Dynamics of two predator-prey models with power law relation

In this paper, we propose a predator-prey model with power law relation based on the model in [Hatton et al, The predator-prey power law: Biomass scaling across terrestrial and aquatic biomes, Science 349(2015), aac6284], and analyze the global dynamics of both models. We obtain that Hatton's model is persistent for power less than 1, and there exists a separa-trix near the origin such that solutions of the model above it are driven to the origin and the ones below it are far away from origin for power greater than 1. However, our model is persistent for all power and has the same singularity as that of Hatton's model at the origin for power greater than 1, which indicate that the prey and predator will coexist or extinct eventually. Furthermore, in our model, the prey will be stable at its carrying capacity and the predator will be extinct for power less than 1, and the prey will be stable at its carrying capacity or both the prey and predator will be extinct for power greater than 1.