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

(1) Lotka-Volterra竞争模型我们考虑具有奇异扰动参数的May-Leonard三物种竞争系统。我们首先讨论了该系统共存稳态稳定性的全貌，并利用AUTO证明了次临界Hopf分岔的发生。通过形式奇异摄动分析和数值模拟进一步研究了3种Lotka-Volterra竞争模型和2种捕食者-猎物模型之间的关系。(2)高阶自催化反应扩散系统的行波证明了任意扩散系数下双组分高阶自催化反应扩散系统的行波存在性。进一步讨论了当反应物不扩散时高阶衰变系统的存在性问题。我们对相空间中矢量场的分析给出了根据自催化阶数和扩散系数估计的最小传播速度。(3)与传染病模型相关的反应-扩散系统讨论了传染病空间传播研究中反应-扩散模型与积分方程模型的关系，并通过反应-扩散模型讨论了传染病的空间传播速度。流行病波的速度通常是从模型方程的线性化中启发式地推导出来的，这被称为“线性猜想”。对于扩散Kermack-McKendrick模型，利用行波解的存在性证明了线性猜想的有效性。然后，给出了反应-扩散流行病模型的行波解的速度大于线性猜想的速度预测值的例子。

(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.