Dynamics of a non-autonomous biocontrol model on native consumer, biocontrol agent and their predator

In this paper, we propose and study the dynamics of a non-autonomous biocontrol model concerned with native consumer A, biocontrol agent B and their common predator L. We assume that some parameters in our proposed model are nonnegative functions instead of being bounded below by positive reals. This assumption is more realistic but makes mathematical proofs more challenging. We first study the positive invariance, permanence, and the global attractivity of bounded positive solution and boundary solution of the proposed model with general time-dependent parameters. Then we focus on the dynamics of the model with periodic parameters which include the existence and uniqueness of positive periodic solutions, and the global asymptotic stability of boundary periodic solution. We also explore and discuss the effects of introducing the biocontrol agent to ecosystem through theoretical analysis and numerical simulations. Our results show that introducing the biocontrol agent to ecosystem has positive effects by promoting its biodiversity and coexistence of all species, also potentially has negative effects by eliminating the predator. In addition, our numerical simulations show that (i) the amplitudes of periodic parameters could affect the permanence of non-autonomous periodic system; and (ii) the non-autonomous periodic system may suppress or improve the permanence of its autonomous version but related to the amplitudes of periodic parameters.