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

（1）Lotka-Volterra竞争模型考虑具有奇异扰动参数的May-Leonard三种群竞争系统。我们首先讨论了该系统共存定态稳定性的全貌，并借助图的方法证明了亚临界Hopf分支的发生。通过形式奇异摄动分析和数值模拟，进一步研究了3种群Lotka-Volterra竞争模型与2种群捕食者-食饵模型之间的关系.（2）高阶自催化反应扩散方程组的行波我们证明了任意扩散系数的两组分高阶自催化反应扩散方程组行波的存在性。当反应物不扩散时，我们进一步讨论了高阶衰变系统的存在性问题。我们的相空间中的矢量场的分析给出了估计的最小传播速度的顺序的自催化和扩散系数。（3）与传染病建模有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。流行病波的传播速度通常由模型方程的线性化推导出来，这被称为“线性猜想”。对于扩散的Kermack-McKendrick模型，我们利用行波解的存在性结果证明了线性猜想的有效性。然后，我们给出了具有行波解的反应扩散传染病模型的例子，这些行波解的速度大于线性猜想所预测的速度值。

(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.