Henry et al. (2012) homing failure formula, assumptions, and basic mathematics: a comment

In March 2012 Henry et al. published a paper that explored whether or not the consumption of thiamethoxam via nectar could be a causal factor of Colony Collapse Disorder (CCD) in honeybees. In the first part of their report, Henry et al. (2012) measured the homing success after “ecologically relevant” thiamethoxam exposure and compared it to non-thiamethoxam exposed, control homing rates [see Guez (2013) for a critique of this aspect of their work]. In the second part of their report, they applied their homing study results to the honeybee population dynamic model devised by Khoury et al. (2011), and from that they concluded that dietary thiamethoxam intoxication may potentially contribute to CCD. Khoury et al.’s (2011) model is build upon the hypothesis that colony failure occurs when bee death rate become unsustainable at the colony level, and the salient assumption that mortality within the hive is negligible. Khoury et al.’s (2011) model allows the evolution of the honeybee hive population to be projected over time. Model outputs are dependent on the total hive population (at the start, and then at any given time), the queen’s egg laying rate (L), an eclosion rate that is directly dependent upon the hive population and modulated via the parameter w (the larger w the lower the eclosion rate), and the forager mortality rate or forager homing failure (m). In Khoury et al.’s (2011) model therefore, the population growth of the colony is controlled mainly by the parameters L and w [but see Cresswell and Thompson (2012) for a critique of the choice of w], whereas population decline is dependent on m, the forager mortality rate or forager homing failure. In order to model the population dynamic under dietary thiamethoxam exposure, Henry et al.’s (2012) undertaking was to calculate the homing failure due to pesticide exposure (mhf). mhf was then used to increase the value of m for population projection under a dietary thiamethoxam exposed scenario, compared to the “normal” homing failure m postulated in the non-thiamethoxam exposed scenario. In my previous critic of Henry et al. (2012) I pointed out that the way in which Henry et al. calculated mhf was incorrect given the authors claim that mhf “[. . . ] estimates the proportion of exposed foragers that might disappear due solely to post-exposure homing failure, all other sources of mortality or homing failure set apart (natural mortality, predation, manipulation stress).” In their answer to my critic (Guez, 2013) Henry and Decourtye (2013) maintain that the formula used in Henry et al. (2012) to calculate homing failure in honeybees post pesticide exposure (mhf) is correct. In this note I show that the calculated mhf value is largely impacted by assumptions, and that regardless of which assumption is taken, Henry et al.’s (2012) formula for calculating honeybee homing failure post-pesticide exposure is incorrect. In their original report Henry et al. (2012) propose that: