The worldwide decline in honeybee colonies during the past 50 years has often been linked to the spread of the parasitic mite Varroa destructor and its interaction with certain honeybee viruses carried by Varroa mites. In this article, we propose a honeybee-mite-virus model that incorporates (1) parasitic interactions between honeybees and the Varroa mites; (2) five virus transmission terms between honeybees and mites at different stages of Varroa mites: from honeybees to honeybees, from adult honeybees to phoretic mites, from honeybee brood to reproductive mites, from reproductive mites to honeybee brood, and from honeybees to phoretic mites; and (3) Allee effects in the honeybee population generated by its internal organization such as division of labor. We provide completed local and global analysis for the full system and its subsystems. Our analytical and numerical results allow us have a better understanding of the synergistic effects of parasitism and virus infections on honeybee population dynamics and its persistence. Interesting findings from our work include: (a) Due to Allee effects experienced by the honeybee population, initial conditions are essential for the survival of the colony. (b) Low adult honeybee to brood ratios have destabilizing effects on the system, generate fluctuated dynamics, and potentially lead to a catastrophic event where both honeybees and mites suddenly become extinct. This catastrophic event could be potentially linked to Colony Collapse Disorder (CCD) of honeybee colonies. (c) Virus infections may have stabilizing effects on the system, and could make disease more persistent in the presence of parasitic mites. Our model illustrates how the synergy between the parasitic mites and virus infections consequently generates rich dynamics including multiple attractors where all species can coexist or go extinct depending on initial conditions. Our findings may provide important insights on honeybee diseases and parasites and how to best control them.

This paper considers a significant problem in biological control of algae issue in ecological environment. A four-dimensional dynamic model is carefully formulated to characterize the interactions among phytoplankton, submerged macrophyte, zooplankton, and general fish class in a lake ecosystem. The predation relationship is modeled by Beddington-DeAngelis functional responses derived from the classical Holling time budget arguments. Qualitative analyses of the global dynamics show that the system can generate very rich dynamics with potentially 10 different equilibria and several bistable scenarios. We perform analysis on the existence and local stability of equilibria and explore the refuge effect of macrophyte on the zooplankton with numerical simulations on aquatic ecosystems. We also discuss effective methods of biological control used to restrain the increase of phytoplankton. Our study shows the proposed model could have rich and complex dynamics including but not limited to bistable and chaotic phenomenon. Numerical simulation results demonstrate that both the refuge constant and the density of the macrophytes are two key factors where refuge effects take place. In addition, the intraspecific competition between the macrophyte and the phytoplankton can also affect the macrophyte's refuge effect. Our analytical and simulation results suggest that macrophytes provide structure and shelter against predation for zooplankton such that it could restore the zooplankton population, and that planting macrophyte properly might achieve the purpose of controlling algae growth.

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