Bifurcations in a Honeybee-Parasite Interaction Model in Constant and Periodic Environments

In this paper, we revisit a honeybee-parasite interaction model with constant and periodic queen egg-laying rates, that is, in constant and periodic environments. In the constant environment, we obtain complete high codimension bifurcations, including a codimension-3 nilpotent saddle bifurcation and a codimension-2 Hopf bifurcation, and find that the nilpotent saddle of codimension 3 acts as the organizing center of the bifurcation sets. Our results indicate that the interacting outcomes (coextinction, survival of only honeybees, and coexistence) can be classified by a threshold value for the semisaturation constant of the honeybee population and two threshold values for the parasitism rate. In the periodic environment, we explore the bifurcation of a stable hyperbolic equilibrium (or a stable limit cycle) into an asymptotically stable periodic solution (or an attracting invariant torus). Numerical simulations are given to illustrate the theoretical results, including multitype bistability, tristability, existence of limit cycles, and invariant tori.