Review article

Population modelling of Varroa jacobsoni Oud.

Johan N.M. Calis Ingemar Fries Stephen C. Ryrie

a Department of Entomology, Wageningen Agricultural University,POB 8031, 6700 EH Wageningen, the Netherlandsb Swedish University of Agricultural Sciences, Bee Division,

Box 7044, S-570 07 Uppsala, Swedenc Department of Mathematical Sciences, University of the West of England,

Frenchay, Bristol BS16 1QY, UK

(Received 20 August 1998; accepted 22 February 1999)

Abstract - To understand population dynamics of the mite Varrroa jacobsoni and to enable computersimulations, Fries et al. [12] incorporated available knowledge into a mite population model. In thispaper, we update and extend this model by incorporating more recent data, in particular on miteinvasion from bees into brood cells. By predicting invasion into and emergence from brood cells, themodel proves to be useful to evaluate the effects of changes in model parameters on the mite popu-lation when the distribution of mites over bees and brood are important. The model predicts that a longerbrood rearing period dramatically increases the mite population size and that a relatively larger num-ber of drone brood cells leads to an increased population growth. As mite control treatments often onlyaffect mites either in brood cells or on adult bees, the model can be used to evaluate their effective-ness and timing. The model indicates that changes in parameters that affect the reproductive successof the mites in brood cells have a large impact on the mite population. © Inra/DIB/AGIB/Elsevier,Pari s

Apis mellifera / Varroa jacobsoni / population dynamics / model

1. INTRODUCTION

The parasitic mite of honey bees, Var-roajacobsoni (Acari: Varroidae), is a majorobstacle to beekeeping throughout most tem-perate and some tropical regions of theworld. Adult female mites feed on bees, and

as honey bee brood becomes available, themites leave the adult bees to reproduceinside the bee brood cells. Reproductivesuccess inside the brood cells depends onvarious factors (reviewed by Donzé et al.[10]). As the bee emerges from the broodcells, the mite mother and her adult female

* Correspondence and reprintsE-mail: Johan.Calis@users.ento.wau.nl

progeny return to the adult bees. If total mite

mortality on the adult bees and in the broodis smaller than the production of new mitesin the brood, the mite population willincrease over time. The life cycle of mites ina honey bee colony is schematically repre-sented in figure 1.

To help understand population dynam-ics of V. jacobsoni in colonies of Europeanhoney bees, models of different aspects ofmite population dynamics have been pre-sented [2, 3, 9, 13, 15, 27, 30]. A review offactors affecting mite population dynamicswas presented by Fries et al. [12] togetherwith differential equations modelling themite population dynamics.

The model described here is an extendedversion of that by Fries et al. [12]. In the

original model, the per capita number ofreproductive cycles was set at 1.4. Second,invasion of mites into brood cells was deter-mined by a mean length of the phoreticperiod. In nature, the number of times amother mite will reproduce is determinedby the invasion rate into brood cells and themortality rate of the mites. In the new versionof the model, we included the equations oncell invasion rates presented by Boot et al. [3]and mite mortality data connected with theemerging infested bee [4], thereby avoidingthe non-biological restrictions in the oldmodel.

The invasion rate into brood cells can becalculated when colony size and numbersof brood cells are known. To simulate colonysize and numbers of brood cells, we designed

a bee colony population model. Thus, thenew model is comprised of two parts, amodel of the bee colony, and a model of themite population, which describe the popu-lations they represent over a period of severalyears. The colony model follows a beecolony throughout the year from broodlesshibernating conditions, via a period of breed-ing and growth in summer, and a period ofdecline in fall, returning to a broodless hiber-nating colony. Its output pattern does notchange from year to year, and is unaffectedby the mite model. The output pattern of themite model does vary from year to year andis influenced by the colony model, and inparticular by the number of bees and thenumber of brood cells present in the hive atany time.

The aim of this paper is to describe and

explore the extended simulation model ofthe interaction between honey bees and mitepopulations. Such a model is used forexploring the effects of changes in variouscharacteristics that relate to mite resistance,effects of beekeeping techniques, and to pre-dict the effect of different control treatments.

2. MATERIALS AND METHODS

2.1. The colony model

The colony model describes the number ofdrone and worker brood cells in the colony oneach day of the year. The basic colony model,like that of Fries et al. [12], is based on field datafrom Scotland on counts of numbers of droneand worker brood cells in colonies during thebrood-rearing season as reported by Allen [1].In addition, we use sets of brood data from mid-European conditions (P. Rosenkranz, unpub-lished data) and from neo-tropical conditions[11]. The use of brood data from different cli-matic regions provides us with an opportunityto simulate V. jacobsoni population dynamics indifferent climatic zones. For the additional twobrood data sets, we have assumed that the num-ber of drone cells is a constant proportion of thenumber of worker cells (4 %; derived mean fromthe work of Allen [1]), because more precise dataare lacking.

In the colony model, the amount of brood andthe development time of the brood are used to

calculate the number of emerging adult bees.The total number of adult bees is calculated byassigning a specified longevity of the bees duringperiods of brood rearing. During winter the deathrate is chosen so that the colony returns to thesame size as 1 year previously (for parametervalues see table I). The brood data used and thenumber of bees simulated are illustrated in fig-ure 2a-c.

2.2. The mite population model

Several processes contribute to changes inthe population level of mites within the colony.Mites are either phoretic on the bees or are inthe brood cells where they reproduce. This dis-tribution depends on the invasion rate of phoreticmites into the brood. Both phoretic mites andmites in the brood are subjected to certain levelsof mortality. The mite population dynamics isdetermined by invasion of mites into brood cells,reproductive factors and mortality factors.

2.2.1. Invasion of mites into brood cellsand mite emergence

Boot et al. [3] measured the rate of invasion ofmites into worker and drone cells in the field,and derived empirical relationships between theinvasion rate per day (rD and rW for drone andworker brood, respectively) and the ratio of thenumber of available cells to the number of adultbees: rD = -6.49D/B and rW = -0.56W/B, whereD and W are the number of available drone andworker brood cells, respectively, and B is theweight in grams of adult bees in the colony.D and W are the number of brood cells that are

capped over 1 day and are calculated from thetotal number of cells of both brood type and thespecific development time; B, the weight of thebees, is calculated by assuming that one beeweighs 0.125 g (J. Calis, unpublished data).

Thus, depending on the weight of the colony,a specific number of brood cells that are beingsealed over 1 day are invaded by:

where P is the number of phoretic mites and I isthe number of mites invading cells on that day.Invading mites divide themselves over workerand drone brood in proportion to the quantitiesrw/(rD + rW) and rD/(rD + rW), respectively. Wehave used these formulae in our model, with val-ues of W, D and B varying with time as deter-mined by the colony model being used.

The rate of invasion of mites into brood cellsthus arises from a dynamic relationship betweenthe number of available brood cells and the num-ber of bees on which the mites are distributed.The invasion rate equations have proved useful indescribing the efficacy of trapping mites inworker brood [7] or in drone brood [8]. Martin[25] recently also made an attempt to integrateknowledge on brood cell invasion into a popu-lation model. However, he interpreted the workof Boot et al. [3] quite differently. Since Martinlet the same phoretic mites invade worker anddrone brood independently, mites have to bedivided artificially between worker and dronebrood cells in his model. In our model, the divi-sion between worker and drone brood follows

directly from the invasion rate equations above.

The number of mites that emerge daily fromcells is derived from the number that invadedthese brood cells one postcapping period earlier(using a delay function), modified to reflect mitereproduction and mortality in the cells.

2.2.2. Reproduction

Once the mites have entered brood cells andthe cells have been sealed, the mother mite maystart reproduction. Some mites die before theyare able to start reproduction, partly because theybecome trapped between the cell wall and thebee cocoon. Other mites may fail to start repro-duction, and of those that do reproduce somemay produce non-viable offspring. In addition

to female juvenile mortality, some mites pro-duce only male progeny. Reproduction may beinterrupted as bees uncap sealed cells and removeparasitised bee pupae. The reproductive successof the mite is also influenced by the duration ofthe postcapping period where a prolonged periodincreases the average number of viable female

offspring. K. Langenbach (unpublished data)found that a reduction in the postcapping periodof worker brood from 12 days to 11 days resultedin a 50 % reduction in the number of offspring.Büchler and Drescher [5] found that a 1-h reduc-tion of the postcapping period in worker broodreduced the mite population growth by 8.7 %.We have linearly interpolated these findings topostcapping periods between 11 and 13 days,thus, ranging from a 50 % reduction to a 150 %increase, respectively. For drone brood weassumed a similar reduction and increase between12 and 16 days.We have considered new data on male mite

mortality in sealed brood cells [10, 23, 24]. Thefact that a surprisingly large number of infestedcells lack male progeny does have implicationsfor population dynamics. Young mites that havenot mated may either parthenogenetically pro-duce only sons or may fail to produce offspringat all. In our model, we consider these mites andregard unmated mites as a part of the mite pop-ulation.

2.2.3. Mite mortality

The natural mortality rate of phoretic mitesin summer is taken to be the sum of mites lostafter falling from bees in the hive, and mites lostvia foragers that do not return to the colony. Therate at which mites fall from bees is taken to be0.6 %/day, following the findings of Boot et al.[4]. Further, we assume that bees that die dur-ing the summer are foragers that fail to return tothe hive and that mites on these bees are lostfrom the mite population. The daily number ofmites lost on non-returning foragers is calculatedassuming that the rate of infestation of the for-agers is one third of the average infestation ofbees [19], their number being calculated by thecolony model and with the lifespan of adult beesas shown in table I. The winter mortality rate isassumed to be 0.4 % per day, as in Fries et al.[ 12].

Boot et al. [4] found that 22 % of emergingmites, mothers and female offspring, fall off thehost bees to the bottom of the colony in the first3 days following emergence, which we haveincluded in our model.

Summarising, mites on bees are subjected tomortality and losses in the field. When theyinvade brood cells, they may reproduce andincrease their numbers. If we define reproduc-tive success as the number of living female mites,including the mother, which emerge from a brood

cell per invading mite, then for worker broodthis is:

where RW is the reproductive success per invad-ing mite in worker brood; fW is the number of

daughters per mite in worker brood; pW is a fac-tor representing influence of postcapping periodon the number of offspring in worker brood; uis the fraction of cells uncapped by bees; jW isthe fraction of infertile female mites in worker

cells; gW is the fraction of female mites in workercells which die before reproducing; mW is thefraction of female mites that produce only maleoffspring; s is the fraction of emerging miteswhich survive emergence.

A similar set of relationships holds for dronebrood.

The values of RW and RD derived from the

parameter values in table I are 1.56 and 2.32 liveemerging mites per invaded mite, respectively.

3. RESULTS AND DISCUSSION

3.1. Mite population growth

The model calculates the number of adultfemale mites present on adult bees and inbrood cells. The following simulationsdemonstrate how the mite population willchange over time using three different sets ofbrood data in colonies that do not swarmand are not subject to mite control.

In figure 3, the mite population growthusing brood data of northern, central Euro-

pean and neo-tropical origin is presentedgraphically. The simulations start on 1 Jan-

uary with an initial population of ten phoreticmites and continue for four successive years.The model does not include the negativeimpact that high mite levels may impose onbee colonies and thus the model allows themite population to increase beyond levelstolerated by bee colonies. This is an impor-tant limitation of the model, because inter-action between the mite population and thecolony is evident at high mite populationlevels.

The duration of the brood-rearing periodhas a dramatic impact on the mite popula-tion. Under neo-tropical brood-rearing con-ditions, colonies are likely to succumb toinfestation within a year if the mite popu-lation is left uncontrolled. This is in agree-ment with available data from southern

Europe and California, where mite growthrates may cause colony death within a yearafter effective mite control measures [ 16,21]. Under northern brood-rearing condi-tions, where mite mortality during brood-less winter conditions causes a decrease inthe mite population, the model predicts rel-atively low mite population levels for thefirst 2 years of the infestation. In the third

year, the mite population may reach dam-aging levels and by the fourth year the infes-tations are likely to cause colony death. Thispattern of development is supported by fielddata from Finland [18]. Central European

brood data give rise to mite populationgrowth that is intermediate to northern andneo-tropical growth. Under these conditions,bee colonies are likely to be severely dam-aged by the mite infestation within 2 yearsof the initial infestation. The presented setsof brood data, besides those from Scotland[I], do not contain actual numbers of dronecells, since they were not known. We usedas numbers of drone brood cells 4 % of thenumber of worker brood cells; a mean ofthe data from Scotland. This extrapolationmay be unrealistic, but the simulations stillillustrate the importance of extended periodsof brood rearing. To make realistic simula-tions for different brood-rearing patterns,however, data on numbers of brood cells ofboth drone and worker brood cells through-out the year are needed.

Mite populations build up relatively fastduring the summer months. Observationsof mite population increase have been madeby Calatayud and Verdu [6] using A. m.iberica. They found that the mite popula-tion growth is exponential, i.e. that the pop-ulation is proportional to exp(rt), where thevalue of the growth parameter r is approxi-mately 0.021 per day. This implies that thepopulation would double in size in a periodof 33 days. The results of our model are inaccordance with this: a logarithmic plot ofmite population versus time shows that pop-ulation growth during the peak of broodrearing is indeed exponential, irrespectiveof what brood data are being used. We findthe value of r to be 0.023 per day, some-what greater than that found by Calatayudand Verdu, and corresponding to a popula-tion doubling time of 30 days. In all furtherruns of our model, we have only used thenorthern European brood data for simplicity.

3.2. Cell passages, distributionover bees and brood, mortality rate

Another way to test model predictions isto compare the number of cell passagesmade by an individual mite. A probabilistic

model of mite behaviour can be used to cal-culate the number of cell passages made bya single mite. Suppose that, on a given day,a mite may be either on a bee or in a cell,and that the daily probabilities of transitionbetween these states are:

then the probability distribution of N, thenumber of cell passages made by an indi-vidual mite, is given by:

and

Further analysis shows that the mean<N> of this distribution is

In our model we have an emergent mite

mortality, e, of 22 %. The death rate duringthe phoretic phase, d, is 0.006 per day as inour model. Values of b and c are calculated

by the model on each day of a simulationas being the proportion of mites whichremain on bees and enter a cell, respectively,as shown in figure 4. The mean number ofcell passages per mite (figure 4), during thebrood-rearing period, is fairly constant atabout 4, an estimate that contrasts with thefindings of Fries and Rosenkranz [13]. Theymade observations on the number of cell

passages and suggest that this figure liesbetween 1.5 and 2.0 per mite. They observedan average loss of about 50 % of the mites

per cell passage, from which we can calcu-late the number of cell passages to be 1.8.

However, in their experiment the loss ofmites during the first cell passage was muchhigher than in the subsequent cell passages.

If we disregard this initial loss of mites, as itmay have been due to the experimental pro-cedure of introducing the mites into thecolonies, the number of cell passages maywell be as high as we have now calculated.

The model also predicts the distributionof mites over adult bees and brood. As soonas brood rearing starts the mites start invad-ing brood cells and a distribution pattern isgenerated as shown in figure 5, which is inagreement with empirically determined dis-tributions [14]. Figure 5 also shows the dailyrate of mite mortality inside the bee colony,including mortality of phoretic mites insidethe colony and mortality upon emergence.The mortality rate inside the colony is espe-cially relevant because it is virtually the onlyfeature of the mite’s population dynamicswhich is easily observable. Daily mortalityis, for almost the entire brood-rearing period,between 1 and 2 % of the mite population.

3.3. Sensitivity analysis

Variation of one parameter at a time will

help to assess influence of that parameteron the model predictions. Parameters thathave the largest impact on mite populationgrowth are those relevant for manipulationby the beekeeper or for obtaining mite-resis-tant bees through selection. In figure 6 weshow the effect on the peak mite population

in year 3 by altering: 1) the mean number offemale offspring in a cell with female off-spring in worker and drone cells (the effec-tive reproductive success of the mitebecomes less because of factors such asmale mortality and infertility); 2) the quan-tity of drone and worker brood cells pre-sent in the colony; and 3) phoretic mite mor-tality during the summer. The values usedare shown in table II.

Reproductive success has a large impacton population growth, and honey bee traitsthat influence the reproductive success maybe relevant for selection of mite-resistantbees. Also, increased phoretic mortality, e.g.due to grooming behaviour of the bees, hasa large impact on population growth. Thequantity of drone brood present in thecolony, relatively easily manipulated by thebeekeeper, also has a strong effect on pop-ulation growth. Decreasing the amount ofworker brood, which gives the same resultsas decreasing the attractiveness of theworker brood, has the counterintuitive effectof increasing somewhat mite population lev-els owing to the higher proportion of miteswhich invade drone brood.

3.4. Responses to treatment regimes

The following three types of V. jacob-soni control methods are used.

1) Biotechnical methods, such as droneculling and queen trapping; these methodsare generally labour-intensive, but carry norisk of contamination of stores and may beused during the summer period.

2) Methods based on (environmentallysafe) organic acids, such as oxalic, lactic orformic acids, or on essential oils; again thesecan be time-consuming to use, but can beeffective.

3) Methods based on synthetic acaricidessuch as Bayvarol (e.g. flumethrin) or Apis-tan (e.g. fluvalinate), which are often highlyeffective, but contaminate bee products andeventually reduce acaricide susceptibility

of the mites through continued use. Syn-thetic acaricides should only be applied atthe end of the season.

Here, we have modelled the effect ofusing drone culling in a systematic way, andof using an acid treatment in both summerand winter.

3.4.1. Drone culling

This control method involves the intro-duction of drone combs and their removalafter sealing during the brood-rearing sea-son. The preference of V. jacobsoni fordrone rather than worker brood (see e.g. [3])means that a large proportion of mites insidebrood cells will be in drone cells, and

removal of relatively small amounts of dronebrood cells therefore provides a way ofdestroying mites.We have modelled drone brood removal

by supposing that a drone comb is intro-duced into the brood nest, that the bees raise1 500 drone larvae in this comb, that thebrood cells are invaded by phoretic mitesas described and are capped evenly dis-tributed over a 1-week period, and that thecomb is removed after sealing together withthe trapped mites. The model predicts a con-siderable effect on the mite population (fig-ure 7). A repeated insertion and removal ofdrone brood, once on 1 June and once on 1

July, reduces the peak mite population inyear 3 from about 16 000 to about 1 750 if

there is no reinvasion. This is in agreementwith field observations from removing dronebrood [28].

3.4.2. Acid treatments

Lactic and oxalic acids have been foundto kill over 80 % of mites on adults (not incells) at the time of treatment [17, 20]. Theyare most useful when applied outside thebrood-rearing period, in late autumn or win-ter, but are also useful at other times of the

year as a ’knock-down’ treatment.

We have modelled the effect of acid treat-ment by assuming that a predetermined pro-portion of mites present on bees are killed atthe time of the application. We have com-pared the effects of a single treatment inautumn after the end of the brood-rearingperiod, with two treatments in summer. Asingle annual treatment on 27 October witha mortality rate of 90 % results in a popula-tion level which is very low and stable pro-vided that there is no reinvasion of mites

(figure 8). Under central European broodconditions, this is not sufficient because ofthe longer breeding period of the mite (datanot shown). However, a mortality rate ofonly 70 % in this late treatment results in apopulation level which increases over a fewyears to levels which are likely to be fatal tothe colony. The use of two treatments insummer is not an acceptable alternative:

even with a mortality rate of 90 %, the mitebuild-up is considerable over several years(figure 8). With brood produced outside ofthe relatively short brood-rearing period nor-mal for Scotland [1], the requirement oftreatment efficacy will be even higher.

3.5. Colony reinvasion

The numbers of new mites reinvading thecolony can be considerable [29]. In figure 9we show how the mite population respondsto a reinvasion of five mites per day duringautumn, from 7 October to 16 November.As would be expected, the populationincreases faster than would be the case withno such reinvasion.

Despite reinvasion, a treatment with amortality rate of 90 % applied on 7 Octo-ber (i.e. before the start of the invasion)reduces the population to a level at whichthe annual increase is sustainable using Scot-tish brood data [1].

4. CONCLUSIONS

In conclusion, since brood cell invasionis crucial in the life cycle of V. jacobsoni, theintegration of knowledge on brood cell inva-sion into the population dynamics model ofFries [12] allows more realistic simulations

of mite populations. The model can be usedas a tool to evaluate control strategies, toevaluate the impact of honey bee traits thatmay lead to resistant bees, and to gaininsight into the mite-bee relationship. Itshould be emphasised that for a realisticsimulation of the V. jacobsoni populationdynamics in a particular area using thismodel, it is vital to use brood data that isrelevant to the situation for which predic-tions are to be made. The model is sum-marised in Appendix 1 and will be madeavailable on disk for researchers and bee-

keepers using the ’Stella’-software packagethat runs on a personal computers.

ACKNOWLEDGEMENTS

SCR gratefully acknowledges financial assis-tance in the form of a travel grant from the C.B.Dennis British Beekeepers’ Research Trust. Wethank J. Beetsma, M. Beekman, W.J. Boot,J.C. van Lenteren and M.W. Sabelis for theirvaluable comments on drafts of the manuscript.

Résumé - Modélisation des populationsde Varroa jacobsoni. L’acarien Varroajacobsoni Oudemans (Mesostigmata, Var-roidae) est un parasite majeur de l’abeillemellifère, Apis mellifera L. Afin de mieuxcomprendre la dynamique des populationsde V. jacobsoni et de permettre des simula-tions mathématiques, Fries et al. [12] ont

intégré les connaissances actuelles dans unmodèle de population qui permet à unepopulation initiale d’acariens de varier quo-tidiennement sur n’importe quelle période.Dans cet article, nous décrivons un déve-loppement de ce modèle qui intégre des don-nées plus récentes portant spécifiquementsur l’invasion des cellules de couvain parles acariens phorétiques. Le modèle popu-lationnel de l’acarien est mis en relationavec un modèle de la colonie d’abeilles.Cela permet une évaluation des évênementsplus réaliste lorsque la répartition des aca-riens sur les abeilles et le couvain est impor-tante. Nous avons utilisé le modèle pourévaluer, entre autres : i) l’influence dediverses conditions climatiques sur la popu-lation d’acariens, ii) les caractéristiques com-portementales des abeilles pouvant conduireà une résistance aux acariens, iii) le nombrerelatif de cellules de couvain, et iv) diversesstratégies de lutte. Le modèle prédit à la foisle nombre d’acariens vivants et d’acariensmorts tombant de la colonie tout au long del’année et peut donc être utilisé pour suivrela population d’acariens afin d’optimiser lesstratégies de lutte. Le modèle, qui fonctionnesur un ordinateur personnel, sera mis à dis-position des scientifiques et des apiculteurs.&copy; Inra/DIB/AGIB/Elsevier, Paris

Apis mellifera / Varroa jacobsoni /

dynamique des populations / modèle

Zusammenfassung - Ein Populations-modell für Varroa jacobsoni. Die parasiti-sche Milbe Varroa jacobsoni ist eine derwichtigsten Krankheiten der HonigbieneApis mellifera L. Um ein besseres Ver-ständnis der Populationsdynamik der Milbenzu erreichen und um mathematische Simu-lationen zu ermöglichen, fa&szlig;ten Fries et al.[12] den Wissenstand jener Zeit in einemPopulationsmodell zusammen. In diesemkonnte die Entwicklung der Milbenpopula-tion über einen Zeitraum von mehreren Jah-ren dargestellt werden. In dem vorliegen-den Artikel beschreiben wir eine Auswei-

tung dieses Modells, in die neuere Daten

besonders über das Eindringverhalten pho-retischer Milben in die Zellen einbezogenwurden. Das Modell der Milbenpopulationist hierbei mit einem Modell des Bienen-volkes verbunden (Abb. 3). Hierdurch wirdeine realistischere Abschätzung des Ein-flusses von Ereignissen ermöglicht, die beider Verteilung von Milben auf die Bienenund die Brut eine Rolle spielen. Das Modellsagt voraus, da&szlig; die Milbenpopulation durcheine längerere Brutperiode dramatischgesteigert wird (Abb. 2a-c), und da&szlig; eineverhältnismä&szlig;ig hohe Anzahl an Brutzel-len zu schnellerem Populationswachstumführt. Ebenso macht das Modell Aussagenzu der Anzahl lebender und toter Milben,die über das ganze Jahr hinweg aus demBienenvolk herunterfallen. Es kann daher

genutzt werden, um die Milbenpopulatio-nen zu überwachen. Dies kann hilfreich fürdie Optimierung von Bekämpfungsstrate-gien sein. Da die Bekämpfungsma&szlig;nahmenzumeist entweder nur die Milben auf denBienen oder nur die Milben in der Brut

erfassen, kann das Modell nützlich zurAbschätzung ihrer Wirksamkeit (Abb. 7-8)oder der günstigsten Anwendungszeitensein. Die Berechnungen weisen darauf hin,da&szlig; Änderungen in den Parametern desReproduktionserfolgs in den Brutzellen dieMilbenpopulation sehr stark beeinflussen(Abb. 6). Das Modell soll Forschern undBienenhaltern zugänglich gemacht werdenund ist auf einem Personal Computer lauf-fähig. &copy; Inra/DIB/AGIB/Elsevier, Paris

Apis mellifera / Varroa jacobsoni /

Populationsdynamik / Modell

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APPENDIX 1: Summary of Varroajacobsoni model

Consider female mites only, and let:

Pn be the number of phoretic mites on day n; In bethe number of phoretic mites which enter a broodcell on day n; En be the number of mites which

emerge from cells with the adult bee on day n(including the original mother mite); and Mn bethe number of phoretic mites which die on day n.

Then the equation governing the V. jacobsonimodel is

with

where Iw,n, Id,n are the number of mites enteringworker and drone brood cells, respectively, onday n, evaluated as:

and

Nw, Nd are the postcapping periods for workerand drone brood, respectively; rw,n is the pro-portion of phoretic mites entering cells on dayn which enter worker cells, evaluated thus:

rd,n is the proportion of phoretic mites enteringcells on day n which enter drone cells, evaluatedthus:

Rw, Rd are fixed parameters representing repro-ductive success, namely the number of livingmites, including the mother, which emerge froma worker and a drone brood cell, respectively; kis the (fixed) daily mortality rate; and where Wn(the number of worker cells available for entryby mites on day n), Dn (the number of dronecells available for entry by mites on day n), Bn(the mass in grams of adult bees in the colony onday n) are pre-determined functions of timewhich constitute the ’colony model’.