Modeling bumble bee population dynamics with delay differential equations

Abstract

Bumble bees are ubiquitous creatures and crucial pollinators to a vast assortment of crops worldwide. Bumble bee populations have been decreasing in recent decades, with demise of flower resources and pesticide exposure being two of several suggested pressures causing declines. Many empirical investigations have been performed on bumble bees and their natural history is well documented, but the understanding of their population dynamics over time, causes for observed declines, and potential benefits of management actions is poor. To provide a tool for projecting and testing sensitivity of growth of populations under contrasting and combined pressures, we propose a delay differential equation model that describes multi-colony bumble bee population dynamics. We explain the usefulness of delay equations as a natural modeling formulation, particularly for bumble bee modeling. We then introduce a particular numerical method that approximates the solution of the delay model. Next, we provide simulations of seasonal population dynamics in the absence of pressures. We conclude by describing ways in which resource limitation, pesticide exposure and other pressures can be reflected in the model.

Lethal and sublethal effects of toxicants on bumble bee populations: a modelling approach

“…We note that larvae were subdivided into age groups and we assumed that consumption was constant across each age group, with nectar being consumed at twice the rate of pollen (Pereboom 2000). The model we built was a system of delay differential equations (DDE's) which is appropriate to use in age structured population models (Murdoch et al,1987;Hartung et al 2006;Banks 2012;Banks et al 2017). The model included time varying larval mortality rates (μ (w) (t), μ (m) (t), μ (g) (t)) which were based on past values of the workers, pollen, and nectar variables.…”

Impacts of Neonicotinoids on the Bumble Bees Bombus terrestris and Bombus impatiens Examined through the Lens of an Adverse Outcome Pathway Framework

“…As an alternative to conducting tests in multiple species to fill these data gaps, some investigators are calling for increased use of in silico tools to support risk assessment performed on non‐ Apis bees (Thompson and Pamminger 2019). Seven bumble bee–specific models have been published (Crone and Williams 2016; Banks et al 2017; Cresswell 2017; Haussler et al 2017). Bumble‐BEEHAVE, the most advanced model available, is designed to simulate colony growth and survival and to predict the effects of multiple stressors on bumble bee survival at the level of the individual, colony, and population (Becher et al 2018).…”

Bumble‐BEEHAVE: A systems model for exploring multifactorial causes of bumblebee decline at individual, colony, population and community level

“…Six contrasting bumblebee models have recently been published (Banks et al., ; Bryden, Gill, Mitton, Raine, & Jansen, ; Cresswell, ; Crone & Williams, ; Häussler, Sahlin, Baey, Smith, & Clough, ; Olsson, Bolin, Smith, & Lonsdorf, ). However, while useful in exploring the impact of individual stressors, such as food availability (Crone & Williams, ) or pesticides (Bryden et al., ; Cresswell, ), none as yet have the structural realism to incorporate multiple stressors or competition, operating at different organisational levels (individual or colony or population).…”

“…Six contrasting bumblebee models have recently been published (Banks et al, 2017;Bryden, Gill, Mitton, Raine, & Jansen, 2013;Cresswell, 2017;Crone & Williams, 2016;Häussler, Sahlin, Baey, Smith, & Clough, 2017;Olsson, Bolin, Smith, & Lonsdorf, 2015).…”

“…They have limited flexibility to incorporate feedback mechanisms that may buffer the colony against environmental stress, or indeed lead to spiralling collapse. This mechanistic richness is essential for deep and broad understanding of risk (EFSA, 2015)-indeed Crone and Williams (2016) and Banks et al (2017) noted that further processes and stage structure need incorporation. Most existing studies do not model multiple colonies (Bryden et al, 2013;Cresswell, 2017;Häussler et al, 2017; although see Banks et al, 2017) or capture the spatio-temporal dynamics of resource availability (although see Häussler et al, 2017;Olsson et al, 2015;Polce et al, 2013) which are essential to make accurate predictions in real landscapes.…”