How Hives Collapse: Allee Effects, Ecological Resilience, and the Honey Bee

Abstract

We construct a mathematical model to quantify the loss of resilience in collapsing honey bee colonies due to the presence of a strong Allee effect. In the model, recruitment and mortality of adult bees have substantial social components, with recruitment enhanced and mortality reduced by additional adult bee numbers. The result is an Allee effect, a net per-individual rate of hive increase that increases as a function of adult bee numbers. The Allee effect creates a critical minimum size in adult bee numbers, below which mortality is greater than recruitment, with ensuing loss of viability of the hive. Under ordinary and favorable environmental circumstances, the critical size is low, and hives remain large, sending off viably-sized swarms (naturally or through beekeeping management) when hive numbers approach an upper stable equilibrium size (carrying capacity). However, both the lower critical size and the upper stable size depend on many parameters related to demographic rates and their enhancement by bee sociality. Any environmental factors that increase mortality, decrease recruitment, or interfere with the social moderation of these rates has the effect of exacerbating the Allee effect by increasing the lower critical size and substantially decreasing the upper stable size. As well, the basin of attraction to the upper stable size, defined by the model potential function, becomes narrower and shallower, indicating the loss of resilience as the hive becomes subjected to increased risk of falling below the critical size. Environmental effects of greater severity can cause the two equilibria to merge and the basin of attraction to the upper stable size to disappear, resulting in collapse of the hive from any initial size. The model suggests that multiple proximate causes, among them pesticides, mites, pathogens, and climate change, working singly or in combinations, could trigger hive collapse.

Sepsis and Hemocyte Loss in Honey Bees (Apis mellifera) Infected with Serratia marcescens Strain Sicaria

“…Dynamics of infectious agents impacting honey bees have been modeled to reveal surprising complexity in disease transmission within hives [57–59]. Similarly, we expect that the number of bees infected by Ss1 within a colony depends upon several variables.…”

Markerless tracking of an entire insect colony

“…Indeed, the colony collapse phenomenon currently threatening bee populations worldwide have been ascribed to a range of factors 42,43 . However, despite a range of theoretical models 44,45 , no quantitative, data-driven assessment has yet been proposed. Unraveling the exact reasons for and mechanism of honey bee colonies collapse would require multiple comparative observations across a range of conditions.…”

On the Effects of Artificial Feeding on Bee Colony Dynamics: A Mathematical Model

“…Its inverse, 1/ m , represents the average flight span of a forager, in days. An alternative mortality expression has been proposed in [15], where bee mortality is described using a function that is higher for a small number of foragers and converges to a constant rate only if the number of foragers is high enough. However, for the sake of simplicity, we have chosen to maintain a constant rate of forager death.…”

“…In spite of their relative mathematical simplicity, these models are more appropriate to analyze general factors that may affect bee population dynamics and lend themselves better to hypothesis testing [15]. This is the case of the framework proposed by Khoury et al [16, 17], which describes a colony population dynamics and has been used as a basis for studies by Russel et al [18], Betti et al [19], and Perry et al [20].…”

“…Dennis and Kemp [15] proposed a model that is governed by a single differential equation; it was used to perform an extensive analysis of Allee effects on the colony, as well as an analysis of ecological resilience.…”