Threshold Dynamics of Honey Bee Colonies with General Eclosion and Social Inhibition

Honey bee colonies are important to worldwide agriculture and the health of natural ecosystems. Understanding the factors behind colony persistence and failure has been a major challenge for both ecologists and mathematicians. In this paper, we present a general mathematical model to explore the threshold dynamics of honey bee colonies. We explicitly define the basic reproduction number in terms of model parameters and demonstrate its critical role in determining whether a colony survives or collapses. We offer a biological interpretation of the basic reproduction number and prove the threshold dynamics for the general model system. Our results are established using dynamical systems techniques, including the comparison principle, persistence theory, linearization, and global analysis. For specifically chosen eclosion and social inhibition functions, we perform a sensitivity analysis to determine how the basic reproduction number depends on the model parameters.