調(diào)性方法不再適用.此外,自治發(fā)展方程的行波解相關(guān)方法很難直接應(yīng)用到周期非單調(diào)方程,所以我們試圖尋求新的方法證明漸近傳播速度和周期行波解.本文通過將給定的出生函數(shù)夾于兩個非降函數(shù)之間從而構(gòu)造原方程的兩個控制方程,進而利用比較方法結(jié)合單調(diào)方程的漸近傳播速度的相關(guān)結(jié)論得到了原方程漸近傳播速度的存在性.然后通過構(gòu)造閉凸集上的一個適當?shù)姆蔷�性非單調(diào)算子,結(jié)合Schauder’s不動點定理證明了周期行波解的存在性.此處非線性算子的構(gòu)造方法與證明自治系統(tǒng)行波解時構(gòu)造的非線性算子非常不同.利用已經(jīng)得到的該模型的漸近傳播速度相關(guān)結(jié)論,我們證明了周期行波解的漸近行為以及周期行波解的非存在性.其次,我們研究一類具有標準發(fā)生率的周期反應(yīng)擴散SIR模型的周期行波解.關(guān)于周期行波解的存在性證明,基本思想與周期單種群模型類似.此時,證明周期行波解的漸近邊界條件成為困難,我們主要使用Laudau型不等式、合作拋物系統(tǒng)的Harnack不等式以及標量方程的比較方法來證明周期行波解滿足的邊界條件.此外,利用標量周期反應(yīng)擴散方程的漸近傳播速度以及標量拋物方程的比較方法,我們對兩種情形證明了周期行波解的非存在性.再次,我們研究一類具有固定潛伏期的周期反應(yīng)擴散SIR模型的動力學.通過考慮季節(jié)變遷、擴散以及潛伏期等因素,我們導出一個有界區(qū)域上周期非局部時滯反應(yīng)擴散系統(tǒng).與自治的時滯微分方程不同,線性周期時滯微分方程的穩(wěn)定性與相應(yīng)的無時滯周期微分方程的穩(wěn)定性不再一致.這對發(fā)展周期時滯模型的基本再生數(shù)?0理論帶來了極大的困難.我們首先利用次代算子方法引進?0,然后結(jié)合線性算子特征值理論進一步得到了?0與相應(yīng)的線性方程的Poincar′e映射的譜半徑之間的關(guān)系.最后,應(yīng)用比較方法和持久性理論證明了閾值動力學.最后,我們研究一類具有固定傳染期的反應(yīng)擴散SIR模型,具體地,由一個周期非局部時滯反應(yīng)擴散系統(tǒng)描述.在該模型中,時滯項是負的且初值滿足一個非線性約束條件,這與以往的反應(yīng)擴散傳染病模型有著本質(zhì)的不同,對分析模型的動力學行為帶來新的數(shù)學上的困難.我們首先利用次代算子方法引入?0,然后通過一個線性積分方程結(jié)合擾動技術(shù)來克服負的時滯項帶來的困難,從而給出了疾病滅絕和持久的充分條件.需要強調(diào)的是,以前的工作通�；谙鄳�(yīng)的線性微分方程的主特征值討論,我們的方法與此明顯不同.
[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.
【學位授予單位】：蘭州大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：O175
，
本文編號：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.
【學位授予單位】：蘭州大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：O175
，
本文編號：2218816
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