Death of the Bees: a Mathematical Model of Colony Collapse Disorder

Transcription

1 Death of the Bees: a Mathematical Model of Colony Collapse Disorder Lindsey Dornberger Christopher Mitchell Brian Hull, Wilber Ventura, Haley Shopp Christopher Kribs-Zaleta Hristo Kojouharov, James Grover Technical Report

2 Death of the Bees: a mathematical model of Colony Collapse Disorder * Lindsey Dornberger, Christopher Mitchell, Brian Hull, Wilber Ventura, Haley Shopp #, Christopher Kribs-Zaleta, Hristo Kojouharov, James Grover A mysterious problem has developed within honey bee populations; in a worst case scenario, bee hives will spontaneously collapse as the entire population disappears from the hive. This phenomenon has been named Colony Collapse Disorder (CCD). The problem is recent and has no known cause, though it is surmised to stem from one or multiple infections. Though the causative agent(s) of CCD remains unknown, there are several defining factors of an infected colony; disappearance of the majority of hive bees while the queen remains, and the avoidance of the abandoned hive by scavenging species. Absence of bee carcasses suggest the forager bees, whose job it is to obtain food for the colony, evacuate once affected. Adolescent hive bees, which care for the larval bees and the queen, are then drawn into premature foraging behavior. In order to gain insight into the possible dynamics of this phenomenon, we created a mathematical model that simulates the dynamics of a healthy hive. To this model we then add equations representing the disorder affecting the population. These equations account for the rapid exodus of forager bees from the hive; the loss of forager bees initiates the collapse of the entire colony. Our model predicts that the transition rate from hive bee to forager bee, along with the queen s reproductive rate, dictates the outcome of an affected colony. Introduction The honeybee industry is one of great economic significance. In the agricultural business, their primary purpose is to pollinate plants. Honeybees are kept in apiaries with millions of bees available to be rented and used for pollination. In the United States alone, honeybee pollination accounts for an estimated fourteen billion dollars per year of agriculture (Oldroyd 2007). Additionally, honeybees produce honey, beeswax, and other products which are used on a day to day basis. A healthy honeybee hive has three classes of bees. A bee begins its life as a member of the brood. All larvae belong to this brood class and are nurtured by adult worker bees in a capped honeycomb compartment through all stages of their larval development. After maturing out of the larvae phase, adult worker bees move into the hive worker class. Hive bees are responsible for maintaining the temperature of the brood chamber in the hive, build the combs of the hive, clean and repair the hive, and ventilate, cool, and heat the hive (Winston 1987). After being a member of the hive worker class for 7-21 days, a bee will mature into the forager class (Winston 1987). Forager bees are responsible for gathering outside the hive. Forager bees make between five and twenty flights a day to gather nectar and pollen from plants surrounding the hive. They return to the hive with nectar, which is stored by the hive bees. After being a forager for days during the foraging season, or up to four months in the winter, a bee will die (Winston 1987). A healthy honeybee hive has between 50,000 and 80,000 bees during its peak foraging season, which lasts approximately from May until October. * This research was supported by an NSF UBM-Institutional grant DUE# as part of the UTTER Program at UT Arlington ( Department of Mathematics, The University of Texas at Arlington, P.O. Box 19408, Arlington, TX Department of Biology, The University of Texas at Arlington, P.O. Box 19498, Arlington, TX # College of Natural Sciences and Mathematics, Dallas Baptist University, Dallas, TX 75211

3 Beginning in the winter of 2006, there has been a massive loss of honeybee colonies. Some apiaries have lost up to 90% of their colonies (Oldroyd 2007). The cause of this colony loss is not yet known, and it has been termed Colony Collapse Disorder. In order for a colony loss to be officially declared the result of Colony Collapse Disorder, there are certain characteristics that must be present. There must be a rapid loss of both forager and hive bees, leaving the queen, capped brood, and food stores in the hive (Oldroyd 2007). There is an absence of dead forager and hive bees around the hive, suggesting that the forager and hive bees have left the hive to die, perhaps in an attempt to protect their hive (Oldroyd 2007). Additionally, other scavengers that generally invade abandoned hives are not found in hives that have fallen to Colony Collapse Disorder for an extended period of time (Oldroyd 2007). Mathematical Model This is a model of Colony Collapse Disorder in a honeybee colony. CCD, as it is known, is a major problem on honeybee farms because of the massive decline in colony numbers. The model uses ODEs to show the change in a colony over a typical maximum foraging season, early spring through late summer. One of the problems with CCD is that there is an insufficient workforce to support the colony, because the foragers are getting infected, thus they leave the hive. This forces an early recruitment of younger hive bees into the foraging class. There are three classes in the model: hive (H), forager (F), and infected forager (I). The brood is not represented as a separate class as they seem to have no effect on the infection of the hive. Creation of this model required thorough examination of hive dynamics and was critical to an accurate depiction of hive functions. Understanding the biology of bee to plant pollination patterns led to an initial model of four dimensions. This fourth equation allowed for a disease vector. Through a series of assumptions based on biological reasoning this extra dimension was subsumed by the k parameter. This reduced the model to three dimensions thus easing the analysis. The model: ( ) ( ) ( ) As is standard with this kind of model the transition rates between the classes are of utmost importance. Hive to forager transition rates appeared to be greatly affected by the disorder and their examination was a good starting point. Since CCD, as modeled, seems to increase this rate we decided to use two rates, a healthy rate and an unhealthy rate. In order for the unhealthy rate to take over there must be some mechanism to describe when to start recruiting faster from the hive class into the foragers. How this term,, works is when the foragers and the infected foragers combined reach some low level threshold, the maximum recruitment rate, α kicks in. The γ term represents healthy dynamics. It is the normal recruitment rate of hive bees into the forager class represented as an average 2

4 amount of time in the hive class. Bees work in the hive for 21 days then become foragers (Amdam 2003). The α term represents the maximum recruitment rate of hive bees to forager bees. It will take over when the foragers reach some threshold and there are not enough to forage for the hive. It takes a minimum of 7 days after a bee hatches before it can actually forage, so (Amdam 2003). L is the queen s average daily egg laying rate, approximately eggs per day (Schmickl et al. 2007). is the healthy death rate of a forager bee with no infection, approximately occurring at 21 days (Amdam 2003). is the average time that an infected bee remains a functional forager before leaving or dying, set at 1/1.25 days. and are half saturation constant. They represent thresholds of bees. is the minimum number of bees needed to rear the brood in order for the hive to survive. is the threshold of the total foragers (healthy and infected) below which you would have maximum recruitment and above which you would have healthy recruitment. is the infection rate of susceptible forager bees, represented as mass action, which is a combined contact rate of bees and rate of transmission of the infection. k is the representation of the infection of flowers derived from a system with flowers being their own vector equation., where p is the total number of plants in the hives foraging radius, c is the clearance rate of the infection from the plants, and b is the contamination rate of flowers by bees. Equilibria Analysis From the model we derived six equilibria of the system. One of them is always biologically irrelevant as we are dealing with negative populations. Of the other five there are two endemic 3

5 equilibria, two disease-free equilibria and one extinction/trivial equilibrium. The first is the extinction/trivial equilibrium: In this case we see that all classes become extinct. This equilibrium is always stable if. This term is important as it arises in the disease-free equilibria feasibility conditions. There are two different disease free equilibria. The smaller disease-free equilibrium,, is feasible when: And the larger disease-free,, is feasible when: Or: Now notice that when the conditions for the smaller equilibrium are met, the larger equilibrium conditions are also met. Therefore both can exist at the same time. We were able to calculate the basic reproductive number,, to determine the conditions for endemic and disease-free states of the hive. Here we have. When, then if both disease-free equilibria exist, the larger one is always stable while the smaller one is unstable. This in turn creates an Allee effect, meaning that if the initial conditions are below the smaller disease-free equilibria, then the hive is not strong enough to support itself and it goes to the stable extinction equilibria due to demographic failure. If however, only one disease-free equilibria exists it is always stable and the extinction equilibria loses its stability. The endemic equilibria are exceedingly complicated but there are some global behaviors that can be discussed. When one or two endemic equilibrium are able to exist in which case the disease persists within the hive. There is also the case where no endemic equilibrium exists and the hive goes extinct due to CCD. Numerical Results A period of 250 days was used to model a hive from the end of winter through the end of summer, which is the maximum foraging season of a typical honey bee colony. To estimate parameter values that had no background in the biological literature values were chosen that coincide with the equilibrium analysis. For a value was chosen that insisted the inequality held, and set it at bees. was set at a value of 1000 bees. Since needed to be higher than the normal death rate of forager bees we assigned a value of 0.8 to represent not just death but the removal rate as well. The infection rate was set at 0.8 as well. k was set to be 100. The model is initiated with 15,000 each of forager and hive bees. In a healthy hive there is maintained an even population size between forager bees and hive bees during the foraging season, something the model accounts for. The model also shows a population growth which again is typical of a healthy foraging season. Healthy hives will have anywhere from 50,000 to 70,000 bees. Figure 1 (left) shows the results of healthy hive dynamics. 4

6 The model of an infected colony started with 1000 infected bees and foragers and hive bees each. The same time frame of 250 days was used. As this figure (Figure 1, right) shows, the hive eventually empties out with no bees left to take care of the brood and queen. Once the forager numbers reach a low enough point the maximum recruitment of hive bees to forager bees takes over and gives a massive decline in hive bees. Discussion The goal of this project was to create a model of colony collapse disorder that accounts for healthy hive dynamics, something that was not previously seen in literature. The model accurately depicts a healthy hive over one typical maximum foraging season without an infection, though it does not demonstrate the natural fluctuation of hize size following foraging and throughout the winter season. When infection is introduced the colony does die out; it also declines due to an emptying of the forager bee class, which leads to an early recruitment of hive bees into the depleted foraging class. This in turn leaves an insufficient workforce of hive bees to care for the brood and queen, something that is typical of a hive that has experienced CCD (Oldroyd 2007). All of the numerical simulations from the model match data found in CCD literature, that there is a massive reduction of activity around the hive, with few hive bees remaining with the queen and brood left behind. Though the results are accurate, there is an issue within the model regarding the parameters that have no biological background in the literature, diminishing the certainty of our results. Further work needs to be done to explore our parameter k, which we used to incorporate the disease vector and transmission possibilities. Other parameter values were established from the equilibria analysis, such as and. Once more evidence has been collected and the exact cause of CCD is identified, the model can be modified to incorporate that data to give a more precise description of the dynamics. Conclusion and Future Work This project is an introductory model to help depict healthy hive dymanics so that future work can be done to study CCD. Though the model does not indicate what causes CCD, it does show how it may be working to destroy the hive. Currently, bifurcation work is being done to aid in understanding the equilibria and how they appear and diappear under certain parameter conditions. These plots will give insight into what parameters can affect the system and how this may help lower the severity, or 5

7 stop entirely the disorder s effect on a hive. The next step in the project is to extend the system to multiple hives and how CCD would spread between them. This may lead to more study into preventive measures to stop its spread once a local colony has the disease. References Amdam, G. The Hive Bee to Forager Transition in Honeybee Colonies: the Double Repressor Hypothesis. Journal of Theoretical Biology 223.4: (2003) Cale, G. R. Jr.; Gowen, J. W. Heterosis in the honey bee (Apis mellifera L.). Genetics, 41, (1956) Ellis, J. D.; Evans, J. D.; Pettis J. S. Colony losses, managed colony population decline and Colony Collapse Disorder in the United States. Journal of Apicultural Research 49(1): (2010) Khoury, D.S.; Myerscough, M.R.; Barron, A.B. A Quantitative Model of Honey Bee Colony Population Dynamics. PLoS ONE 6(4): e (2011) Oldroyd BP What's Killing American Honey Bees? PLoS Biology 5(6): e168. (2007) Paxton, R. J. Does infection by Nosema ceranae cause Colony Collapse Disorder in honey bees (Apis mellifera)? Journal of Apicultural Research 49(1): (2010) Schmickl, Thomas; Crailsheim, Karl HoPoMo: A Model of Honeybee Intracolonial Population Dynamics and Resource Management. Ecological Modelling : (2007) Thom, Corinna; Seeley, Thomas Dyer; Tautz, Jurgen. A Scientific Note on the Dynamics of Labor Devoted to Nectar Foraging in a Honey Bee Colony: Number of Foragers versus Individual Foraging Activity. Apidologie 31.6: (2000) VanEngelsdorp, Dennis; Evans, Jay D.; Saegerman, Claude; Mullin, Chris; Haubruge, Eric; Nguyen, Bach Kim; Frazier, Maryann; Frazier, Jim; Cox-Foster, Diana; Chen, Yanping; Underwood, Robyn; Tarpy, David R.; Pettis, Jeffery S. "Colony Collapse Disorder: A Descriptive Study." Ed. Justin Brown. PLoS ONE 4(8): e6481. (2009) Winston ML The biology of the honey bee. Cambridge: Harvard University Press. 281 p. (1987) 6