Diffusion-mediated persistence in three-species competition models with heteroclinic cycles

We consider a model composed of two patches. One patch has three competing species forming a heteroclinic cycle within the path. The other is a refuge for one of the three species, which can diffuse between the two patches. The remaining two competitors are confined to the competitive patch and cannot diffuse. A new heteroclinic cycle can exist in the model, and the underlying cycle in the competitive patch cannot appear with a positive diffusion rate. It is proved that the model can be made persistent under appropriate diffusion conditions even if the underlying heteroclinic cycle is an attractor in the competitive patch and the patch is not persistent without the refuge. Further it is shown that the model with a specific structure is globally stable if the underlying cycle is a repeller.