PASCUA YAQUI SUMMER PROGRAMmath.arizona.edu/~jwatkins/beepop14.pdfabsconding - departure from the...

PASCUA YAQUISUMMER PROGRAM

A Joint Endeavor ofPasqua Yaqui Tribe

Carl Hayden Bee Research CenterUniversity of Arizona, School of Mathematical Sciences

June 9 - July 11, 2014

BEEPOP

The Population Dynamics of theHoney Bee

in the Hive and in the Wild

I. Sizing up the Population

Reading:

Bee Hives

Materials:

• Map of New Pascua

• Frames with comb, brood, and honey

• Frames with foundation

Classroom Activities:

Census Methods

• Estimating the number of saguaros on a reserve

• Estimating the number of fish in a lake

• Estimating the population of a village

1

Bee Hives

Long ago, people from many parts of theworld discovered that they could increasehoney production by creating a special envi-ronment for bee hives. Today, apiculture, thescience of beekeeping, is a comprehensive sci-entific and technological enterprise. Beekeep-ers study topics from the anatomy and physi-ology of bees to their evolution, genetics, andecology. The best beekeepers are those whocan combine scientific knowledge with practi-cal skills to manage their hives and to processand market honey.

We know, from pictures on the of the Egyp-tian sun temple of Nyuserre Ini from the 5thDynasty, dated earlier than 2422 BC, thatbee have been kept in artificial hives for morethat 4500 years. The pictures on these wallsdepict workers blowing smoke into hives asthey remove honeycombs.

The first human-made hives were quitevariable in size and shape. They were madeby these ancient beekeepers from materialthat was readily available. For example, earlyMediterranean and Egyptian hives were oftenpipes made of sun-baked mud, hives in Africawere pipes made from tree bark, and the ear-liest northern European hives were verticallystanding hollowed out logs. In Western Eu-rope, the hives were constructed from basket-work plastered with mud and cow dung. The

Figure 1: Medieval beekeeper working withbees in a skep, conical basket for housing sees

3

4 Bee Hives

material was made into a long tube severalinches in diameter. This tube was laid uponitself into a spiral to form a cylinder with acap. The final construction was then boundtogether using split bramble stems. Thesehives held about five gallons of honey. Whenthe time to harvest arrived, honey was col-lected by first killing the colony of bees insidethe hive and then removing all of the honeycomb.

Beginning in the seventeenth century, bee-keepers began to experiment with hives madefrom wooden boxes. With this strategy,boxes could be stacked or placed side by sideto extend the size of the original hive. Divid-ing plates could be placed between the boxesto make addition and removal of boxes easyto do while still having control of the activ-ities of the bees inside the hive. With theseinnovations, honey could be removed withoutdestroying the colony.

Since the middle of the eighteenth century,bee hives have had movable parts and spacesfor bees to move around the hive. The inte-rior design of the beehive is a series of ver-tically hanging rectangular frames. Theseframes look like a shallow dresser drawers.In building a frame, a piece is cut out fromthe base to allow the bees to move around.The remaining base is then coated with a waxfoundation. The coating has an imprint of thehexagonal design of the honey comb to en-courage the bees to build their comb so thatthe frames can be easily removed.

Figure 2: Elements of a modern beehive

Quadrat Method

The quadrat method is frequently used to count plantpopulations over a large region.

For this method:

• Drop some split peas on a checkerboard.

• Choose 4 squares at random using the TI-82 programSELECT.

• Count the split peas in those 4 squares.

• Divide your answer by 4 to find the average numberof split peas per square.

• Multiply your answer by 64 to estimate the totalnumber of split peas.

How can this be used to estimate the total population?

To take an example, say you count 15 split peas in thefour squares.

What does that tell you?

• The total number of peas on 4 squares is 15.

• The average number of peas per square is about 15/4.

• Thus the total number of seeds is about 15/4 × 64= 240.

5

If you start with a map of a region:

• Place a clear rectangular grid over the map.

• Number the rectangles in the grid: 1, 2, · · · , N .

• Decide how many rectangles you plan to use. Callthis number C.

• Choose C rectangles at random using the TI-82 pro-gram SELECT.

• Count the population in the selected rectangles.

• Multiply this count by the total number of rectan-gles.

• Divide by the number of chosen rectangles.

TI-82 Program SELECT

This program uses the TI-82’s random number gener-ator to select a simple random sample from the number1, 2, · · · , N . First give the total N , then continue punch-ing the ENTER key to obtain the desired sample size.

PROGRAM:SELECT

:Disp “TOTAL”

:Input N

:Lbl 1

:iPart(rand*N)+1 → R

:Disp R

:Pause

:Goto 1

6

Capture-Tag-Recapture

Capture-tag-recapture is a popular method for esti-mating a population of animals like birds, fish, or largemammals that make long range movements.

This census method works well for animals that thor-oughly explore a habitat. The time between the captureand the recapture should be long enough so that the in-habitants of the population have had time to explore thehabitat. However, this interval of time should not beso long that the inhabitants have a significant chance ofdying or that a sizable new population has been born.

For this activity:

• Take a large number of Pepperidge Farm cheddargoldfish and place them in a resealable plastic bag.

• Draw out a handful and count them. This is thenumber in the first capture.

• Place in the bag a number of Pepperidge Farm origi-nal fish equal to the number of cheddar fish captured.These are the tagged fish.

• Thoroughly mix the fish so that the tagged fish areevenly dispersed in the bag.

• Draw a second handful of fish and determined thenumber of fish that each tagged fish represents.

• Multiply this number by the total number of taggedfish to obtain an estimate of the total.

• Count the fish to see how close the estimate is to theactual total.

7

• Eat the fish.

How can this be used to estimate the total population?

To take an example, say that you remove 27 cheddarfish and you tag them. In the second capture, you remove42 fish, with 6 of them tagged.

What does that tell you?

• The number of tagged fish is 27.

• Each tagged fish represents about 42/6 = 7 fish inthe entire population.

• Thus the total number of fish is about 27 × 7 = 189.

8

II. Practical Knowledge of theEuropean and Africanized HoneyBeesReading:

Africanized Honey Bees in Arizona

Materials:

• Glossary


Slide show and video presentation on the Africanizedhoney bee.

For more:

http://gears.tucson.ars.ag.gov/

9

Glossary

• abdomen - the rear section of a insect body containing the digestiveand reproductive organs.

• absconding - departure from the hive by the entire colony.

• anther - the part of the flower that contains pollen.

• apiarist - a beekeeper.

• apiary - a collection of managed bee colonies.

• apiculture - the science of beekeeping.

• Apis mellifera - the genus and species name for the honey bee. Thisname, given by Linnaeus in 1758, is Latin for honey bearer.

• beeswax - a substance secreted on the underside of the abdomen ofworker bees used to build comb.

• brood - developing bees (eggs, larvae, pupae) that have not yetemerged from their cells.

• cap - a covering that closes a cell containing a pupa or honey.

• cell - a single hexagonal unit of comb.

• colony - a community of bee having a single queen, thousands ofworker bees, and for many parts of the year, drones.

• colony division - exiting of a part of a bee colony to form a newhive.

11

12 Southwest RIMS Class Notes

• comb - the hexagonal structure used to store honey and raise brood.

• dance - a series of movements made by a forager bee or a scout beeto communicate the location and type of resource.

• drone - a male bee.

• entomology - the science of insects.

• foraging - the act of gathering pollen and nectar from flowers byworker bees.

• forager bee - a foraging worker bee.

• feral - domesticated animals that have escaped captivity.

• foundation - wax coating in the base of a frame. The coating hasan imprint of the hexagonal design of the comb to encourage bees tobuild their comb in line with the design.

• frame - wooden rectangle with a sheet of foundation to support acomb.

• head - the front section of a insect body containing antennae and othersensory apparatus.

• hive - the home for a bee colony.

• honey - sweet viscous material produced from nectar.

• house bee - a young worker bee whose activities are confined to thehive.

• larva (plural, larvae) - middle stage of a developing bee; unsealedbrood.

Glossary 13

• mating flight - an excursion taken by drones in order to mate witha queen.

• nectar - a sweet liquid secreted in flowers and on leaves of plants.

• nuptial flights - a series of mating excursions made by a young queen.

• orientation flights - flights taken by house bees in preparation forbecoming foragers.

• pollen - dust like grains on the tops of anthers containing the flower’ssperm

• pollination - the transfer of pollen from an anther to a stigma of aflower.

• pupa (plural, pupae) - final stage of a developing bee; sealed brood.

• queen - a female bee that lays all the eggs in the colony

• queen cell - a special vertically hanging cell used to place an egg thatwill become a queen.

• royal jelly - food for queen larvae.

• Schwirrlauf - a whir dance made by scout bees to announce thetime for colony division.

• scout bees - bees that search and select a new hive site.

• skep - a straw hive without movable frames.

• spermatheca - a pouch-like stucture on a queen’s abdomen forstoring sperm.


• stinger - 1/8” long hollow tube with a barbed tip attached to a pocketat the end of the abdomen used to eject venom.

• super - section of a mangaed hive used for honey storage, typicallyabove the brood chamber.

• supercedure - the taking over of an old queen by a daughter queen.

• swarm - a colony of bees found outside the hive, may be abscondingor in colony division.

• thorax - the middle section of an insect body to which the wings andlegs are attached.

• worker bee - an unmated female bee.

AFRICANIZED HONEY BEES

IN ARIZONA

15

Africanized Honey Bees in Arizona∗

Bee experts believe that the African-ized honey bee is here to stay. Studiesshow that as the regular honey bees andthe Africanized bees interbreed, theAfricanized strain appears to be domi-nant. So, the movement of Africanizedhoney bees into Arizona constitutes apermanent change in our state’s envi-ronment. As a result, all of Arizona’scitizens and visitors will need to perma-nently change their view of honey bees.

Africanized honey bees (AHB) area more temperamental relative of thecommon garden honey bee, which isknown as the European honey bee(EHB). Honey bees, whether they areEuropean or African, only sting defen-sively. They do not go out of their wayto sting. But some AHB colonies de-fend their colonies more intensively andwith less provocation than other bees.

Scientists at the USDA Carl HaydenBee Research Center in Tucson antici-pate that the AHB will continue to col-onize the lower regions of Arizona andthe United States. So, we are now deal-

∗from the Carl Hayden Bee ResearchCenter

ing with a different sort of honey beethat will remain different. And just aswe Arizonan’s have learned to walk inthe desert - ever mindful of jumpingcholla or rattlesnakes or scorpions - wemust now display that kind of cautionwith respect to bees.

This article will introduce you tothe AHB and discuss the following fivemain topics concerning honey bees:

1. Why honey bees are important.

2. How the Africanized honey bee isdifferent from any other domestichoney bee.

3. What safety precautions must nowbe routinely followed to avoid astinging incident.

4. How to bee proof your property.

5. What you must do if you inadver-tently agitate and/or encounter anangry AHB hive or swarm.

Education plays a critical role in re-ducing the threat of the AHB to thehealth and safety of the public. Peo-ple can coexist with Africanized honey

17


bees by learning about the bee and itshabits, taking a few precautions, andby supporting managed beekeeping ef-forts.

The University of Arizona Entomol-ogy Department and Cooperative Ex-tension, in cooperation with Carl Hay-den Bee Research Center, have under-taken coordination of Arizona’s AHBeducational activities as a project of theIntegrated Pest Management Program.The statewide efforts focus exclusivelyon the development and disseminationof AHB educational materials to educa-tors and volunteer presenters through-out Arizona.

The Arizona AHB Education Projecthas developed an AHB Education Kitconsisting of a comprehensive trainingmanual, a script and 37-slide presen-tation, sample bees in resin, a plas-tic honeycomb, eight 4-color laminateddisplay posters, and a safety video en-titled “What Arizonans Need to KnowAbout Africanized Honey Bees”. Allitems are contained in a silk-screenedheavy-duty tote bag. Cost of the kit is$80, which includes shipping and han-dling.

Africanized Honey Bee in Arizona 19

1. Why Honey Bees areImportant

To understand the threat of African-ized honey bees, it is necessary to knowsomething in general about honey beesand their behavior.

Honey bees are important beneficialinsects and we would be in big trouble ifthey were all suddenly destroyed. Un-less a honey bee colony is in a locationthat is close to people, pets or farm an-imals, it should be left alone.

Most people appreciate the mainproduct of the hive - honey. Honey pro-duction, however, isn’t the only use forhoney bees. They are also very impor-tant to Arizona agriculture, a sophisti-cated business that impacts the state’seconomy by about $6.3 billion annually.

In fact, one-third of our daily dietcomes from crops pollinated by honeybees. Without the pollen that honeybees transport, many plants can’t pro-duce fruits, vegetables and seeds. Imag-ine walking into your neighborhood su-permarket and finding a third of thefood currently available not on theshelves!


2. How the AHB Differs fromthe EHB

The behavior - not the appearance -of the AHB is different from the EHBin several major ways:

• The AHB swarms much more fre-quently than other honey bees. Acolony is a group of bees with comband brood. The colony may ei-ther be managed (white hive boxesmaintained by professional bee-keepers) or wild (feral).

A group of bees that are in the pro-cess of leaving their parent colonyand starting a nest in a new lo-cation is called a “swarm”. Usu-ally a new queen is reared to staywith the parent colony and theold queen flies off with the swarm.Scout bees often locate potentialnest sites prior to swarming, butthe swarm may spend a day ortwo clustered in impressive, hang-ing clumps on branches or in othertemporary locations until the beessettle on a new nesting site. If theycan’t find a suitable location, thebees may fly several miles and clus-ter again.

Typically an EHB hive will swarmonce every 12 months. However,the AHB may swarm as often asevery six weeks and can producea couple of separate swarms eachtime. This is important for you to

know, because if the AHB swarmsmore often, the likelihood of yourencountering an AHB swarm in-creases significantly.

Contrary to myths, Africanizedhoney bees do not fly out in an-gry swarms randomly to attack un-lucky victims. However, the AHBcan become highly defensive in or-der to protect their hive, or home.Again, it is now better to con-sistently exercise caution with re-spect to all bee activity. So keepyour distance from any swarm ofbees.

• The AHB is far less selectiveabout what it calls home andwill occupy a much smaller spacethan the EHB. Known AHB nest-ing locations include water meterboxes, metal utility poles, cementblocks, junk piles, and house eaves.Other potential nesting sites in-clude overturned flower pots, oldtires, mobile home skirts, andabandoned structures. Holes inthe ground and tree limbs, mailboxes, even an empty soda pop canor bottle, can could be viewed as“home” to the AHB.

• The Africanized honey bee is ex-tremely protective of their hive andbrood. The AHB’s definition oftheir “home turf” is also muchlarger than the European honey


bee. So, try to allow ample physi-cal distance between the hive. Atleast 100 feet, or the width of afour-lane highway, is a good dis-tance. The best advice is that ifyou see a bee hive, start movingaway immediately.


3. How To Avoid a StingingIncident

Things to remember:

• Stay away from honey bee colonies.There are estimated to be about250,000 wild honey bee coloniesin Arizona. Because honey beesnest in such a wide variety of lo-cations, be alert for groups of fly-ing bees entering or leaving anentrance or opening. Listen forbuzzing sounds. Be especially alertwhen climbing, because honey beesoften nest under rocks or withincrevices within rocks. Don’t putyour hands where you can’t seethem.

• If you find a colony of bees, leavethem alone and keep others away.Do not shoot, throw rocks at, tryto burn or otherwise disturb thebees. If the colony is near a trailor near areas frequently used byhumans, notify your local officeof the Parks Department, ForestService, Game and Fish Depart-ment, even if the bees appear tobe docile. Honey bee colonies varyin behavior over time, especiallywith changes in age and season.Small colonies are less likely to bedefensive than large colonies, soyou may pass the same colony forweeks, and then one day provokethem unexpectedly.

• Wear appropriate clothing. Whenhiking in the wilderness, wearlight-colored clothing, includingsocks. Avoid wearing leatherclothing. When they defend theirnests, Honey bees target objectsthat resemble their natural preda-tors (such as bears and skunks), sothey tend to go after dark, leath-ery or furry objects. Keep in mindthat bees see the color red as black,so fluorescent orange is a betterclothing choice when hunting.

• Avoid wearing scents of any sortwhen hiking or working outside.Africanized honey bees communi-cate to one another using scentsand tend to be quite sensitive orodors. Avoid strongly scentedshampoo, soaps, perfumes, heavilyscented gum, etc. If riding, avoidusing fly control products on yourhorse with a “lemony” or citrusodor. Such scents are also knownto provoke or attract honey bees.

• Be particularly careful when us-ing any machinery that producessound vibrations or loud noises.Bees are alarmed by the vibra-tion and/or loud noises producedby equipment such as chain saws,weed eaters, lawn mowers, tractorsor electric generators. Honey beesmay also be disturbed by strongsmells, such as the odor of freshly


cut grass. Again, check your envi-ronment before you begin operat-ing noisy equipment.

• Pet safety. When hiking it is bestto keep your dog on a leash orunder close control. A large an-imal bounding through the brushis likely to disturb a colony andbe attacked. When the animal re-turns to its master, it will bringthe attacking bees with it. Athome, be careful not to tie orpen animals near honey bee hives.Even the mild-mannered Euro-pean honey bee has been knownto attack animals tied near theirhives. The animals receive numer-ous stings because they can’t es-cape the bees. If your animals orpets are being stung, try to releasethem without endangering your-self.


4. How to Bee Proof YourProperty

The best way to prevent bees fromestablishing a colony on your propertyis to not provide them with an ideal en-vironment for survival. Honey bees re-quire three things in order to survive:food, water and shelter.

Remember, Africanized honey beesalso nest in a wide variety of locationsand may enter openings as small as3/16-inch in diameter (about the sizeof a pencil eraser) as long as there is asuitable-sized cavity behind the open-ing for a nest.

• Eliminate shelter. To preventhoney bees from settling in yourhouse or yard, you will need tobe vigilant in preventing potentialnesting sites.

• Caulk cracks in walls, in the foun-dation and in the roof.

• Fill or cover all holes 1/8-inch indiameter or larger in trees, struc-tures and/or block walls.

• Check where the chimney meetsthe house for separation, and makesure chimneys are covered prop-erly.

• Put small-mesh screen (such aswindow screen) over attic vents, ir-rigation valve boxes and water me-ter box key holes.

• Remove any trash or debris thatmight serve as a shelter for honeybees.

• Fill or cover animal burrows in theground.

• Make sure window and sun screensare tight fitting.

• Keep shed doors tightly closed andin good repair and exercise cautionwhen entering buildings that arenot used frequently.

• Inspect your home and yard reg-ularly for signs of bee colonies. Asingle bee or just a few bees in youryard does not necessarily mean youhave an established colony on yourproperty because bees will fly somedistance in search of food and wa-ter. Although honey bees use nec-tar and pollen from flowers as food,removing flowers as a source offood is generally not an effectivebee deterrent.

• Look for large numbers of beespassing into and out of or hover-ing in front of an opening. Listenfor the hum of active insects. Looklow for colonies in or at groundlevel, and also high for colonies un-der eaves or in attics.

• If you find a colony on your prop-erty, consult a bee expert. If youdo find an established bee colony


in your neighborhood, don’t panic.On the other hand, don’t ignorethem either. Small colonies thathave recently swarmed may bedocile at first, but tend to becomemore defensive with age. Havecolonies located around the houseremoved as soon as possible.

• Keep everyone away from thecolony. Look in the Yellow Pagesunder “bee removal” or “pest con-trol” for the names of beekeepersor pest control operators in yourarea who are qualified to removethe colony. Do not try to removecolonies yourself!


5. What To Do If Attacked byAfricanized Honey Bees

Remember these important steps:

• RUN away quickly. Do not stop tohelp others. However, small chil-dren and the disabled may needsome assistance.

• As you are running, pull your shirtup over your head to protect yourface, but make sure it does not slowyour progress. This will help keepthe bees from targeting the sensi-tive areas around your head andeyes.

• Continue to RUN. Do not stoprunning until you reach shelter,such as a vehicle or building. Donot jump into water! The bees willwait for you to come up for air.If you are trapped for some rea-son, cover up with blankets, sleep-ing bags, clothes, or whatever elseis immediately available.

• Do not swat at the bees or flailyour arms. Bees are attracted tomovement and crushed bees emit asmell that will attract more bees.

• Once you have reached shelter orhave outrun the bees, remove allstingers. When a honey beesstings, it leaves its stinger in theskin. This kills the honey bee so it

can’t sting again, but it also meansthat venom continues to enter intothe wound for a short time.

• Do not pull stingers out withtweezers or your fingers. This willonly squeeze more venom into thewound. Instead, scrape the stingerout sideways using your fingernail,the edge of a credit card, a dullknife blade or other straight-edgedobject.

• If you see someone being attackedby bees, encourage them to runaway or seek shelter. Do notattempt to rescue them yourself.Call 911 to report a serious sting-ing attack. The emergency re-sponse personnel in your area haveprobably been trained to handlebee attacks.

• If you have been stung more than15 times, or are feeling ill, or if youhave any reason to believe you maybe allergic to bee stings, seek med-ical attention immediately. Theaverage person can safely toler-ate 10 stings per pound of bodyweight. This means that although500 stings can kill a child, the av-erage adult could withstand morethan 1100 stings.

III. Hands on Models of Pop-ulation Dynamics

Reading:

Introduction to the Honey Bee

Materials:

• Pennies

• Styrofoam cup with lids

• Lots of beads in two different colors

• Programable calculator


• Coin models of exponential growth and decay.

• Bead models of exponential and logistic growth.

27

Introduction to the Honey Bee

The Apis mellifera, commonly referred toas honey bees, are perhaps the most intenselystudied of all insects. Their economic impor-tance to the agricultural industry has driventhe need for scientific research. As a result,a wealth of valuable and interesting informa-tion has accumulated. This mass of data hasshown honey bees to be highly social crea-tures with complicated behaviors and intrigu-ing population dynamics.

The economic importance of honey bees isdue to the products they produce as well asservices they perform. Although honey beesproduce honey, they serve a more importantrole as pollinators. What makes the honeybee so special is that unlike many insects, thehoney bee will seek out pollen and not nec-tar. Honey bees commonly pollinate agricul-tural crops such as apples, cherries, melons,and almonds. In fact, many farmers hire bee-keepers to raise and maintain bee colonies ontheir farms entirely for this purpose. Honeybees also produce wax used for polishes andcandles. The importance of honey bees is nota new discovery. Pictographs depicting beesand their hives have been found painted onthe walls of caves believed to be many thou-sands of years old.

Although honey and bees play an enor-mous role in the United States agricultural

Figure 1: Rock painting depicting honeygathering. Discovered in the Cuevas de laArana near Bicorp in Valencia, Spain, fromE. Hernendez-Pacheco, Museo nacional deciences naturales, Madrid

29


industry, they are not native to North Amer-ica. Interestingly, they were brought here byearly European colonizers. The indigenoushabitat of the Apis mellifera ranges from thetip of Southern Africa to Southern Scandi-navia, and from continental Europe to West-ern Asia. Thus the honey bee is a highlyadaptable insect able to adjust to a wide va-riety of climes and geographic regions.

The desert regions of the southwesternUnited States and northern Mexico are hometo the richest variety of bees in all the world.According to the Carl Hayden Bee ResearchCenter, located in Tucson, Arizona there areapproximately 1000 to 1200 species of beeswithin a one hundred mile radius of Tucson.Yet, none of these are native honey bees.

Approximately 25,000 species of bees havebeen identified, with almost 40,000 still yet tobe catalogued. That is, entomologists knowthat they are out there, but have yet to placethem in a specific genus. However, out of this25,000, only 8 to 10 species are consideredhoney bees. Yet, this number is growing asmore species are identified.

Honey bees are not only classified by genusand species, but by strain. The strain denotesthe bees’ place of origin. The most com-mon strains of honey bees currently foundin the United States are the Apis mellif-era ligustica, the Italian bees, and the Apismellifera carnica, the Carniolan bees. How-ever, many common lines of honey bees havebeen allowed to interbreed. The goal of suchcrossings has been to develop hybrid beeswith specific morphological and behavioraltraits that will enhance their honey produc-ing and pollinating traits. The most famous

attempt at creating such a hybridized linewas the crossing of the European lines andthe African lines.

The goal of crossing the European and theAfrican lines was to mate the docile but highhoney yield European bees with their ag-gressive but low yield African counterparts.In 1956, Brazilian researchers hoped that aharder working bee that made more honeywould result. However, the experiment didnot succeed. What the researchers foundwas that the aggressive traits dominated andessentially masked the European character-istics. The experiments have become fa-mous because a worker in the apiary wherethe hybridized lines were being kept acciden-tally removed protective screens that keptthe queens in their hives. As a result, atleast 26 swarms of the Africanized bees es-caped. The descendants have been mov-ing northward ever since. Today. 40 yearslater, Africanized honey bees are found in thesouthwestern United States and are a cause

Introduction to the Honey Bee 31

for concern due to their aggressive nature,and their ability to take over and replaceestablished colonies of productive Europeanlines.

A major product of the general scientificresearch into the ecology of Apis Mellifera is agreater understanding of the honey bee’s “so-cial structure” and population dynamics. Inexamining the population of the colony, scien-tists have uncovered the existence of a highlyordered caste system. The queen reigns atthe top of the caste, with the male dronesand female workers below.

The queen’s primary duties are to popu-late the colony by mating with drones (malehoney bees), and direct the activities of theworkers. The queen mates during the earlysummer months, generally within the firstweek after having emerged from her cham-ber. She takes what is referred to as “nup-tial flights” where she may mate with severaldrones per day over the course of several days.She collects their sperm inside a large bodycavity called the “spermatheca.” After thisflight has been completed the queen has ac-cumulated enough sperm to sustain her entirecareer as the egg-layer of the hive. However,if the queen’s egg-laying capacity is deficient,or if she is unresponsive to the needs of thecolony in some way, she may be attacked bythe workers and replaced.

The workers are the second caste in thecolony and perform many crucial tasks withinthe hive. Most importantly, they are re-sponsible for tending to the queen by bring-ing to her a special food, and grooming her.Additionally, workers build comb, tend thebrood, seal and cap comb cells containing ei-

ther honey or bees, remove debris from thehive, store pollen, ripen and store honey, andguard the hive.

The male drones are the third caste in thecolony. They are responsible for mating withqueens from other colonies and perform vir-tually no other useful task within the hive.After mating, drones die from the ruptur-ing of their abdomens and genital apparatus.However, many drones die before they get achance to mate with the queen because theymay be killed or thrown out of the nest byworker bees when food resources are low.

Examining the behavior and ecology ofbees is a worthy task. Bees are importanteconomically as well as ecologically. By un-derstanding their behavior, morphology, andpopulation dynamics, scientists, beekeepers,and farmers may work together to developstrategies that enhance the productivity ofour agricultural industries.

Coin Models of ExponentialDecay and Growth

Exponential Decay Gather some coins with eachgroup choosing between 100 and 200 pennies. Enter thatnumber in generation 0. Flip each coin, discarding theheads. Enter the number of coins remaining in genera-tion 1. Continue this process until all of your coins aregone.

Exponential Growth. Start with a single coin andflip it.

• If the coin lands tails, add no more coins and enterthe number 1 for the number of coins in generation1.

• If the coin lands heads, add 2 more coins and enterthe number 3 for the number of coins in generation1.

• For generation 2, take all of the coins in generation1, flip them and add 2 coins for every head.

• Continue this process until you use all of your coins.

So if we have three coins in generation 1 and we flip“tails, heads, heads”, then we add 4 coins to the pileresulting in 7 coins in generation 2. Letting “H” denoteheads and “T” denote tails, we can record each historyof coin tosses in a “family tree”.

The potential for exponential growth of population wasrecognized by Thomas Malthus in the early nineteenth

33

century. Malthus was concerned with societies ability toincrease the production of food at an exponential rate tokeep pace with the needs of an increasing population.

The two coin experiments in this exercise are exam-ples of simple branching processes. This name was chosenbecause of the “branches” on a family tree. I. J. Bien-ayme created this mathematical model in the 1840’s tostudy the extinction of family lines. A TI-82 programBIENAYME will allow you to simulate these branchingprocesses.

generation 0

generation 1

generation 2

c c

c c

c c

c c

c cc c c c c c

T H

H T H H

34

TI-82 Program BIENAYME

This program allows you to simulate simple branchingprocesses.

• Give the fraction of the population that has 0, 1, 2, · · ·children. If the fractions do not add up to one, theprogram will report a IMPROPER DISTRIBUTION.

• After the distribution of family sizes has been en-tered the program will give you

– the MALTHUSIAN PARAMETER, that is the meannumber of offsprings for each individual, and

– the STANDARD DEVIATION, a measure of thespread of the offspring distribution.

• Enter the INITIAL POPULATION, the number of in-dividuals at the beginning of your simulation.

• Enter the FINAL GENERATION, the number of gen-erations you want to follow your simulated popula-tion.

• Use the ENTER key to see the population for succes-sive generations.

• You can see the table of populations by choosing theSTAT key, then the EDIT key and looking at list L1for the generation and list L2 for the population total.

35

PROGRAM:BIENAYME

:ClrList L1,L2,L3,L4,L5,L6

:Disp “ENTER OFFSPRING”

:Disp “DISTRIBUTION”

:0→I

:0→Q

:Lbl 1

:Disp “PROBABILITY”

:Disp I

:Disp “OFFSPRINGS”

:Input A

:A→L3(I+1)

:If I=0

:Then

:A→L6(I)

:Else

:A+L6(I)→L6(I+1)

:End

:I*L3(I+1)→L4(I+1)

:I2*L3(I+1)→L5(I+1)

:L6(I+1)→Q

:I+1→I

:If Q<0.9999999

:Then

:Goto 1

:End

:If Q>1.0000001

:Then

:Disp “IMPROPER”

:Disp “DISTRIBUTION”

36

:Goto 2

:End

:sum(L4)→M

:Disp “MALTHUSIAN”

:Disp “PARAMETER”

:Disp M

:√(sum(L5)−M2)→S

:Disp “STANDARD”

:Disp “DEVIATION”

:Disp S

:Pause

:Disp “INITIAL”

:Disp “POPULATION”

:Input P

:0→G

:Disp “FINAL GENERATION”

:Input F

:For(G,1,F,1)

:0→N

:For(J,1,P,1)

:rand→R

:For(K,1,I,1)

:If L6(K)≤R:Then

:N+1→N

:End

:End

:End

:N→P

:G→L1(G)

:P→L2(G)

37

:Disp “GENERATION”

:Disp G


:Disp “TOTAL”

:Disp P

:Pause

:Lbl 2

:End

38

Exponential Growth

Exponential growth follows from to statement:

The population change between consecutives censusesis proportional to the population total of the current cen-sus.

Place 100 beads - varying amounts of yellow and black- in a styrofoam cup. Put a lid on the cup with a smallhole notched out of the edge to facilitate pouring 1 beadout of the cup. The yellow beads in the cup representedresources.

Here are the rules:

1. At census 0, the population total is 1.

2. Each member of the population looks for resourcesthat will allow it to add one to the population.

3. Pour out a bead for each member of the total pop-ulation. If it is yellow, add one to the populationchange for the next census. If the bead is black, donot add to the population change.

4. Return the bead to the cup.

5. Record the population change.

6. Compute the population total for the next censusby adding the population change to the populationtotal.

39

TI-82 Program EXPONENT

This program will simulate the bead model for expo-nent growth. The program asks for an input of the num-ber of censuses and a probability - a number between 0and 1. Each member of the current census adds one tothe current population with the chosen probability, in-dependent of the other members. The census number isstored in L1, the population total is stored in L2 and thepopulation change is stored in L3.

PROGRAM:EXPONENT

:Disp “CENSUSES”

:Input C


:Input P

:0→L1(1)

:1→L2(1)

:For(I,1,C,1)

:0→J

:For(K,1,L2(I),1)

:If rand≤P:J+1→J

:End

:J→L3(I)

:I→L1(I+1)

:J+L2(I)→L2(I+1)

:End

40

Logistic Growth

Logistic growth follows from the statement:

The population change between consecutive censusesis proportional to both the current population total andto the difference between the population capacity andthe current population total.

Place 50 yellow beads in a styrofoam cup. Put a lidon the cup with a small hole notched out of the edge tofacilitate pouring 1 bead out of the cup at a time. Theyellow beads in the cup represent resources.

Here are the rules:

1. At census 0, the population total is 1.

2. Each member of the population looks for resourcesthat will allow it to add one to the population.

3. Pour out a bead for each member of the total pop-ulation. If it is yellow, add one to the populationchange for the next census. If the bead is black, donot add to the population change.

4. Replace the bead poured out of the cup with a blackbead.

5. Record the population change.

6. Compute the population total for the next censusby adding the population change to the populationtotal.

41

TI-82 Program LOGISTIC

This program will simulate the bead model for logisticgrowth. The program asks for an input of the populationcapacity of the environment. Each member of the currentcensus adds one to the current population with proba-bility proportional to the difference of the present popu-lation and the population capacity. The census numberis stored in L1, the population total is stored in L2 andthe population change is stored in L3.

PROGRAM:LOGISTIC

:Disp “CAPACITY”

:Input C

:1→I

:0→L1(1)

:1→L2(1)

:1→J

:While J<C

:For(K,1,L2(I),1)

:If rand≤(C−J)/C:J+1→J

:End

:J−L2(I)→L3(I)

:J→L2(I+1)

:I→L1(I+1)

:I+1→I

:End

42

IV. A Month in the Hive

Reading:

Life in the Hive

Materials:

• Programable calculator or

• Personal computer.


• Developing a flow chart of hive dynamics

queen daylength�Egg laying increases with daylength.

• Working with a day to day model of the populationdynamics.

• Finding the change in hive population over the courseof a selected month in a selected location.

43

Life in the Hive

The Matriarchy and Caste System

The social structure of a bee hive is thatof a matriarchal family headed by a queen.Soon after the queen emerges from her cell,she takes a series of nuptial flights. Duringthese flights, the queen will mate with 7 to17 male drones. Thus, the bees in her hivewill have between 7 and 17 different fathers.This provides the genetic diversity necessaryso that all of the hive tasks can be acom-plished successfully.

It is estimated that the queen may receiveup to 5,000,000 individual sperm during thisshort period of mating, all of which is storedin a pouch-like structure on her abdomencalled the spermatheca. The queen accessesthese sperm throughout her life, fertilizingeggs at a rate determined by the needs of thehive.

The queen has a potential life span of threeyears and her only job is to lay the eggs nec-essary to establish and maintain the colonypopulation. A mature hive can have twenty-five to fifty thousand bees. Almost 95% ofthe queens offspring are what are referred toas worker bees, with the remaining 5% devel-oping into drones.

Figure 1: Queen surrounding by workers

The Workers

The worker bees, who make up the ma-jority of the population, are all females.They differ from the queen by being shorterand by having an undeveloped reproductivesystem. The worker-females take care ofthe queen and perform virtually all of thetasks necessary for the support of the hive.These worker-females have a short lifespanof approximately 30 days, and during thistime will go through different developmentalstages. Younger worker bees may be clas-

45


sified as housekeepers, or simply house beesand are responsible for the upkeep of the hive.After 21 days, these house bees become for-agers and take on the job of collecting thenectar, pollen, and water necessary to sustainthe hive.

The primary cause of death for a forag-ing bee is wing burnout. Their wing musclesoverheat after 500 miles of flight. When for-agers can no longer take care of themselves,they die.

The basis for the division of labor withinthe hive is the age of the worker. The workerbegins its life cleaning the storage cells of thehive. A worker then moves on to brood careand food storage, and ends its life as a for-ager. This strategy maximizes the averagelife of a worker. The young worker bees spendthe majority of their time inside the protectedenvironment of their hive. The older foragersmust deal with the hazards of predation, badweather, and wing burnout.

Figure 2: Worker, queen, and drone honeybee

Drones

The drones are the only males producedby the queen. Although few in number,they serve an important role as mates forthe queens of other hives. The lifespan ofthe male drone is very short, for after mat-ing their abdomens explode which results inrapid death. Drones only serve as mates forthe queen, and are not involved in feedingthe colony, or the upkeep of the hive. Thus,if resources are scarce, worker bees do notlike to keep the seemingly lazy males aroundand will often force them from the hive orkill them directly. Further, since the dronesspend a great deal of time outside of thehive they are more susceptible to predationor death. All this results in an average dronelifespan of less than 25 days.

The drones will often begin mating flightseight days after emerging from their cells.Within twelve days they may perform up tofive mating flights per day.

Population Dynamics of the Hive

The caste system functions as an inte-grated feedback system of interdependent el-ements. In creating a mathematical modelto simulate the population dynamics in thehive, we must be able to gather some specificinformation..

First, we want to be abel to estimate thenumber of eggs that may be laid by the queenon any given day. Through many observa-tional studies, scientists have determined howthe queen’s age, the outside temperature, and

Life in the Hive 47

the adult population size effect the number ofeggs laid. For example, scientists know thatthe maximum number of eggs that strongqueen may lay in a day is 3000. However,a more realistic value for an average queen is2000 eggs per day.

We also know that the bees make the tran-sition from brood to adult female bees after21 days and that females remain house beesfor 21 days. Drones are a little larger thanworker bees and take a little longer, 24 days,to mature to male adult bees

We still need to see how the environmentalconditions can be used to estimate how longbees can forage before perishing. In an envi-ronment rich in resources, bees make shorterforaging trips and so can live longer beforeusing their 500 mile allotment.

Let’s look at these individual factors morecarefully.

Temperature

Temperature plays a pivotal role in hivepopulation dynamics for two reasons. Thequeen responds to a temperature increase bylaying more eggs. Also, honey bees will for-age only when the average daily temperatureexceeds 54 degrees Fahrenheit.

Photoperiod

The photoperiod, length of day, plays animportant role.

In the spring time, queens begin laying eggsas soon as the daylength reaches 10 hours.

This happens even if the hive is covered withsnow. This egg laying strategy prepares thecolony for its intense foraging activities in thespring and early summer when a large num-ber of flowers are in bloom.

Beekeepers know that as the daylengthshortens, eggs laying slows down. This de-crease in egg laying is necessary with the ap-proach of winter. During the winter, withits low temperatures and little or no pollenor nectar, the hive cannot continue to growand meet its needs for food. A colony thatdoes not store enough food will not survivethe winter.

Other Weather Factors

Honey bees will not forage if the wind ve-locity is greater than 22 miles per hour or ifit is raining. High winds impare bees abil-ity to maneuver. As a consequence, the beesmay not be able to return to their hive fromforaging. Rain results in wet bees with thewater increasing their weight to the extentthat they can no longer support themselvesin flight.

Weather conditions affect the size andfunctioning of the hive resulting in differentpopulation dynamics for different geographicregions of the United States. For example,in the midwest, photoperiod may range from9.1 (December) to 15.25 (June) hours of lightper day, with the average daily temperaturesranging from 0 to 88 degrees Fahrenheit. Bycontrast, the photoperiod in the southwestmay range from 10 (December) to 14.5 (June)


hours of light per day, with the average tem-perature ranging from 32 to 100 degrees cel-sius.

Under the midwestern conditions, thecolony may be confined to the hive and willnot produce brood from autumn to mid-winter, and nor will they forage from lateOctober to late April. In contrast, coloniesin the Southwest may produce brood and for-age year round. From this information, it iseasy to predict what times of the year willpromote colony growth or decline.

Queen Maturation

Now that the research has given us a math-ematical expresion for the number of eggslaid, we have to find out how many of theseeggs will become worker bees and how manywill become drones. Bee biologists have longknown that a queen lays many more droneeggs when her spermatheca is nearly empty.This is logical - if the spemantheca is nearlyour of sperm, then the queen is near the endof life. The drones that she lays will pass onher genetic material.

Thus, entomologists monitored egg layingcarefully, watching for an increase in droneegg laying as the queen ages. With enoughstudy, the researchers could also write downa mathematical equation for the proportionof drones laid. For the first year of a queen’slife this proportion is nearly zero.

How can we use these observations to makeprediction about the dynamics of a bee hive?

“Free-Body Modeling”

The first task in creating a populationmodel for honey bees is to take all the inter-dependent factors and determine their rela-tionship to one another. This can be accom-plished by creating a concept-map, or flowchart, visually depicting how these factors re-late to one another, this is called free-bodymodeling. To do this, one writes out the vari-ables that govern the functioning of the sys-tem, then draw lines between them to showhow they influence each other. Further, onemay then write on the lines just how it is thatthe factors affect each other. This is an ex-tremely important tool that allows the mod-eler a graphic visualization of how the modelwill work.

After the flow chart is drawn, equationscan be created to describe the behavior ofeach variable. Finally, these equations can beentered into a computer program or spreadsheet to see if they accurately represent whatis seen in the empirical data.

A Month in the Hive

The information thus far presented can beused to predict how the population of a beecolony may change each day over the courseof a month.

• The first consideration will be estimatingthe initial population. We cannot deter-mine the population at the end of themonth without knowing the populationat the beginning.

Life in the Hive 49

• The next consideration is to determinethe ratio of workers to drones. This de-pends primarily on the age of the queenand her egg laying capacity, as well as se-men availability. Recall, during the firstyear of a queen’s life, virtually no droneeggs are laid.

• Thirdly, one must determine the season.In the winter months the population willbe low, whereas in mid-summer the pop-ulation may be peaking. In spring andfall, heavy winds and frequent rains canlimit the forager bees opportunity to col-lect resources. Inherent in looking atthe season is analyzing the interdepen-dencies of the weather and photoperiod.Consequently, the weather is a strongcontributor to the rate at which thequeen lays eggs which in turn affects thenumber of worker bees. The number ofworker bees affects the amount of forag-ing as well as the number of brood thatmay be reared.

• Lastly, one must know the developmen-tal rates of the different castes of bees.For the number of worker bees able toforage is of extreme importance in de-termining whether the colony is ableto feed itself. Further, the lifespan ofworker bees is dependent upon the sea-son in which they emerge - a worker thatemerges in the autumn will live longerthan one emerging in the late spring orsummer. This is due to foraging activity,in the late spring and summer there willbe more resources available than in the

fall.

A lot of careful thought must go into mak-ing the model. A model that has too muchinfomation cannot be used to make predic-tions simply because the mathematics is toocomplicated. On the other hand, a modelthat has too little information can make themathematics easier, but it will not say muchabout the biology.

INSIDE THE HIVE

daylengthEgg laying increases with daylength.

temperature

Bees forage intemperaturesover 54◦ F.

Egg laying increaseswith temperature.

wind/rainBees do not flyin wind or rain.

OUTSIDE THE HIVE

queenDrones matewith queen.

Queen lays eggs. Workersserve queen.

brood

24 daystomature

21 daystomature

21 days

Housebeesmaintainhive.

�A

drones�

� -

� ��

�A

�A

house bees� ��

�A

foragers �Foragersprovidenectarand pollen.

��

��

Foragers fly500 miles,then die.

� �� A

�

�

6

��

� �

51

Charting the Dynamics of aHive for a Month

• Obtain a copy of the worksheet “A Month in theHive”.

• Choose a city in the United States and a month.

• Choose the strength of the queen’s egg laying poten-tial.

• For your city, find the latitude and the mean dailytemperature for the month chosen.

• Determine the hours of daylight and enter it on theworksheet

• Determine the temperature and the daylength scoreand enter it on the worksheet.

• Multiply these two numbers together and enter it onthe worksheet.

• Decide if bees forage. If the mean temperature isabove 54◦F , then the answer is yes. If the meantemperature is below 54◦F , then the answer is no.

• Decide on the level of resources - nectar and pollen -high, medium, or low and use the following table. Inthe winter, the resources are likely to be low. At thepeak of flowering season, the resources are likely tobe high.

53

Forager FractionResource Lifetime Deaths d

Low 4 0.25

Medium 8 0.13

High 12 0.08

The forager lifetime is in days. It is based on thefact that a foraging bee’s wings burn out after ap-proximately 500 miles.

• Enter the population of BROOD, HOUSE BEES,and FORAGERS in the hive at the beginning of themonth in thousands of bees.

• Run the program BEPOPITA. Enter 1 for the prompt-ing of MONTHS SIMULATED. The table of dailypopulations for brood, house bees, and foragers isdisplayed in the STAT lists L1, L2, and L3.

• Graph the hive population dynamics for the monthand think about these population changes and theadaptability of honey bees to this environment.

• Try a new situation. You can

– pick a new city for the same month,

– pick a different month for the same city,

– change the populations at the beginning of themonth, or

– see what happens if resources increase or decrease.

54

A Month in the Hive

City

Queen Strength Hive PopulationMonth (in Thousands of Bees)Latitude Initial FinalTemperature Score BroodHours daylight Score House Bees

Brood Score b ForagersBees Forage? TOTALResourceForager LifetimeFraction Deaths d

-

6

Day

30252015105

POPULATION IN THOUSANDS

55

A Month in the Hive

City

Queen Strength Hive PopulationMonth (in Thousands of Bees)Latitude Initial FinalTemperature Score BroodHours daylight Score House Bees

Brood Score b ForagersBees Forage? TOTALResourceForager LifetimeFraction Deaths d

-

6

Day

30252015105


56

Latitudes and Monthly Mean Temperatures for

Selected Cities

The United States

Lat. J F M A M J J A S O N DWashington, D.C. 39 31 34 42 53 62 71 76 75 67 55 45 35AlabamaMobile 31 51 54 60 68 75 81 82 82 78 69 59 53Birmingham 34 42 46 54 63 70 77 80 80 74 62 52 45AlaskaAnchorage 61 13 18 24 35 46 54 58 56 48 35 22 14Fairbanks 65 -13 -4 9 30 48 59 62 57 45 25 4 -10Juneau 58 22 28 31 39 46 53 56 55 49 42 33 27Nome 65 9 3 7 18 36 45 51 50 52 28 16 4ArizonaFlagstaff 35 28 31 34 37 50 58 66 64 58 47 37 30Phoenix 34 52 56 61 68 77 87 92 90 85 73 61 53Tucson 32 50 54 59 66 74 84 86 84 80 70 61 52ArkansasLittle Rock 35 40 44 52 62 71 79 82 81 74 63 51 43CaliforniaEureka 41 47 49 48 49 52 55 56 57 57 54 51 48Fresno 37 46 51 54 60 68 75 81 79 74 65 53 45Los Angeles 34 57 59 60 62 65 69 74 75 73 69 63 58San Diego 33 57 58 59 61 63 66 70 72 71 68 62 57San Francisco 38 49 52 53 55 58 61 62 63 64 61 55 49ColoradoDenver 40 30 34 38 47 57 67 73 71 63 52 39 33Grand Junction 39 26 34 42 52 62 72 79 76 67 55 40 28ConnecticutHartford 42 25 28 37 49 59 69 73 71 63 52 42 29DelawareWilmington 40 31 33 42 52 62 71 76 75 68 56 46 36

57


Lat. J F M A M J J A S O N DFloridaJacksonville 30 53 55 61 68 74 79 81 81 78 70 61 55Miami 26 67 68 72 75 79 81 83 83 82 78 73 69Tampa 28 60 61 66 72 77 81 82 82 81 74 67 61GeorgiaAtlanta 34 42 45 53 59 69 76 79 78 73 62 52 45Savannah 32 49 52 58 66 73 79 81 81 77 67 58 51HawaiiHonolulu 21 73 73 74 76 78 79 80 81 81 80 77 74IdahoBoise 44 30 36 41 49 57 66 75 72 63 52 40 32IllinoisChicago 42 21 26 36 49 59 69 73 72 65 54 40 28IndianaIndianapolis 40 26 30 40 52 63 72 75 73 67 55 42 32IowaDes Moines 42 19 25 35 51 62 72 76 74 65 54 39 26KansasDodge City 38 30 35 42 54 64 75 80 78 69 58 53 34KentuckyLexington 38 32 35 44 55 64 72 76 75 69 57 45 36Louisville 38 33 36 45 57 65 74 78 76 70 58 46 37LouisianaNew Orleans 30 52 55 61 69 75 80 82 82 79 69 60 55MainePortland 44 22 23 32 43 53 62 68 67 59 49 38 26MarylandBaltimore 39 33 35 43 54 63 72 77 76 69 57 46 37MassachusettsBoston 42 30 31 48 49 59 68 74 72 65 55 45 34MichiganDetroit 42 23 26 35 47 58 68 72 71 63 52 40 39Grand Rapids 43 22 24 33 46 58 67 71 70 62 51 39 27Sault Ste. Marie 47 13 14 24 38 50 58 64 63 55 45 33 20MinnesotaDuluth 47 6 12 23 38 50 59 65 63 54 44 28 14Minneapolis 45 11 18 29 46 59 68 73 71 61 50 33 19MississippiJackson 32 46 49 56 65 73 79 83 81 76 65 55 49MissouriKansas City 39 26 32 42 55 65 76 79 77 68 58 43 32St. Louis 39 29 34 43 56 66 75 79 77 70 58 45 34Springfield 37 32 36 45 56 65 73 78 77 70 58 45 36

Latitudes and Mean Temperatures 59

Lat. J F M A M J J A S O N DMontanaHelena 47 18 26 32 42 52 60 68 66 56 45 31 23NebraskaOmaha 41 19 25 35 50 62 71 76 74 64 54 38 26NevadaReno 40 32 37 41 46 55 62 70 67 60 50 40 33New HampshireNew JerseyNewark 41 31 33 41 52 62 72 77 76 68 57 47 36New MexicoAlbuquerque 35 35 39 46 55 64 75 79 76 69 57 44 36Las Cruces 32 43 48 54 62 71 79 82 81 75 65 50 44New YorkAlbany 43 21 23 34 47 58 67 71 69 61 51 39 26Buffalo 43 24 25 33 45 56 66 71 69 62 52 40 29New York 41 32 33 41 53 62 71 77 75 68 58 47 36Syracuse 43 23 24 33 46 57 66 71 69 62 51 41 28North CarolinaAsheville 36 37 39 46 56 63 70 73 73 70 5 46 39Raleigh 36 40 42 49 59 67 74 78 77 71 60 50 42North DakotaBismark 47 7 15 26 43 55 64 70 69 57 46 29 15OhioCleveland 41 26 27 37 48 58 68 72 70 64 53 42 31Columbus 40 27 30 40 51 61 70 74 72 66 54 42 31OklahomaOklahoma City 36 36 41 49 58 67 76 78 78 72 61 52 44OregonPortland 46 39 43 46 50 57 63 67 67 63 54 46 41PennsylvaniaPhiladelphia 40 31 33 42 53 63 72 77 75 68 57 46 36Pittsburgh 40 27 29 39 50 60 68 72 71 64 53 42 31Puerto RicoSan Juan 18 77 77 78 80 79 80 82 82 82 81 80 78Rhode IslandProvidence 41 28 29 37 48 58 67 73 71 64 53 43 32South CarolinaCharleston 33 49 51 57 66 73 79 82 81 77 68 59 52South DakotaRapid City 44 21 26 33 45 56 65 73 71 61 50 35 26


Lat. J F M A M J J A S O N DTennesseeKnoxville 36 38 42 50 60 67 74 78 77 72 60 49 41Memphis 35 40 44 52 63 71 79 82 81 74 63 51 43Nashville 36 37 40 49 60 68 76 79 78 72 60 49 41TexasEl Paso 32 35 48 50 63 66 80 82 80 74 64 52 44Dallas-Ft.Worth 33 44 49 56 66 74 82 86 86 79 68 56 48Houston 30 51 55 61 69 75 81 83 83 78 70 60 54San Antonio 29 50 54 62 70 76 82 85 84 79 70 60 53UtahSalt Lake City 41 29 34 41 49 59 68 78 75 65 53 40 30VermontBurlington 43 17 18 29 43 55 65 70 67 59 48 37 23VirginiaNorfolk 37 40 41 49 58 67 76 78 78 72 61 52 44Richmond 38 37 39 47 58 66 74 78 77 70 59 49 40WashingtonSeattle 48 39 43 44 49 55 60 65 64 60 52 45 41Spokane 48 26 32 38 46 54 62 70 68 60 58 45 36West VirginiaWisconsonMilwaukee 43 19 23 32 45 55 65 71 69 62 51 37 25Wyoming

Latitudes and Hours of Daylight

-

6

JANUARY - JUNE

20 30 40 50 60

degrees North latitude

hours of daylight

8

9

10

11

12

13

14

15

16

June

a a a a a a a a a a a a a a a a May

a a a a a a a a a a a a a a a a a a a

April

a a a a a a a a a a a a a a a a a a a a a a

March


February

a a a a a a a a a a a a a a a a a a a a a aJanuary

a a a a a a a a a a a a a a a a a a

61


-

6

JULY - DECEMBER

20 30 40 50 60

degrees North latitude

hours of daylight

8

9

10

11

12

13

14

15

16

July

a a a a a a a a a a a a a a a a aAugust


September


October


November

a a a a a a a a a a a a a a a a a a a aDecember

a a a a a a a a a a a a a a a a a

The information on daylengths was taken from The World Almanac andBook of Facts Astronomy - Daily Calendar. The daylight hours are for thefifteenth day of each month.

Computing Brood Score

-

6

mean daily temperature - degrees Fahrenheit20 30 40 50 60 70 80 90 100

MEAN DAILY TEMPERATURE SCORE

0.0

0.2

0.4

0.6

0.8

1.0

a a a a a a a a a aa a a

a a aa a a a

a a a aa a a a a

a a a a a a aa a a a a a a a

-

6

hours of daylight8 9 10 11 12 13 14 15 16

HOURS OF DAYLIGHT SCORE

0.0

0.2

0.4

0.6

0.8

1.0

a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a

63

TI-82 Program BEPOPITA

• Choose a strength in egg laying potential for youqueen.Input 1 for a weak queen, laying a maximum of 1000eggs per day.Input 3 for a strong queen, laying a maximum of3000 eggs per day.Input some number between 1 and 3 to indicate anappropriate intermediate stregth in egg laying po-tential.

• Input the initial population of BROOD, HOOUSEBEES, and FORAGERS.

• Input the number of months in your simulation.

PROGRAM: BEPOPITA

:Disp “QUEEN STRENGTH”

:Input Q

:Disp “INITIAL”


:Disp “(THOUSANDS)”

:Disp “BROOD”

:Input U

:Disp “HOUSE BEES”

:Input V

:Disp “FORAGERS”

:Input W

:Disp “MONTHS SIMULATED”

:Input T

65

:For(M,1,T,1)

:Disp “MONTH”,M

:Disp “BROOD SCORE”

:Input B

:Disp “DEATH RATE”

:Input D

:For(I,1,30,1)

:20*U/21+0.571*

B*Q*log(W+1)→L1(I)

:U/21+20*V/21→L2(I)

:V/21+(1-D)*W→L3(I)

:L1(30)→ U

:L2(30)→ V

:L3(30)→ W

:End


:Disp U,V,W

:L1(30)→L4(M)

:L2(30)→L5(M)

:L3(30)→L6(M)

:End

66

V. A Year in the Hive

Reading:

Life in the Wild

Materials:

• Observational Hive

• Programable calculator or

• Personal computer.


• Following the population dynamics in a hive over thecourse of a year.

• Investigating the response of a hive to an environ-mental catastrophe

67

Life in the Wild

For long periods of time, life goes on com-fortably inside the hive. On long, warm,sunny days, the queen is busy laying eggs.They house bees keep the hive in good work-ing order. They do this by cleaning and pol-ishing cells and packing them with honey andpollen. This honey is produced from the nec-tar collected by the forager bees. The housebees add enzymes to the nectar to change itschemical composition. As the water in thehoney evaporates and the bees respire, thehive becomes humid and warm. The housebees are responsible for fanning in order tocool and freshen the hive. They must alsodevote considerable attention to the needs ofthe queen and the queen’s egg laying. Afterthe queen lays eggs, the house bees must capthe comb cells, and rear the brood. Housebees must also secrete wax and build combin anticipation of the need for future broodand food storage. In preparation for theirjobs as foragers, house bees will take orienta-tion flights. Honey bees can raid another hiveas a method of collecting resources. Some ofthe older house bees are assigned to guardthe hive and ward off attacks. During thedaylight hours, forager bees are going aboutthe business of collecting nectar and pollen.Drones looking to mate make their daily af-ternoon flights.

These conditions cannot continue forever.At some point, the hive will become over-crowded or will be disturbed. Environmentalconditions may change - the hive may be un-der attack by predators like a bear, by ants,or by natural disasters like fire. In these in-stances, the hive must make a critical deci-sion.

Swarming and Colony Division

When the population of a colony becomestoo large for its nest site, the bees begin per-forming several special activities. Some ofthese activities are visible to someone observ-ing the hive. Bees can be seen clustering out-side the hive. Scout bees begin to search fora new site. Even on a clear warm day, a hivepreparing to divide into two or more coloniesdoes little foraging. Inside the hive, the housebees begin to construct large vertically hang-ing queen cells. The workers begin to engorgethemselves on the honey reserve in prepara-tion for a swarm.

The scouts begin by making an exhaustivesearch for nearby sites. The typical choice fora new hive is a quarter to a half mile away.However, the bees are willing to look consid-erably farther if it is necessary. They seekout cavities having on opening less than 12

69


square inches situated 3 or more feet abovethe ground. Ideally, the entrance to the cav-ity is well below the roof. The preferred sizesare between 600 and 6000 square inches. Thisvolume is small enough so that the colony islikely to outgrow it within a year, but largeenough so that the hive has a good chancefor survival. The shape of the cavity doesnot seem to matter.

If a scout finds a potential site for the newhive, she will communicate the location of herfinding in much the same way a forager com-municates the location of a food source. Thescouts inspect each others findings and reacha consensus. If the weather conditions areright, the hive is now ready to make its move.

Swarming is initiated by a special dance,the Schwirrlauf, the German word for whirdance. During the Schwirrlauf, workers movewithout stopping in straight lines across thecomb. Every couple seconds, they vibratetheir partially spread wings. These dancersalso make occasional five second contactswith other worker bees. During this contactthe bee makes a continuous piping sound.

For situations that are not emergencies, thecolony may divide. Part of the hive stays be-hind with the new queen, which may still bein her larval state. Although scientists arenot certain on this point, the bees that swarmseem to be a random selection from the en-tire population. Thus, the swarm contains amixture of bees of all ages. The majority ofswarms take place within an hour of the mid-dle of the day. Because drones do most oftheir flying in the afternoon, morning swarmsare likely to contain drones.

When a swarm first emerges from the hive,

it chooses a nearby bush or tree to settle.This is a particularly dangerous time for thecolony. For example, if rain begins falling atthis point, the bees cannot fly to their newnesting site and resume their usual activities.Consequently, the colony might starve. If theweather remains suitable, the bees make surethat their queen is present. The scouts thenleave the swarm to find their choice for a newhive and to verify that the area is still favor-able. Most of the scouts then return and re-port to the others its location by performinga wagtail dance on the surface of the swarm.At first the hive moves very slowly - taking 5minutes to move 100 yards. At this time, thecolony again checks that the queen is withthem. The swarm then spreads out to fill aspace about 50 yards in diameter and speedsup to 6 miles per hour - flying between 3and 10 feet above the ground over and notaround obstacles. A couple hundred yardsbefore reaching the site of their new hive, the

Figure 1: Honey bee swarm

Life in the Wild 71

swarm slows down. The scout bees at thenew site are performing a “breaking” danceas a guide to the scouts that are escorting theswarm.

At this time, the scouts take positions nearthe hive entrance and leave a scent to attractthe queen and workers. Most of the workerswill enter the site before the queen enters.

For hives having a high population, addi-tional swarms of bees may leave at the timethat virgin queens are making their nuptialflights. In this manner, the original colonymay divide to form several new colonies.

Swarming and Superceding

If the swarm takes with it an older queen,then shortly after establishing the new hive,the queen is replaced by a daughter. Thisevent, termed supercedure, also takes place inany colony that has an old or failing queen.The queen produces a chemical substancethat inhibits her replacement. As time goeson, she produces less and less of this sub-stance. As a consequence, her ability to fore-stall being superceded becomes weaker andweaker. Like the case in which house beesdetect that the colony is about to divide, thebees begin to produce queen cells. Typically,they will not make quite as many queen cellsas they make in preparation for colony divi-sion. Supercedure is necessary because thequeen has a diminished capacity for layingbrood. Also the lack of spermatozoa in thespermatheca means that the queen is morelikely to lay unfertilized eggs, that is, eggsthat will become drones.

Figure 2: Emerging adult honey bee queen

When a young queen hatches into a colonyhaving a queen, the older queen is usuallykilled. In some situations, the older queencontinues living and producing brood untilher successor begins to lay. Sometimes theworkers will kill an old queen with no otherqueen present or before the young queenstarts producing brood. In either circum-stance, because young queens produce lit-tle of the chemical that inhibits supercedure,the worker will begin to develop emergencyqueen cells. These cells are made by enlarg-ing worker cells.

After the young queen emerges, the hivemay choose to kill any of the remainingqueens while they still reside in their cell.This can be a risky strategy. If the youngqueen fails to return to the hive after her nup-tial flight, the the colony will perish. The hivemay also choose to attack other queens afterthey have emerged. The bees can also pre-vent the first young queen from attacking theyoung queens still in their cells by leaving or


adding wax to the queen cell capping. Theycan also feed these young queens while theyare still in their cells. For very large colonies,this delay in queen emergence allows the hiveto divide into more than two colonies.

Swarming and Absconding

Bees can face a variety of enviromentalemergencies - food or water resources becomescarce or the bees sense a threat to the hive.Under these circumstances, the entire colonycan leave the hive at a moment’s notice andtake off to find a new nest. This type of be-havior is call absconding.

The European continent has cold winters.This environment favors those bees who canstore food for long periods of time and sur-vive in the hive having little activity. Thisstrategy favors colony division as a methodof migration.

On the other hand, scarcity of resources isa common occurrence in Africa. During hardtimes, one method of survival for the hive isto raid another hive. A second method isto move the entire colony to more favorablesurroundings. Thus the honey bees in Africathat have been selected over the centuries arethose that have the propensity to abscondand the ability to move long distances to finda new home. These bees are extremely pro-tective of their hive and brood and are muchmore sensitive to hive disturbances. This pro-tective behavior is the source of the tales ofthe “killer bee”.

Absconding is the central strategy for mi-gration for African honey bees. Because they

use a hive for a shorter period of time, per-haps as short as six weeks, the honey beesof Africa are far less selective in their choiceof hives and will settle on a much smallerspace to build the hive. These traits persistin the hybrid European-African honey bee.These hybrid bees have spread through largeparts of Central and South America becauseof their tendency to make frequent moves.

At the present time, the Africanized honeybee is well established in southern Arizona.We are witnessing its continued migrationin Arizona and its introduction to Califor-nia and Nevada. We can see that bees aremoving quickly into places that are favorablefor their survival - habitats that have suffi-cient water and pollen resources and are suf-ficiently warm. Their eventual habitat in Ari-zona, California, Nevada, and in other partsof the United States is now a topic of activeresearch.

Charting the Dynamics of aHive for a Year

• Obtain a copy of the worksheet “A Year in the Hive”.

• Choose a city in the United States.

• Choose the strength of the queen’s egg laying poten-tial.

• Find the latitude of your city and the mean dailytemperature for each month of the year.

• Determine the hours of daylight for each month andenter it on the worksheet

• Determine the temperature and the daylength scoresfor each month and enter it on the worksheet.

• Multiply these two numbers together and enter it onthe worksheet.

• Decide if bees forage. If the mean temperature isabove 54◦F , then the answer is yes. If the meantemperature is below 54◦F , then the answer is no.

• Decide on the level of resources - nectar and pollen- for each month - high, medium, or low and usethe following table. In the winter, the resources arelikely to be low. At the peak of flowering season, theresources are likely to be high.

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Forager FractionResource Lifetime Deaths d

Low 4 0.25

Medium 8 0.13

High 12 0.08

The forage lifetime is in days. It is based on the factthat a foraging bee’s wings burn out after approxi-mately 500 miles.

• Run the program BEPOPITA.

• Enter the QUEEN STRENGTH.

• Enter the population of BROOD, HOUSE BEES, andFORAGERS in the hive at the beginning of the yearin thousands of bees.

• Enter 12 for the MONTHS SIMULATED.The table of monthly populations for brood, housebees, and foragers is displayed in the STAT lists L4,L5, and L6.

• Graph the hive population dynamics for the year andthink about these population changes and the adapt-ability of honey bees to this environment.

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A Year in the HiveThe Good Life

City LatitudeQueen Strength

Hours of Mean Brood Forage DeathMonth Light Score Temp. Score Score ? Resource Rate

123456789101112

Brood Death Population in ThousandsMonth Score Rate Brood House Bees Foragers

0 − − −123456789

101112

75

-

6

Month


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The Bee Hive’s Response toDramatic Events

Stable conditions inside a hive cannot continue for-ever. At some point, the hive becomes overcrowded oris disturbed. The queen becomes old and is replaced. Adisease may affects the hive. Environmental conditionsmay change - resources may vanish, or the area near thehive may be sprayed by pesticide. We will use BEPOPITAto simulate the effects of these dramatic events.

Below are five critical situations that a hive may face.

• a weak queen

– When the queen has a daily egg laying capacityaround 1000, the hive struggles throughout theyear.

• a swarm

– Approximately 25% to 30% of the house bees andforaging bees leave with a swarm. You can sim-ulate this by keeping 70% to 75% of these popu-lations and continuing the model.

• resource depletion,

– This will force the foraging bees to look harderfor resources. You can simulate this by settingD equal to 0.25 for the month in which resourcesare depleted.

• supercedure of the queen, or

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– With supercedure, the colony can go without eggsfor 28 days. You can simulate this with a broodscore B equal to 0 for the month in which thequeen is superceded.

• death by pesticide of a fraction of the foraging pop-ulation.

– Having 25% of the foraging population die froma pesticide spray is typical. This number cango higher if the foraging bees bring the pesticideback to the nest.

To simulate the three hive events - swarm, death bypesticide, death from a brood disease - choose the monthof the event by making an appropriate choice for theMONTHS SIMULATED in the BEPOPITA program. Makethe changes in the hive population, and return to BE-POPITA. The brood population is stored under U, thehouse bee population is stored under V, and the foragerpopulation is stored under W.

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A Year in the HiveDeath and Destruction

City LatitudeQueen Strength

Hours of Mean Brood Forage DeathMonth Light Score Temp. Score Score ? Resource Rate

123456789101112

Brood Death Population in ThousandsMonth Score Rate Brood House Bees Foragers

0 − − −123456789

101112

79

-

6

Month


80

VI. Birth, Death, andMigration

Materials:

• Styrofoam cup with lids

• Lots of beads in three different colors

• Remote sensing maps of Tucson, Arizona, and Sonoradesigned for honey bee migration.

• Data on Africanized honey bee sitings in Arizona andSonora


• Bead models of birth and migration.

• Predicting the spread of the Africanized honey beein Arizona

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Simulating the Migration ofHoney Bee Colonies

Consider an 11 mile stretch of the Colorado River anddivide it into 1 mile segments. Imagine placing a colonyof bees in the middle segment of the 11. Number thissegment 0, to indicate the middle. Number the segmentsupstream 1 through 5 and number the segments down-stream −1 through −5.

• Flip a coin.

• If the coin lands heads, move one mile upstream.If the coin lands tails, move one mile downstream.

• Keep track of the number of coin flips necessary tomove 1, 2, 3, 4, and 5 miles away from the startingpoint.

• If you run this experiment several times or if othersrun this experiment, you will discover the speed ofmigration with no new bee colonies.

• Migration may move faster in one direction than an-other. This can be silulated by placing the appropri-ate number of black and yellow beads in a styrofoamcup or by using the TI-82 program WALK.

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TI-82 Program WALK

Mathematicians call this model a simple random walk.A TI-82 program WALK will produce examples of thesewalks. After the PROBABILITY ? prompt, give the prob-ability for movement upstream. The walk makes one stepper time unit. You are also asked to chose the LASTTIME for the random walk. Follow the successive posi-tions for the random walk by punching the ENTER key.The sequence of postions for the random walk are dis-played in the STAT list L1.

PROGRAM:WALK


:Input P

:Disp “LAST TIME STEP”

:Input T

:0→L1(1)

:For(I,1,T,1)

:rand→Q

:L1(I)+(P-Q)/abs(Q-P)→L11(I+1)

:Disp “TIME”,I-1

:Disp “POSITION”,L1(I)

:Pause

:End

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Simulating the Birth andMigration of Honey Bee

Colonies

Now we combine into a model the actions of birthand movement. Begin with a collection of pennies anda styrofoam cup having black and yellow beads. Thefraction of yellow beads is the probability that a colonydivides.

• Begin with a coin on the board at square 0. Thisindicates one colony at this location.

• Flip each coin on the board. If the coin lands heads,move the coin up one square. If the coin lands tails,move down one square.

• For each coin on the board, pour a bead from thestyrofoam cup. If the bead is yellow, place a newmarker on the square with the marker. This indi-cates that a colony has divided.

• Return to the second step and continue until a markerreaches either square +5 or square -5.

• If the probability of moving upstream and down-stream is different from 1/2, then we will need a sec-ond styrofoam cup with bead. We can go to threecolors of beads to include the possibility that thecolony does not move.

• Colony migration depends on the season, so we maywant to have a different set of styrofoam cups fordifferent times of the year.

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