Additive Allee effect on prey in the dynamics of a Gause predator–prey model with constant or proportional refuge on prey at low or high densities

Assuming that the intrinsic growth of prey is affected by an additive Allee effect, and from which a proportion to the population or critical size of prey hide from the predator, and thus avoid being consumed by it, if the quantity of prey is above or below the critical size, respectively, this paper proposes a predator–prey Gause model composed of two vector fields separated by the critical prey population size. Since the proposed model is not discontinuous, it could have a single interior equilibrium, belonging to one of the vector fields of the model, and a stable limit cycle formed between the two vector fields, and a possible extinction of the prey considering a strong Allee effect.