PASCUA YAQUI SUMMER PROGRAMmath.hawaii.edu/~sarah/beepop_combined.pdf · 2. How the Africanized...

PASCUA YAQUISUMMER PROGRAM

A Joint Endeavor ofPasqua Yaqui Tribe

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

June 2 - 21, 2013

BEEPOP

The Population Dynamics of the

Honey Bee

in the Hive and in the Wild

June 3-10, 2013

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

I. Sizing up the Population

Reading:

Bee Hives

Materials:

• Photographs of bees on a frame

• 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 partsof the world discovered that they couldincrease honey production by creatinga special environment for bee hives.Today, apiculture, the science of bee-keeping, is a comprehensive scientificand technological enterprise. Beekeep-ers study topics from the anatomy andphysiology of bees to their evolution,genetics, and ecology. The best bee-keepers are those who can combine sci-entific knowledge with practical skills tomanage their hives and to process andmarket honey.The first human-made hives were

quite variable in size and shape. Theywere made by these ancient beekeepersfrom material that was readily avail-able. For example, early Mediterraneanand Egyptian hives were often pipesmade of sun-baked mud, hives in Africawere pipes made from tree bark, andthe earliest northern European hiveswere vertically standing hollowed outlogs. In Western Europe, the hives wereconstructed from basketwork plasteredwith mud and cow dung. The mate-rial was made into a long tube sev-eral inches in diameter. This tube was

laid upon itself into a spiral to forma cylinder with a cap. The final con-struction was then bound together us-ing split bramble stems. These hivesheld about five gallons of honey. Whenthe time to harvest arrived, honey wascollected by first killing the colony ofbees inside the hive and then removingall of the honey comb.

Beginning in the seventeenth century,beekeepers began to experiment withhives made from wooden boxes. Withthis strategy, boxes could be stacked orplaced side by side to extend the size of

3

4 Bee Hives

the original hive. Dividing plates couldbe placed between the boxes to makeaddition and removal of boxes easy todo while still having control of the ac-tivities of the bees inside the hive. Withthese innovations, honey could be re-moved without destroying the colony.Since the middle of the eighteenth

century, bee hives have had movableparts and spaces for bees to movearound the hive. The interior design ofthe beehive is a series of vertically hang-ing rectangular frames. These frameslook like a shallow dresser drawers. Inbuilding a frame, a piece is cut outfrom the base to allow the bees to movearound. The remaining base is thencoated with a wax foundation. Thecoating has an imprint of the hexagonaldesign of the honey comb to encouragethe bees to build their comb so that theframes can be easily removed.

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 cheddar

goldfish 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.

7

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

• 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 HoneyBees

Reading:

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 Research

Center

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 to

the 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 Project

has 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 beneficial

insects 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 main

product 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 diet

comes 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 bees

also 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 re-

ferred to as honey bees, are perhaps

the most intensely studied of all insects.

Their economic importance to the agri-

cultural industry has driven the need

for scientific research. As a result, a

wealth of valuable and interesting infor-

mation has accumulated. This mass of

data has shown honey bees to be highly

social creatures with complicated be-

haviors and intriguing population dy-

namics.

The economic importance of honey

bees is due to the products they pro-

duce as well as services they perform.

Although honey bees produce honey,

they serve a more important role as

pollinators. What makes the honey

bee so special is that unlike many in-

sects, the honey bee will seek out pollen

and not nectar. Honey bees commonly

pollinate agricultural crops such as ap-

ples, cherries, melons, and almonds. In

fact, many farmers hire beekeepers to

raise and maintain bee colonies on their

farms entirely for this purpose. Honey

bees also produce wax used for pol-

ishes and candles. The importance of

honey bees is not a new discovery. Pic-

tographs depicting bees and their hives

have been found painted on the walls of

caves believed to be many thousands of

years old.

Rock painting depicting honey gathering. Discovered in the

Cuevas de la Arana near Bicorp in Valencia, Spain, from

E. Hernendez-Pacheco, Museo nacional de ciences naturales,

Madrid

Although honey and bees play an

enormous role in the United States agri-

cultural industry, they are not native

29


to North America. Interestingly, they

were brought here by early European

colonizers. The indigenous habitat of

the Apis mellifera ranges from the tip

of Southern Africa to Southern Scan-

dinavia, and from continental Europe

to Western Asia. Thus the honey bee

is a highly adaptable insect able to ad-

just to a wide variety of climes and ge-

ographic regions.

The desert regions of the southwest-

ern United States and northern Mexico

are home to the richest variety of bees

in all the world. According to the Carl

Hayden Bee Research Center, located

in Tucson, Arizona there are approx-

imately 1000 to 1200 species of bees

within a one hundred mile radius of

Tucson. Yet, none of these are native

honey bees.

Approximately 25,000 species of bees

have been identified, with almost

40,000 still yet to be catalogued. That

is, entomologists know that they are

out there, but have yet to place them

in a specific genus. However, out of this

25,000, only 8 to 10 species are consid-

ered honey bees. Yet, this number is

growing as more species are identified.

Honey bees are not only classified by

genus and species, but by strain. The

strain denotes the bees’ place of origin.

The most common strains of honey bees

currently found in the United States are

the Apis mellifera ligustica, the Italian

bees, and the Apis mellifera carnica,the Carniolan bees. However, many

common lines of honey bees have been

allowed to interbreed. The goal of such

crossings has been to develop hybrid

bees with specific morphological and

behavioral traits that will enhance their

honey producing and pollinating traits.

The most famous attempt at creating

such a hybridized line was the crossing

of the European lines and the African

lines.

The goal of crossing the European

and the African lines was to mate the

docile but high honey yield European

bees with their aggressive but low yield

African counterparts. In 1956, Brazil-

ian researchers hoped that a harder

working bee that made more honey

would result. However, the experi-

ment did not succeed. What the re-

searchers found was that the aggres-

sive traits dominated and essentially

masked the European characteristics.

The experiments have become famous

because a worker in the apiary where

the hybridized lines were being kept

accidentally removed protective screens

that kept the queens in their hives.

As a result, at least 26 swarms of the

Africanized bees escaped. The descen-

dants have been moving northward ever

since. Today. 40 years later, African-

ized honey bees are found in the south-

western United States and are a cause

for concern due to their aggressive na-

ture, and their ability to take over and

replace established colonies of produc-

tive European lines.

Introduction to the Honey Bee 31

A major product of the general sci-

entific research into the ecology of ApisMellifera is a greater understanding of

the honey bee’s “social structure” and

population dynamics. In examining the

population of the colony, scientists have

uncovered the existence of a highly or-

dered caste system. The queen reigns

at the top of the caste, with the male

drones and female workers below.

The queen’s primary duties are to

populate the colony by mating with

drones (male honey bees), and direct

the activities of the workers. The queen

mates during the early summer months,

generally within the first week after

having emerged from her chamber. She

takes what is referred to as “nuptial

flights” where she may mate with sev-

eral drones per day over the course of

several days. She collects their sperm

inside a large body cavity called the

“spermatheca.” After this flight has

been completed the queen has accumu-

lated enough sperm to sustain her en-

tire career as the egg-layer of the hive.

However, if the queen’s egg-laying ca-

pacity is deficient, or if she is unrespon-

sive to the needs of the colony in some

way, she may be attacked by the work-

ers and replaced.

The workers are the second caste in

the colony and perform many crucial

tasks within the hive. Most impor-

tantly, they are responsible for tend-

ing to the queen by bringing to her a

special food, and grooming her. Addi-

tionally, workers build comb, tend the

brood, seal and cap comb cells contain-

ing either honey or bees, remove debris

from the hive, store pollen, ripen and

store honey, and guard the hive.

The male drones are the third caste

in the colony. They are responsible for

mating with queens from other colonies

and perform virtually no other useful

task within the hive. After mating,

drones die from the rupturing of their

abdomens and genital apparatus. How-

ever, many drones die before they get

a chance to mate with the queen be-

cause they may be killed or thrown out

of the nest by worker bees when food

resources are low.

Examining the behavior and ecology

of bees is a worthy task. Bees are im-

portant economically as well as ecologi-

cally. By understanding their behavior,

morphology, and population dynamics,

scientists, beekeepers, and farmers may

work together to develop strategies that

enhance the productivity of our agricul-

tural 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”.

33

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

❝ ❝

❝ ❝

❝ ❝

❝ ❝

❝ ❝❝ ❝ ❝ ❝ ❝ ❝

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

– theMALTHUSIAN PARAMETER, that is themean

number 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”

36

:Disp “DISTRIBUTION”

: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

37

:G→L1(G)

:P→L2(G)

: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 black

bead.

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 CasteSystem

The social structure of a bee hiveis that of a matriarchal family headedby a queen. The queen has a poten-tial life span of three years and dur-ing this time may continually lay eggsthereby establishing and maintaininga total colony population of approxi-mately twenty five thousand bees. Al-most 95% of the queens offspring arewhat are referred to as worker bees,with the remaining 5% developing intodrones. The queen mates during oneperiod of her life in a series of excur-sions called nuptial flights. She takesthese flights shortly after emerging fromher egg, and may mate with 7 to 17male drones. It is estimated that thequeen may receive up to 5,000,000 in-dividual sperm during this short periodof mating, all of which is stored in apouch-like structure on her abdomencalled the spermatheca. The queen ac-cesses the sperm throughout her life,fertilizing eggs at a rate dictated by theneeds of the hive.

The Workers

The worker bees make up the ma-jority of the population and are all fe-males, but with undeveloped, or staticreproductive systems. The worker-females are altruistic, for they take careof the queen at the expense of their abil-ity to reproduce, and perform virtuallyall of the tasks necessary for the sup-port of the hive. These worker-femaleshave a short lifespan of approximately30 days, and during this time will gothrough different developmental stageswhich dictates their role in the hive,as well as giving the hive its hierarchalcharacter. Worker bees may be classi-fied as housekeepers, which are respon-sible for the upkeep of the hive, or asforagers whose role is to collect the nec-tar, pollen, and water necessary to sus-tain life.The primary cause of death for a

worker bee is burnout. That is,their wing muscles only have a certainamount of flight, generally 800 kilo-meters, and when this point has beenreached they are incapable of fulfillingtheir role and taking care of themselves.

45


The basis for the division of laborwithin the hive is the age of the worker.The worker begins its life taking care ofthe storage cells of the hive, then moveson to brood care and food storage, andends its life as a forager. The adap-tive significance of this labor scheduleis that it extends the life of the workerby establishing a system whereby theyoung worker bees spend the majorityof their lives inside the hive where theylive in the protected environments oftheir colony. Once they become for-agers, they are susceptible to predation,bad weather, and wing burnout.

Drones

The drones are the only males pro-duced by the queen. Although few innumber, they serve a singular, but im-portant, role as mates for the queen.The lifespan of the male drone is veryshort, for after mating their abdomensexplode which results in rapid death.Drones only serve as mates for thequeen, and are not involved in feedingthe colony, or the upkeep of the hive.Thus, if resources are scarce, workerbees do not like to keep the seeminglylazy males around and will often forcethem from the hive or kill them di-rectly. Further, since the drones spenda great deal of time outside of the hivethey are more susceptible to predationor death. All this results in an average

drone lifespan of less than 25 days. Thedrones will often begin mating flightseight days after emerging, and withintwelve days they may perform up to fivemating flights per day.

Population Dynamics of theHive

The caste system functions as an in-tegrated feedback system of interde-pendent elements. The growth of thecolony, and the rearing of brood dependon the reproductive state of the queen,that is, the number of eggs she is capa-ble of laying each day, and the workerpopulation that is responsible for tend-ing to the larvae. The amount of broodthat can be effectively reared is directlyproportional to the number of workerbees, resulting in a cycle whereby thegreater the number of workers, themore a population may grow (assum-ing an abundance of resources). How-ever, the ability of the colony to obtainresources, and the reproductive stateof the queen are strongly influenced byfactors such as the queen’s age, the pho-toperiod (amount of light in a day),mean daily temperature, and the avail-ability of resources.These factors may be used to emulate

the feedback system of interdependentelements and allow ecologists to createa mathematical model which may pre-dict the population of a hive given a

Life in the Hive 47

specific set of conditions. The mostimportant factors used are the initialpopulation size, the queen’s reproduc-tive state, and the weather. Yet, moreinformation is needed to determine thesignificance of these factors. One mustalso know the developmental rates ofthe worker bees through their life cy-cle, the lifespan of drones, rate of broodproduction, amount of sperm obtainedby queen during mating, and the aver-age worker age before becoming a for-ager. This information has been devel-oped by entomologists and reported inthe scientific literature.

“Free-Body Modeling”

The first task in creating a popula-tion model for honey bees is to takethe interdependent factors and deter-mine their relationship to one another.This can be accomplished by creating aconcept-map, or flow chart, visually de-picting how these factors relate to oneanother, this is called free-body mod-eling. To do this, one can write outthe variables that govern the function-ing of the system, then draw lines be-tween them to show how they influ-ence each other. Further, one may thenwrite on the lines just how it is that thefactors affect each other. This is an ex-tremely important tool that allows themodeler a graphic visualization of howthe model will work. Next, equations

can be created to describe the behaviorof each variable. Finally, these equa-tions can be entered into a computerprogram to see if they accurately repre-sent what is seen in the empirical data.

In the Hive

In creating a model representing amonth in the hive, we must be able topredict the number of eggs that maybe laid by the queen on any given day.This is accomplished by determiningthrough many observations the maxi-mum number of eggs that a queen maylay in a day, and how factors such asthe queen’s age (the number of daysthat a queen has been laying eggs),ambient temperature (in degrees cel-sius or Fahrenheit), the photoperiod (inhours), and the adult population size.The maximum number of eggs that aqueen may lay in a day has been esti-mated to be as large as 3000 eggs perday. However, a more realistic valueunder average conditions may be 1500eggs per day.

Egg Maturation

After determining the number of eggslaid per day one must then determinethe proportion of eggs that develop intoeither workers or drones. This is in partestablished by the amount of sperm


that the queen has left in her spermath-eca, which is dependent upon the num-ber of days that the queen has beenlaying eggs, for as the queen nears theend of her supply she predominantlyfertilizes drones. The second factor inpredicting the number of drones is theproduct of egg fertilization responsesto photoperiod and the population sizeof the colony. As the population be-comes very large, the queen may reduceher egg laying to prevent an unneces-sary strain on available resources. Fur-ther, as the photoperiod decreases, thecolony anticipates the advent of win-ter and signal the queen to reduce orstop egg laying so that the hive willachieve a population size that is sus-tainable through the winter months.Using these factors it has been deter-mined that fertilized drone brood maycomprise no greater than 5% of the to-tal brood, with the remaining 95% des-ignated as workers. The different eggshave different rates by which they de-velop into adults. Worker brood takesan average of 21 days to mature intoadults, whereas the drone brood takes24.

Temperature

Temperature plays a pivotal role inhive population dynamics. Honey beeswill forage only when the average dailytemperature exceeds 12 degrees celsius.

Photoperiod

The photoperiod plays an importantrole as noted above. It has a strongaffect on the number of eggs that aqueen may lay. It is believed that a de-crease in the photoperiod sends a signalto the hive that winter is approaching.This is supported by the logic that asdaily low temperatures fall below freez-ing, the amount of available resources,such as pollen and nectar, falls dramat-ically. The decrease in resources meansthat the hive may not continue to grow,and may even need to reduce its popu-lation to sustain itself through the win-ter.

Other Weather Factors

Honey bees will not forage if the windvelocity is greater than 34 km/hr, or ifit is raining. The high winds preventbees from being able to maneuver whichmay result in their being unable to re-turn to the hive from foraging. Rainresults in wet bees with the water in-creasing their weight to the extent thatthey can no longer support themselvesin flight.Weather conditions affect the size

and functioning of the hive resulting indifferent population dynamics for dif-ferent geographic regions. For example,in the midwest, photoperiod may rangefrom 9.1 (December) to 15.25 (June)

Life in the Hive 49

hours of light per day, with the tem-perature ranging from 0 to 31 degreescelsius. In contrast, the photoperiod inthe southwest may range from 10 (De-cember) to 14.5 (June) hours of lightper day, with the temperature rangingfrom 0 to 39 degrees celsius. Under themidwestern conditions, the colony maybe confined to the hive and will not pro-duce brood from autumn to mid-winter,and nor will they forage from late Octo-ber to late April. In contrast, coloniesin the Southwest may produce broodand forage year round. From this infor-mation, it is easy to predict what timesof the year will promote colony growthor decline.

A Month in the Hive

The information thus far presentedcan be used to predict how the popu-lation of a colony may change over thecourse of a month.

• The first consideration will be es-timating the initial population.

• The next consideration is the egglaying strength of the queen. Thisdepends primarily on genetics, theage of the queen and semen avail-ability.

• Thirdly, one must consider the sea-son. In the winter months thepopulation will be low, whereas in

mid-summer the population maybe peaking. In spring and fall,heavy winds and frequent rains canlimit the forager bees opportunityto collect resources. Inherent inlooking at the season is analyz-ing the interdependencies of theweather and photoperiod. Conse-quently, the weather is a strongcontributor to the rate at whichthe queen lays eggs which in turnaffects the number of worker bees.The number of worker bees affectsthe amount of foraging as well asthe number of brood that may bereared.

• Lastly, one must know the de-velopmental rates of the differentcastes of bees. For the numberof worker bees able to forage is ofextreme importance in determin-ing whether the colony is able tofeed itself. Further, the lifespan ofworker bees is dependent upon theseason in which they emerge - aworker that emerges in the autumnwill live longer than one emergingin the late spring or summer. Thisis due to foraging activity, in thelate spring and summer there willbe more resources available than inthe fall.

INSIDE THE HIVE

daylengthEgg laying increases with daylength.

temperature

Bees forage in

temperatures

over 54◦F.

Egg laying increases

with temperature.

wind/rainBees do not fly

in wind or rain.

OUTSIDE THE HIVE

queenDrones mate

with queen.

Queen lays eggs.Workers

serve queen.

brood

24 days

to

mature

21 days

to

mature

21 days

House

bees

maintain

hive.

✁❆

drones✞

✝ ✲

✝ ✆✝ ✆✝ ✆

✁❆

✁❆

house bees

✛ ✆☎

✁❆

foragers ☎Foragers

provide

nectar

and pollen.

✆✛

✆✛

Foragers fly

500 miles,

then die.

✝ ✆✝ ✆✝ ✆✁❆

✛

✛

✻

✆✛

✛ ☎

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 0.25Medium 8 0.13High 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

✲

✻

Day

30252015105

POPULATION IN THOUSANDS

55

Latitudes and Monthly Mean Temperatures forSelected 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

✲

✻

JANUARY - JUNE

20 30 40 50 60

degrees North latitude

hours of daylight

8

9

10

11

12

13

14

15

16

June

❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ May

❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛

April

❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛

March

❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛

February

❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛January

❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛

61


✲

✻

JULY - DECEMBER

20 30 40 50 60

degrees North latitude

hours of daylight

8

9

10

11

12

13

14

15

16

July

❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛August

❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛

September

❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛

October

❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛

November

❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛December

❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛

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

✲

✻

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

❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛❛ ❛ ❛

❛ ❛ ❛❛ ❛ ❛

❛ ❛ ❛❛ ❛ ❛ ❛

❛ ❛ ❛ ❛ ❛❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛

✲

✻

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

❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛

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”

65

:Input T

: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

Esther Widiasih

should beL1(I) L2(I)L3(I)

Esther Widiasih

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 oncomfortably inside the hive. On long,warm, sunny days, the queen is busylaying eggs. They house bees keep thehive in good working order. They dothis by cleaning and polishing cells andpacking them with honey and pollen.This honey is produced from the nec-tar collected by the forager bees. Thehouse bees add enzymes to the nectarto change its chemical composition. Asthe water in the honey evaporates andthe bees respire, the hive becomes hu-mid and warm. The house bees are re-sponsible for fanning in order to cooland freshen the hive. They must alsodevote considerable attention to theneeds of the queen and the queen’s egglaying. After the queen lays eggs, thehouse bees must cap the comb cells,and rear the brood. House bees mustalso secrete wax and build comb in an-ticipation of the need for future broodand food storage. In preparation fortheir jobs as foragers, house bees willtake orientation flights. Honey bees canraid another hive as a method of col-lecting resources. Some of the olderhouse bees are assigned to guard the

hive and ward off attacks. During thedaylight hours, forager bees are goingabout the business of collecting nectarand pollen. Drones looking to matemake their daily afternoon flights.These conditions cannot continue

forever. At some point, the hive will be-come overcrowded or will be disturbed.Environmental conditions may change -the hive may be under attack by preda-tors like a bear, by ants, or by naturaldisasters like fire. In these instances,the hive must make a critical decision.

Swarming and Colony Division

When the population of a colony be-comes too large for its nest site, thebees begin performing several specialactivities. Some of these activities arevisible to someone observing the hive.Bees can be seen clustering outside thehive. Scout bees begin to search fora new site. Even on a clear warmday, a hive preparing to divide intotwo or more colonies does little forag-ing. Inside the hive, the house bees be-gin to construct large vertically hanging

69


queen cells. The workers begin to en-gorge themselves on the honey reservein preparation for a swarm.The scouts begin by making an ex-

haustive search for nearby sites. Thetypical choice for a new hive is a quar-ter to a half mile away. However, thebees are willing to look considerablyfarther if it is necessary. They seekout cavities having on opening less than12 square inches situated 3 or morefeet above the ground. Ideally, the en-trance to the cavity is well below theroof. The preferred sizes are between600 and 6000 square inches. This vol-ume is small enough so that the colonyis likely to outgrow it within a year, butlarge enough so that the hive has a goodchance for survival. The shape of thecavity does not seem to matter.If a scout finds a potential site for the

new hive, she will communicate the lo-cation of her finding in much the sameway a forager communicates the loca-tion of a food source. The scouts in-spect each others findings and reach aconsensus. If the weather conditionsare right, the hive is now ready to makeits move.Swarming is initiated by a spe-

cial dance, the Schwirrlauf, the Ger-man word for whir dance. Duringthe Schwirrlauf, workers move with-out stopping in straight lines acrossthe comb. Every couple seconds, theyvibrate their partially spread wings.These dancers also make occasional five

second contacts with other worker bees.During this contact the bee makes acontinuous piping sound.For situations that are not emergen-

cies, the colony may divide. Part of thehive stays behind with the new queen,which may still be in her larval state.Although scientists are not certain onthis point, the bees that swarm seemto be a random selection from the entirepopulation. Thus, the swarm containsa mixture of bees of all ages. The ma-jority of swarms take place within anhour of the middle of the day. Becausedrones do most of their flying in the af-ternoon, morning swarms are likely tocontain drones.When a swarm first emerges from the

hive, it chooses a nearby bush or treeto settle. This is a particularly danger-ous time for the colony. For example,if rain begins falling at this point, thebees cannot fly to their new nesting siteand resume their usual activities. Con-sequently, the colony might starve. Ifthe weather remains suitable, the beesmake sure that their queen is present.The scouts then leave the swarm to findtheir choice for a new hive and to verifythat the area is still favorable. Most ofthe scouts then return and report to theothers its location by performing a wag-tail dance on the surface of the swarm.At first the hive moves very slowly -taking 5 minutes to move 100 yards. Atthis time, the colony again checks thatthe queen is with them. The swarm

Life in the Wild 71

then spreads out to fill a space about50 yards in diameter and speeds up to6 miles per hour - flying between 3 and10 feet above the ground over and notaround obstacles. A couple hundredyards before reaching the site of theirnew hive, the swarm slows down. Thescout bees at the new site are perform-ing a “breaking” dance as a guide to thescouts that are escorting the swarm.At this time, the scouts take posi-

tions near the hive entrance and leave ascent to attract the queen and workers.Most of the workers will enter the sitebefore the queen enters.For hives having a high population,

additional swarms of bees may leave atthe time that virgin queens are makingtheir nuptial flights. In this manner,the original colony may divide to formseveral new colonies.

Swarming and Superceding

If the swarm takes with it an olderqueen, then shortly after establishingthe new hive, the queen is replaced bya daughter. This event, termed su-percedure, also takes place in any colonythat has an old or failing queen. Thequeen produces a chemical substancethat inhibits her replacement. As timegoes on, she produces less and less ofthis substance. As a consequence, herability to forestall being superceded be-comes weaker and weaker. Like the

case in which house bees detect that thecolony is about to divide, the bees beginto produce queen cells. Typically, theywill not make quite as many queen cellsas they make in preparation for colonydivision. Supercedure is necessary be-cause the queen has a diminished ca-pacity for laying brood. Also the lack ofspermatozoa in the spermatheca meansthat the queen is more likely to lay un-fertilized eggs, that is, eggs that will be-come drones.When a young queen hatches into a

colony having a queen, the older queenis usually killed. In some situations,the older queen continues living andproducing brood until her successor be-gins to lay. Sometimes the workers willkill an old queen with no other queenpresent or before the young queen startsproducing brood. In either circum-stance, because young queens producelittle of the chemical that inhibits su-percedure, the worker will begin to de-velop emergency queen cells. Thesecells are made by enlarging worker cells.After the young queen emerges, the

hive may choose to kill any of the re-maining queens while they still residein their cell. This can be a risky strat-egy. If the young queen fails to returnto the hive after her nuptial flight, thethe colony will perish. The hive mayalso choose to attack other queens afterthey have emerged. The bees can alsoprevent the first young queen from at-tacking the young queens still in their


cells by leaving or adding wax to thequeen cell capping. They can also feedthese young queens while they are stillin their cells. For very large colonies,this delay in queen emergence allowsthe hive to divide into more than twocolonies.

Swarming and Absconding

Bees can face a variety of enviro-mental emergencies - food or water re-sources become scarce or the bees sensea threat to the hive. Under these cir-cumstances, the entire colony can leavethe hive at a moment’s notice and takeoff to find a new nest. This type of be-havior is call absconding.The European continent has cold

winters. This environment favors thosebees who can store food for long peri-ods of time and survive in the hive hav-ing little activity. This strategy favorscolony division as a method of migra-tion.On the other hand, scarcity of re-

sources is a common occurrence inAfrica. During hard times, one methodof survival for the hive is to raid an-other hive. A second method is tomove the entire colony to more favor-able surroundings. Thus the honey beesin Africa that have been selected overthe centuries are those that have thepropensity to abscond and the ability tomove long distances to find a new home.

These bees are extremely protective oftheir hive and brood and are much moresensitive to hive disturbances. Thisprotective behavior is the source of thetales of the “killer bee”.Absconding is the central strategy

for migration for African honey bees.Because they use a hive for a shorterperiod of time, perhaps as short assix weeks, the honey bees of Africaare far less selective in their choice ofhives and will settle on a much smallerspace to build the hive. These traitspersist in the hybrid European-Africanhoney bee. These hybrid bees havespread through large parts of Centraland South America because of theirtendency to make frequent moves.At the present time, the African-

ized honey bee is well established insouthern Arizona. We are witness-ing its continued migration in Arizonaand its introduction to California andNevada. We can see that bees aremoving quickly into places that arefavorable for their survival - habitatsthat have sufficient water and pollenresources and are sufficiently warm.Their eventual habitat in Arizona, Cal-ifornia, Nevada, and in other parts ofthe United States is now a topic of ac-tive research.

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.

73

Forager FractionResource Lifetime Deaths d

Low 4 0.25Medium 8 0.13High 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.

74

A Year in the Hive

The Good Life

City LatitudeQueen Strength

Hours of Mean Brood Forage Death

Month Light Score Temp. Score Score ? Resource Rate

1

2

3

4

5

6

7

8

9

10

11

12

Brood Death Population in Thousands

Month Score Rate Brood House Bees Foragers

0 − − −1

2

3

4

5

6

7

8

9

10

11

12

75

✲

✻

Month


76

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.

77

• supercedure of the queen, or

– 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.

78

A Year in the Hive

Death and Destruction

City LatitudeQueen Strength

Hours of Mean Brood Forage Death

Month Light Score Temp. Score Score ? Resource Rate

1

2

3

4

5

6

7

8

9

10

11

12

Brood Death Population in Thousands

Month Score Rate Brood House Bees Foragers

0 − − −1

2

3

4

5

6

7

8

9

10

11

12

79

✲

✻

Month


80

VI. Birth, Death, andMigration

Reading:

Introduction to Remote Sensing

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


• Interpreting remote sensing maps

• Bead models of birth and migration.

• Predicting the spread of the Africanized honey beein Arizona

81

Remote Sensing

Remote sensing is defined as thetechnique of obtaining informationabout objects through the analysis ofdata collected by special instrumentsthat are not in physical contact withthe objects of investigation. This al-lows us to make decisions, predictions,and to model environments and situ-ations quickly and efficiently withoutever having to visit the area to be stud-ied. Remote sensing allows us to “see”things we normally could not with thenaked eye.The first use of remote sensing was in

military operations. One could fly a spyplane and take reconnaisance photos ofthe enemy’s territory and thus gatherinformation about it without ever be-ing on the ground. When it becameapparent that information was beinggathered from above, camoflage meth-ods were used. However, with the in-vention and use of thermal infrared andultraviolet films, identification of “hot”and inorganic items became easily at-tainable.Remote sensing then evolved into the

use of space satellites to obtain im-ages of the ground. These images

could be recorded in different portionsof the electromagnetic or “light” spec-trum. Humans view objects in the vis-ible portion of the spectrum. That is,with our own eyes, we see green plantsas having a green color, or a red sweateras looking red. In different portions ofthe electromagnetic spectrum, picturesof objects do not always appear to betheir “normal” color. In the infraredportion of the spectrum, the colors ap-pear shifted so that green plants appearred, and a red sweater appears yellow.Parts of the visible spectrum are brokeninto different regions. At the highestenergy level are the gamma rays andx-rays. The middle energy levels arethe ultraviolet, the visible, and the in-frared. Microwaves and radio waves areat the low end of the spectrum. Remotesensing allows us to take advantage ofthe entire electromagnetic spectrum asa method of detecting objects.Today, many countries have many

satellites that orbit the earth. Datais recorded in pixels. The word pixelwas formed as a word by combiningpicture and element. The pixels arethen put together in a rectangular ar-

83


ray to form a picture. The smallest de-tail of data obtained from these remotesensing satellites that is presently avail-able to the public is a square 30 feet by30 feet. So, using a satellite that has aresolution of 30 feet square within eachpixel, the human eye can recognize onthe picture generated by remote sens-ing data objects that are longer andwider than 30 feet. The military is cur-rently working with resolutions that are3 feet square or less. This will allowyou to recognize not only a car on theground but also its license plate fromdata gathered by an orbiting satellite.

Acquiring Information aboutBees from Remotely Sensed

Data

Remote sensing allows us to mapthe vegetation on the ground from asatellite which is orbiting the earth atan altitude over 500 miles above thesurface of the earth. This allows usto gather information on a large areawithout having to visit each and everysquare inch of it. As far as mappingthe movement of honey bees outside ofthe hive, we can use satellites to exam-ine the types and densities of vegeta-tion, the riparian or water areas, andthe elevation. Using what is called afalse color composite map, we make amap in which healthy and dense vege-tation appears dark red and water ap-pears black. Using overlay analysis, we

can then map elevation in relation tovegetation and locate the areas whichhave the most favorable conditions forhoney bee migration.Here, in Arizona, we can match

this information with information fromknown Africanized honey bee sightings.These sightings are records collectedfrom the public and from scientists whohave discovered bees in swarm traps.So far, we see that the bees have beenfollowing riparian areas at low eleva-tion having low laying desert vegeta-tion. Africanized honey bees tend notto migrate into colder areas at higherelevation. Instead, they are movinginto Arizona’s southwest deserts. TheAfricanized honey bees have also beenfound in areas where water is used forirrigation and agriculture.Where the Africanized honey bees

will go next and at what rate is a topicof research. Since the first Africanizedbee swarm was detected in Hildalgo,Texas in October, 1990, their rateof spread has increased to more than375 miles per year in the southwesternUnited States. Some researchers sug-gest that the Africanized honey bee willdisperse almost as far north as Canada;others say that they will go no fartherthan the southwestern and southeast-ern corners of the United States. Whatis known is that the Africanized honeybee is in the United States to stay andwill continue to have some impact onits native plants and animals.

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.

85

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

86

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.

87

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

88

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