compare and contrast a high cycle versus low cycle fatigue models?

Answer 1

The distinction between high-cycle fatigue and low-cycle fatigue is made by determining whether the dominant component of the strain imposed during cyclic loading is elastic (high cycle) or plastic (low cycle), which in turn depends on the properties of the material and on the magnitude of the stress.
(From ref. 1)
Ref. 2 provides a bit more detail on typical LCF and HCF environments.