近年来，随着研究手段的发展和认识的深入，生物学家和数学家在对宏观生态系统中种群演化规律以及微观细胞膜中棒状体蛋白及钙离子浓度的变化等问题的研究中建立了一些非局部反应系统。这些非局部反应系统比传统的反映扩散系统更能准确地反映实际问题中状态变量的真实变化过程。然而，对这类系统中由于非局部项而导致的复杂的动力学尤其是高余维分支问题的研究才刚刚起步。本项目通过对反映项中具有空间非局部、时滞非局部、具有空间记忆扩散的这三类非局部系统的稳定性、空间强（弱）共振分支、双Hopf分支、空间共振-Hopf分支以及一些余维3的分支问题进行深入研究，揭示非局部效应、空间记忆扩散、时空时滞联合作用而导致的复杂时空动力学的理论机制。

Recently, the biologists and mathematicians have established some nonlocal systems during the study on the population evolution in the macroscopic ecological system and microscopic on the change of the concentrations of rhoptry protein and calcium ion in the microscopic cell membrane with the development of research method and the improvement of epistemology. Comparing with the classical reaction-diffusion system, these nonlocal systems can more accurately describe the changing process of the corresponding state variables. However, the researchers have only just begun to investigate the complex dynamics, especially the higher codimension bifurcation，induced by nonlocality and diffusion. This project deeply study the stability、 spatially strong/weak resonant bifurcation、 double Hopf bifurcation、spatially resonance-Hopf bifurcation and some codimension-3 bifurcations in the following three kinds of nonlocal reaction system: spatial nonlocal system、delayed nonlocal system and the spatial memory diffusion system with nonlocal term, to reveal the theoretical mechanism of the complex spatiotemporal dynamics induced by the effects of the interaction of nonlocal effect, spatial memory and diffusion.