Migration Effects on Population Dynamics of the Honeybee-mite Interactions

¹ Simon A. Levin Mathematical and Computational Modeling Sciences Center, Arizona State University, Tempe, AZ 85281, USA
² Carl Hayden Bee Research Center, United States Department of Agriculture-Agricultural Research Service, Tucson, AZ 85719, USA
³ Departamento de Ingenieria Biomédica, Universidad de los Andes, Bogota, Colombia
⁴ Sciences and Mathematics Faculty, College of Integrative Sciences and Arts, Arizona State University, Mesa, AZ 85212, USA

^* Corresponding author. E-mail: yun.kang@asu.edu

Abstract

Honeybees are an amazing and highly beneficial insect species that play important roles in undisturbed and agricultural ecosystems. Unfortunately, honeybees are increasingly threatened by numerous factors, most notably the parasitic varroa mite (Varroa destructor Anderson and Trueman). A recent field study showed that migrations of mites into hives on foraging bees greatly contributes to the rapid growth of mite population in colonies, and increases the mortality of honeybees. Motivated by this, we propose a simple two-patch honeybee-varroa model to explore how foraging behavior of honeybees in the presence of varroa mite infestations affect the population dynamics of honeybees and mites, respectively. We provide a full analysis on the local and global dynamics of our proposed two-patch model, which incorporates mite migration generated by honeybee foraging activities. Our analytical and numerical studies reveal the dynamical outcomes of migration including: (a) Mite's extinction cannot be prevented by mite migration when mite population in each patch goes extinct in the absence of mite migration, however, mite migration could drive mite extinct under proper conditions. (b) Under proper conditions, high rates of mite migration could have the following effects: (1) save one honeybee colony from collapsing when both honeybee colonies are going extinct without migration; (2) drive honeybee extinct in at least one patch. (c) Intermediate migration rate could generate multiple locally stable honeybee-mite coexistence equilibria, and drive mite's extinction under proper environments. (d) An increase in migration rate causes a growth of the varroa population, which in return has a negative feedback on the colony population. (e) Increasing mite migration from a healthy patch to a collapsing patch could reduce the extinction time in the collapsing patch. Our results provide novel insights on the effects of foraging and Varroa migration on colony survival.