Complex dynamical behaviors of a honeybee-mite model in parameter plane

The honeybee has a significant impact on ecosystem stability, biodiversity conservation, and pollinating crops. However, during the past few years, there has been a sharp reduction in both the honeybee population and their colonies. It has been found that the parasitic mite Varroa destructor is responsible for the decline in honeybee colonies around the world, which in turn immensely affects the economic growth of a country. To investigate how the dynamics of a system is influenced by the growth of honeybees and the parasitism of mites, we study a nonlinear honeybee-mite population model in a discrete-time setup. We observe that an increase in the value of the queen’s egg-laying rate drives the system towards chaos. However, chaos can be controlled as well if the parasite attachment effect is increased. The intrinsic dynamical properties of the proposed system are investigated with the simultaneous variation of the queen’s egg-laying rate and the mite’s parasite attachment effect by constructing several largest Lyapunov exponent and isoperiodic diagrams. The investigation reveals the existence of several periodic structures in the quasiperiodic and chaotic regimes of the parameter plane, including Arnold tongues, saddle area, spring area, connected shrimp-shaped structure, and connected saddle area. We also find the appearance of a novel ‘jellyfish’-shaped periodic structure. One of the most fascinating findings of this study is the appearance of Arnold tongues along the inner boundary region of another Arnold tongue. In addition, this work also reveals different types of multistability, e.g., the coexistence of two, three, and even four attractors. What is more interesting is that the current analysis unveils the coexistence of five attractors as well, more specifically, four different periodic attractors coexist with the trivial fixed point attractor, which is quite rare in ecological systems. The structures of the basins of these coexisting attractors are either smooth or very complex in nature. Furthermore, the present study also discloses the fact that variation in the initial condition of the system can significantly change the appearance of the periodic structures in the parameter plane.