空间异质交错扩散系统的稳态解和分支

A large number of diffusion systems with cross-diffusion and spatially heterogeneous environment have been proposed in the area of population ecology, epidemiology etc, and this kind of systems can better describe cross-diffusion and spatially heterogeneous for practical problems. Researching the existence, the quantity, the stability of the positive solutions and Hopf bifurcation problem of the mentioned systems above has important biological and theoretical meanings. However, cross-diffusion and spatially heterogeneous environment will induce the difficulties on the study of mathematical theory. Based on the above reasons, this work will be devoted to investigating the stationary solutions and bifurcations of reaction-diffusion systems with cross-diffusion and spatially heterogeneous environment. We will obtain the methods that can overcome the difficulties resulting from the sensibility of the spatially heterogeneous for the present mathematical methods and the estimates of upper and lower bounds, and solve the problem of the system without variational structure. Then the application of the obtained results in population ecology and epidemiology will be discussed.

在种群生态学和流行病学等学科领域中，大量具有交错扩散、空间异质的扩散系统被建立，其能够更好地描述实际问题中的交错扩散、空间异质现象。研究这类系统正解的存在性、存在个数与稳定性以及Hopf分支等问题具有重要的理论和实际意义。然而，交错扩散和空间异质将导致数学理论研究的困难。基于此，本项目将致力于具交错扩散和空间异质的扩散系统的稳态解和分支的研究。寻找克服已有的数学方法对空间异性变化太敏感和上、下界估计所造成困难的方法；解决这类系统不具有变分结构的困难。并探讨这些结果在生态学和流行病学中的应用。

在种群生态学和流行病学等学科领域中，具有交错扩散、空间异质的扩散系统能够更好地描述实际问题中的一些重要现象。本项目主要研究了这类系统中具有重要理论和实际意义的正解的存在性、稳定性和Hopf分支等问题。运用能量积分方法得到了非常数正解不存在的充分条件，然后利用Leray-Schauder度理论证明了稳态系统非常数正解的存在性。利用线性化和Kato的特征值扰动理论，得到一类交错扩散系统正稳态解的局部稳定性。同时证明了交错扩散对平衡解的稳定性存在影响，并获得非常数正解的存在性和不存在性，Hopf分支的一些相关性质。并对得到的结果给出合理的生态学解释。