Persistence analysis of mathematical ecological models

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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).