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Honeybee populations

The modelling of eusocial insect colonies differs significantly from the modeling of other population dynamics, because they have to deal with:

  • Only one (or only a few) reproductive individuals
  • Division of labor (castes ...)
  • Age structured populations
  • Collective decisions
  • Resource storage, resource processing and resource management
  • Reproduction on colony level
  • Highly specialized parasites and enemies
  • Competition on an inter-colonial and on an intra-colonial level

We produced a set of honeybee population models, that try to incorporate the first 5 points yet (and will maybe incorporate all points in future).

The models are all “offsprings” of “SimBee”, which was a tabular (spreadsheet) model describing the population dynamics that result from a specific (season and intra-colonial) state dependent egg laying rate. The model already included weather data to predict foraging efforts and foraging success, behavioral data (on a group level) by modeling the collective division of labor into nurses and foragers. It models a complete brood demography but tracks adult bees only in two age groups (young bees and old bees). Also  nectar collection and the processing of nectar into honey is not part of this model. And the expected maximum life span of an adult bee was deducted from the season (external) and not due to internal feedback loops (life history of the workers and colony status). But the effective mean life span was already regulated by internal feedbacks, as several task specific mortalities affect this value. The model was able to simulate also a set of “events” like the leaving of a swarm, rain periods, removal of brood, removal of pollen, removal of honey, pollen traps etc.

One of the major impacts of SimBee was that we developed the so called “population backbone” of the colony. Speaking in terms of “Stock&Flow” modeling, the bees follow a flow from one stock to the other when they age day by day. Each stock represents one age group and (of course) flow between the stocks is only unidirectional. But as there are bees (and brood) dying in this process, each stock must also have a certain “leakage” which represents the specific mortality of that age. This mortality is not only age-dependent, it also depends on the day in season, the environment and the colony’s total status.

You see a picture of such an population backbone created in Vensim below (click on the figure to zoom it to full size):

All further improvements of the basic model (SimBee) kept this “population backbone” and just improved the way the resource state, the task decisions and other factors act upon these mortalities.

The first improvement of SimBee was an application programmed by Andreas Gobiet called “SimBee.lesslie ”. It was an approach based on Lesslie-matrices implemented in IDL, which is a mathematical programming language. One major improvement of this attempt was to solve the problem with the maximum life span of adult bees. In this attempt, bees were “born” with a certain amount of days “on stock” and consume life span accordingly to the tasks they perform each day. Additional, this “life stock” is changed throughout the season, but not at the order this was done in SimBee.

Based on SimBee (which was implemented in Excel) we generated another version of this model in Mathematica. This was done because some technical limitations of Excel prevented us from further improvement of the model. Although we changed some basic ideas by transferring the model (e.g. making the model able to deal with different time spans for developmental stages), this difference-equations based new model mainly represents the former table-based model. It is called “HoPoMo” (Ho neybee Population Model) and it was the new base for further improvements.

Based on HoPoMo we implemented “HoPoMo.basic”, which incorporated now also nectar collection and processing. This brought the need for a new task allocation module into the model, a new submodel that decides for each day, how the available workforce is split up into the tasks that have to be performed (nursing, pollen-foraging, nectar-foraging, nectar-processing). But this model still tracked all adult bees only in two age classes. The model was formulated rather complicated, because it assumed that young adults mainly perform nursing and old adults mainly perform foraging. To keep the model valid in spring and in autumn (when age demography differs significantly), we needed a variety of extra formulations that allowed also young bees to forage (if needed) and older bees to nurse.

Based on “HoPoMo.basic”, which was formulated rather complicated as said before, we performed a total simplification of the model. This model cutted away the adult demography and was able to produce similar results with significantly less equations with very short calculation time. In this model, we did not assume a maximum life span for adult bees anymore, the daily mortality of bees was only calculated by internal colony factors and by the mean task histories of the bees. This model was called “HoPoMo.simple”.

Based on “HoPoMo.basic”, we developed also in another direction and produced a model that tracks all adult bees in one-day-wide age classes. This gave us a full demography of the colony, both for the brood and for the adults. This way, we could experiment with several task assignment algorithms that predict division of labor as seen in several real experiments. The price for this is a very long calculation time, as there are more than 280 age classes to be calculated per day per task group.  This model was called “HoPoMo.demographic” (not published yet).

Please note that all HoPoMo models use the same 95% of code (equations) for modeling the hive conditions, the brood demography and for the resource management. They only differ in the way they track adult bees and in the way they assign tasks to these bees.

Beside the branch of HoPoMo models, we also generated a VERY, VERY downscaled version of SimBee in Excel again, which was called “MiniBee”. This mini model is used in our seminars to demonstrate to the students how the population dynamics of eusocial insects can be described and explored in a spreadsheet program.

Here you can see a scheme of the development history:

 

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