双噪声扰动下随机Fisher-KPP方程的行波及动力学性质研究

行波是Fisher-KPP方程的重要特征，对物种扩散现象具有很好的解释。由于现实世界中噪声干扰无处不在，本项目将考虑双噪声扰动下随机Fisher-KPP方程的行波问题。基于物种扩散的背景，将双噪声扰动的类型分为三种，分别为具有随机切向流和随机栖息地承载量的扰动、具有随机切向流和随机物种繁殖速率的扰动以及具有随机栖息地承载量和随机物种繁殖速率的扰动。从双噪声如何共同作用对Fisher-KPP方程原有行波造成影响的角度出发，对于一维空间变量情形，本项目拟获得随机行波存在时噪声强度和相关性参数所需满足的条件，以及波速与噪声强度、相关性参数之间的函数关系，进一步得到解的动力学性质关于噪声强度、相关性参数的分支。对于高维空间变量情形，由于没有随机行波的定义，将以具有紧支集的非负函数作为初始函数获得解的演化特征，以此作为高维行波的传播特征，特别是刻画传播特征的扩散面函数与噪声强度、相关性参数间的关系。

Traveling wave is an important feature of Fisher-KPP equation. It can give well interpretation for diffusion of species. Since perturbations of noises are common in real world, the project will consider traveling waves of Fisher-KPP equations perturbed by double noises. Under the background of diffusion of species, the combinations of double noises consist of three types including stochastic shear flow and stochastic capacity of habit, stochastic shear flow and stochastic production rate of species, stochastic capacity of habit and stochastic production rate of species. To see the effects of noises on the original traveling wave of deterministic Fisher-KPP equation, in the case of one dimensional space, the project plans to obtain the conditions of intensity and correlated parameter of noises under which the existence of stochastic traveling waves can be established. Moreover, the project will present relations between wave speeds and intensity and correlated parameter of noises and bifurcations of dynamical properties of solutions with respect to intensity and correlated parameter of noises. In the case of high dimensional space, due to the lack of definition of stochastic traveling waves, the project plans to obtain the evolutive features of solutions at the initial conditions of nonnegative functions with compacted supports. Especially, the project will present the relations between functions of diffusing surface and intensity and correlated parameter of noises.