两类种群模型行波解的存在性
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摘要
本文首先建立了三个反应扩散方程模型：霍乱模型,营养-细菌模型及具有治疗的流感模型.然后用打靶法及Schauder不动点定理研究了这三个模型行波解的存在性及不存在性,得到了模型的最小波速,从而为传染病控制及细菌种群控制提供了理论依据.下面分章介绍.
第一章是引言部分,主要介绍了问题研究的背景,所用的数学方法及不同方法的比较.
第二章建立了具有污染物扩散的霍乱传播模型.首先忽略因病死亡率,考虑霍乱传播的两种方式(人与环境之间的传播及人与人之间的传播).把原系统行波解的存在性问题转化为极限系统行波解的存在性问题,然后通过打靶法得到了模型行波解存在的充要条件,给出了最小波速的计算公式.接着考察因病死亡率对霍乱传播的影响,但是忽略人的自然出生与死亡过程.通过常数变易法把原系统降维,转化为具有分布时滞的反应扩散系统,然后构造一对有界的上下解,从而得到了一个正锥.对这个正锥用Schauder不动点定理得到行波解的存在性.我们用双边Laplace变换方法来排除行波解的存在性,为了应用双边Laplace变换就首先要说明行波解至少是指数衰减的,受到稳定流形定理证明过程的启发,结合行波解的平移不变性,提出了一种新的证明指数衰减性的方法.
第三章基于Mimura的营养-细菌模型,建立了一个简化的反应扩散模型.通过常数变易法把原系统转化为一个具有分布时滞的发展系统.首先考虑线性化特征值问题,通过把两个高次多项式与同一个一次多项式进行比较得到了最小波速的计算公式.为了证明行波解的存在性,进一步构造一个辅助系统.对辅助系统构造一对上下解,使用Schauder不动点定理得到辅助系统行波解的存在性,这样就得到一列行波解,通过Arzela-Ascoli定理就证明了辅助系统行波解序列的极限就是原系统的行波解.为了证明行波解的不存在性,我们定义了”负向单边Laplace变换”,通过”负向单边Laplace变换”证明行波解的不存在性.
第四章建立了具有治疗的流感扩散模型.通过分析线性化特征值问题得到最小波速的计算方法.类似于第三章的方法,首先构造一个辅助系统,通过Schauder不动点定理得到辅助系统行波解的存在性,进一步使用Arzela-Ascoli定理就证明了原系统的行波解的存在性.行波解不存在性的证明与第二章的第二个模型的方法类似.
本段给出论文的创新点.创新点分为两个方面：模型对现实问题的解释及行波解证明方法上的创新.本文通过反应扩散方程模型的行波解揭示了传染病传播及细菌扩散的内在机理,得到了最小波速,为霍乱与流感的控制提供了理论基础,有助于分析细菌的扩散模式.证明方法上的创新点如下：
1.第一个创新点就是线性化问题特征方程的分析.第三章的特征值问题是一个三次多项式方程,为了得到最小波速c*的计算,我们把两个高次多项式与一个共同的一次多项式相比较得出了c*的存在性,这样的方法对三次多项式方程具有较大的适用性,目前没有见到有文献这样用过.
2.第二个创新点是”负向单边Laplace变换”概念的引入.为了证明行波解的不存在性,Wang和Wu(2010)使用了双边Laplace变换,我们引入的负向单边Laplace变换使证明变得更加简洁明了.
3.第三个创新点在于我们引入了辅助系统,构造的上下解是有界的,从而所得到的正锥也是有界的,这与Wang和Wu(2010)的无界上解有根本的区别.对非合作系统来说,利用所构造的有界上下解来得到最小波速是很不容易的.
4.第四个创新点就是提出了一个证明行波解指数衰减性的新的方法.Wang和Wu(2010)是使用分析的方法证明行波解的指数衰减性的,但是Wang和Wu的线性化方程只有一个,而本论文第四章中模型的线性化方程却有两个,故使用分析的方法就行不通了.受到稳定流形定理证明过程的启发,我们提出了一个证明指数衰减性的新方法,这种方法不受方程个数的限制,所以具有更广泛的适用性.
In this paper, three diffusion-reaction models are established, which are cholera model, nutrient-bacteria model and influenza model with treatment. By shooting method and Schauder's fixed point theorem, the existence and non-existence of traveling wave solutions are proved and the minimal wave speed is obtained. These results provide a theoretical basis for the control of diseases and bacteria.
The first chapter devoted into the introduction on the background of models, mathematical methods and the comparison of different methods.
In Chapter2, a cholera model with contaminants diffusive is established. Firstly, we study the case with man-man and man-environment transmissions considered and mortality due to illness ignored. The existence of traveling wave solutions are proved by changing the original system into its limit one and, furthermore, using shooting method. At the same time, the minimal wave speed is obtained. In the second case, the influence of illness death on cholera is considered with natural birth and death pro-cess ignored. By constant variation method, original system is transformed reaction-diffusion system with distributed delay. The existence of traveling wave solutions are get by constructing upper and lower solutions and using Schauder's fixed point the-orem. To prove the non-existence of traveling wave solutions by two-sided Laplace transform, it is necessary to show the exponential decay of traveling wave solutions and a new method is proposed for this aim.
In Chapter3, based on Mimura's nutrient-bacteria model, s simple model is pro-posed. Similar to the second chapter, original system is transformed reaction-diffusion one with distributed delay. The formula for minimal wave speed is deduced by com-paring two high-order polynomials with a linear function. An auxiliary system is con-structed and the existence of a sequence of traveling wave solutions for the auxiliary system is proved by Schauder's fixed point theorem. The limit of the sequence of trav-eling wave solutions for the auxiliary system is shown to be the traveling wave solution of original system. The non-existence of traveling wave solutions is proved be defining a negative one-sided Laplace transform.
In Chapter4, a diffusive influenza model with treatment is established. Firstly, we construct an auxiliary system and prove the existence of traveling wave solutions by the methods similar to that of Chapter3. However, it is more technical since the dimension of this system can not be reduced.
The innovation consists of two aspects:interpretation of the real world and the methods for traveling wave solutions. The results of this paper reveal the internal mechanism of the spread of infectious diseases and the spread of bacteria and provide a theoretical basis for the control of cholera, influenza and bacteria. The innovation on mathematical methods are as follows.
1. The first innovation is the analysis of linearization. The formula for minimal wave speed is deduced by comparing two high-order polynomials with a linear function. This method can be applied to most of cubic polynomials
2. The definition of "negative one-sided Laplace transform" is the second innovation which is motivated by Wang and Wu(2010) and makes proofs simpler.
3. The third innovation lies in the introduction of an auxiliary system and the bound-edness of upper and lower solutions, which is different from that of Wang and Wu (2010). It is difficult to get the minimal wave speed by constructing bounded upper and lower solutions for non-cooperative systems.
4. The fourth innovation lies in the new method to prove the exponential decay for traveling wave solutions. The method used by Wang and Wu (2010) is applicable for one equation and not for a system. However, our method can be applied to general systems.
第一章是引言部分,主要介绍了问题研究的背景,所用的数学方法及不同方法的比较.
第二章建立了具有污染物扩散的霍乱传播模型.首先忽略因病死亡率,考虑霍乱传播的两种方式(人与环境之间的传播及人与人之间的传播).把原系统行波解的存在性问题转化为极限系统行波解的存在性问题,然后通过打靶法得到了模型行波解存在的充要条件,给出了最小波速的计算公式.接着考察因病死亡率对霍乱传播的影响,但是忽略人的自然出生与死亡过程.通过常数变易法把原系统降维,转化为具有分布时滞的反应扩散系统,然后构造一对有界的上下解,从而得到了一个正锥.对这个正锥用Schauder不动点定理得到行波解的存在性.我们用双边Laplace变换方法来排除行波解的存在性,为了应用双边Laplace变换就首先要说明行波解至少是指数衰减的,受到稳定流形定理证明过程的启发,结合行波解的平移不变性,提出了一种新的证明指数衰减性的方法.
第三章基于Mimura的营养-细菌模型,建立了一个简化的反应扩散模型.通过常数变易法把原系统转化为一个具有分布时滞的发展系统.首先考虑线性化特征值问题,通过把两个高次多项式与同一个一次多项式进行比较得到了最小波速的计算公式.为了证明行波解的存在性,进一步构造一个辅助系统.对辅助系统构造一对上下解,使用Schauder不动点定理得到辅助系统行波解的存在性,这样就得到一列行波解,通过Arzela-Ascoli定理就证明了辅助系统行波解序列的极限就是原系统的行波解.为了证明行波解的不存在性,我们定义了”负向单边Laplace变换”,通过”负向单边Laplace变换”证明行波解的不存在性.
第四章建立了具有治疗的流感扩散模型.通过分析线性化特征值问题得到最小波速的计算方法.类似于第三章的方法,首先构造一个辅助系统,通过Schauder不动点定理得到辅助系统行波解的存在性,进一步使用Arzela-Ascoli定理就证明了原系统的行波解的存在性.行波解不存在性的证明与第二章的第二个模型的方法类似.
本段给出论文的创新点.创新点分为两个方面：模型对现实问题的解释及行波解证明方法上的创新.本文通过反应扩散方程模型的行波解揭示了传染病传播及细菌扩散的内在机理,得到了最小波速,为霍乱与流感的控制提供了理论基础,有助于分析细菌的扩散模式.证明方法上的创新点如下：
1.第一个创新点就是线性化问题特征方程的分析.第三章的特征值问题是一个三次多项式方程,为了得到最小波速c*的计算,我们把两个高次多项式与一个共同的一次多项式相比较得出了c*的存在性,这样的方法对三次多项式方程具有较大的适用性,目前没有见到有文献这样用过.
2.第二个创新点是”负向单边Laplace变换”概念的引入.为了证明行波解的不存在性,Wang和Wu(2010)使用了双边Laplace变换,我们引入的负向单边Laplace变换使证明变得更加简洁明了.
3.第三个创新点在于我们引入了辅助系统,构造的上下解是有界的,从而所得到的正锥也是有界的,这与Wang和Wu(2010)的无界上解有根本的区别.对非合作系统来说,利用所构造的有界上下解来得到最小波速是很不容易的.
4.第四个创新点就是提出了一个证明行波解指数衰减性的新的方法.Wang和Wu(2010)是使用分析的方法证明行波解的指数衰减性的,但是Wang和Wu的线性化方程只有一个,而本论文第四章中模型的线性化方程却有两个,故使用分析的方法就行不通了.受到稳定流形定理证明过程的启发,我们提出了一个证明指数衰减性的新方法,这种方法不受方程个数的限制,所以具有更广泛的适用性.
In this paper, three diffusion-reaction models are established, which are cholera model, nutrient-bacteria model and influenza model with treatment. By shooting method and Schauder's fixed point theorem, the existence and non-existence of traveling wave solutions are proved and the minimal wave speed is obtained. These results provide a theoretical basis for the control of diseases and bacteria.
The first chapter devoted into the introduction on the background of models, mathematical methods and the comparison of different methods.
In Chapter2, a cholera model with contaminants diffusive is established. Firstly, we study the case with man-man and man-environment transmissions considered and mortality due to illness ignored. The existence of traveling wave solutions are proved by changing the original system into its limit one and, furthermore, using shooting method. At the same time, the minimal wave speed is obtained. In the second case, the influence of illness death on cholera is considered with natural birth and death pro-cess ignored. By constant variation method, original system is transformed reaction-diffusion system with distributed delay. The existence of traveling wave solutions are get by constructing upper and lower solutions and using Schauder's fixed point the-orem. To prove the non-existence of traveling wave solutions by two-sided Laplace transform, it is necessary to show the exponential decay of traveling wave solutions and a new method is proposed for this aim.
In Chapter3, based on Mimura's nutrient-bacteria model, s simple model is pro-posed. Similar to the second chapter, original system is transformed reaction-diffusion one with distributed delay. The formula for minimal wave speed is deduced by com-paring two high-order polynomials with a linear function. An auxiliary system is con-structed and the existence of a sequence of traveling wave solutions for the auxiliary system is proved by Schauder's fixed point theorem. The limit of the sequence of trav-eling wave solutions for the auxiliary system is shown to be the traveling wave solution of original system. The non-existence of traveling wave solutions is proved be defining a negative one-sided Laplace transform.
In Chapter4, a diffusive influenza model with treatment is established. Firstly, we construct an auxiliary system and prove the existence of traveling wave solutions by the methods similar to that of Chapter3. However, it is more technical since the dimension of this system can not be reduced.
The innovation consists of two aspects:interpretation of the real world and the methods for traveling wave solutions. The results of this paper reveal the internal mechanism of the spread of infectious diseases and the spread of bacteria and provide a theoretical basis for the control of cholera, influenza and bacteria. The innovation on mathematical methods are as follows.
1. The first innovation is the analysis of linearization. The formula for minimal wave speed is deduced by comparing two high-order polynomials with a linear function. This method can be applied to most of cubic polynomials
2. The definition of "negative one-sided Laplace transform" is the second innovation which is motivated by Wang and Wu(2010) and makes proofs simpler.
3. The third innovation lies in the introduction of an auxiliary system and the bound-edness of upper and lower solutions, which is different from that of Wang and Wu (2010). It is difficult to get the minimal wave speed by constructing bounded upper and lower solutions for non-cooperative systems.
4. The fourth innovation lies in the new method to prove the exponential decay for traveling wave solutions. The method used by Wang and Wu (2010) is applicable for one equation and not for a system. However, our method can be applied to general systems.
引文
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