Persistence analysis of mathematical ecological models

本研究的目的是寻找一个数学生态模型持续生存的条件。主要研究结果如下：1.我们考虑一个由两个通过扩散连接的斑块组成的系统，每个斑块有两个竞争者。给出了系统持续生存的条件。（文件1）对于由两个竞争者组成的系统，其中一个竞争者可以在两个斑块之间扩散，另一个竞争者仅限于其中一个斑块，证明了在适当的扩散条件下，即使竞争斑块在没有扩散的情况下是不持久的，系统也可以持久（文献2）.我们考虑一个由两个斑块组成的模型。一个有三个竞争物种，形成一个异宿周期内的补丁。另一个是三个物种之一的避难所。证明了即使竞争斑块中的潜在周期是吸引子，该模型也是持续的（论文3）.考虑了一个由三个避难所和一个竞争补丁组成的具有异宿环的模型。证明了当我们改变扩散常数的值时，系统可能发生Hopf分支，并且在特定情况下可能存在周期轨道（文献6）。

The purpose of this research project is to find out tlie condition for a mathematical ecological model to be persistent. Main neiv results obtained, are as follows :1. We consider a system composed of two patches connected by diffusion and each patch has two competitors. Conditions for persistence of the system are given. (paper 1).2. For a system composed of two competitors, one can diffuse between two patches and the other is confined to one of the patches, it is proved that the system can be made persistent under appropriate diffusion conditions, even if the competitive patch is not persistent without diffusion (paper 2).3. We consider a model composed of two patches. One has three competing species forming a heteroclinic cycle within the patch. The other is a refuge for one of the three species. It is proved that the model can be made persistent even if the underlying cycle is an attractor in the competitive patch (paper 3).4. We consider a model composed of three refuges and one competitive patch with a heteroclinic cycle. It is shown that Hopf bifurcation is possible when we change the value of diffusion constant and periodic orbits may exist in a specific case (paper 6).