Mathematical analysis of emergence of a rich variety of solutions in reaction-diffusion systems

反应扩散系统中丰富多样的解的出现的数学分析

基本信息

批准号：
14540143
负责人：
HOSONO Yuzo
金额：
$ 2.18万
依托单位：
Kyoto Sangyo University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2005
项目状态：
已结题

项目摘要

(1)The Lotka-Volterra competition modelsWe consider the May-Leonard 3-species competition system with a singular perturbation parameter. We first discuss the complete picture of the stability of the coexistence steady state of this system, and show the occurrence of the subcritical Hopf bifurcation with the help of AUTO. We further investigate the relation between the 3-species Lotka-Volterra competition model and the 2-species predator-prey model by the formal singular perturbation analysis and the numerical simulations.(2)Traveling waves for the higher order autocatalytic reaction-diffusion systemsWe prove the existence of traveling waves for the two component higher order autocatalytic reaction-diffusion systems for any diffusion coefficients. We further discuss the existence problem for the system with higher order decay when the reactant does not diffuse. Our analysis of the vector fields in the phase space gives the estimate of the minimal propagation speeds in terms of the order of autocatalysis and the diffusion coefficients.(3)The reaction-diffusion systems related to epidemic modelingWe discuss the relation between the reaction-diffusion models and the integral equation models in the study of the spatial spread of epidemic and to discuss the speeds of spatial spread of an infectious disease through the reaction-diffusion models. The speeds of the epidemic waves are often derived heuristically from the linearization of the model equations, which is called ‘the linear conjecture.' For the diffusive Kermack-McKendrick model, we show the validity of the linear conjecture by the use of the existence results of traveling wave solutions. Then, we give the examples of the reaction-diffusion epidemic models which have traveling wave solutions with the speed greater than the predicted values of the speed by the linear conjecture.

（1）Lotka-Volterra竞争Modelswe考虑了具有单数扰动参数的May-Leonard 3种竞争系统。我们首先讨论了该系统共存稳态的稳定性的完整图，并在自动方面显示了亚临界型霍夫夫分叉的发生。我们进一步研究了通过形式的单数扰动分析和数值模拟的三种物种Lotka-volterra竞争模型与2种特种捕食者捕食者模型。（2）高阶自体催化反应 - 扩散系统的行进波证明了两种组成级别的自动化自动化系统的行进波的存在，以供任何diffife diffiffipy diffiffife diffiffife diffiffifion diffiffifion diffiffife。当反应物没有扩散时，我们进一步讨论系统的存在问题。我们对相空间中矢量场的分析给出了最小传播速度的估计，从自催化的顺序和扩散系数方面进行了估计。（3）与流行模型相关的反应 - 扩散系统在反应 - 扩散模型之间相关的关系与反应模型与整体疾病的整体疾病模型之间的关系之间的关系进行了反应，从而通过流经量的传播来进行传播，以进行流行速度传播。流行波的速度通常是从模型当量的线性化中得出的，这称为“线性猜想”。对于扩散的kermack-mckendrick模型，我们通过使用波动波解决方案的存在结果来显示线性概念的有效性。然后，我们给出了反应扩散流行模型的示例，这些模型的速度大于线性概念的速度值大于预测的速度值。