Abstract:

A diffusive predator-prey model with Beddington-DeAngelis function response and harvesting is studied. Firstly, the sufficient conditions for the existence of positive solutions are obtained by the fixed point index theory, and the multiplicity of positive solutions is discussed by the bifurcation theorem. Then, by virtue of the combination of the perturbation theory for linear operators, degree theory and stability theory, the uniqueness and stability of positive solution are investigated when the interaction among predators is large. In addition, we analyze the extinction and permanence of the two species by means of the comparison principle of parabolic systems. Finally, we make some numerical simulations to validate and complement the theoretical results. If the maximal growth rate of the prey is large and low density mortality of the predator is small, the findings suggest that the model has only one unique asymptotically stable positive solution provided that the effect of the interaction among predators is sufficiently large and the harvesting rate of the predator lies in a certain range; and has at least two positive solutions when the harvesting rate of the predator belongs to another range.

Key words: Harvesting, Nonconstant mortality rate, Uniqueness, Stability, Multiplicity