1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A mathematical model to estimate the seasonal change in apparent longevity of bee colony Yasuhiro Yamada1†, Toshiro Yamada2††*, Kazuko Yamada2†† 1
Department of Applied Physics, Graduate School of Engineering, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-8656, Japan
2
Graduate School of Natural Science & Technology, Kanazawa University, Kakuma-machi, Kanazawa 920-1192, Japan
17
†Present address: Department of Physics, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka,
18
560-0043, Japan
19
†† Present address: 2-10-15, Teraji, Kanazawa, Ishikawa, 921-8178, Japan
20
*Corresponding author. Email:
[email protected]. Present situation: professor
21
emeritus of Kanazawa University
22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
Keywords: mathematical model, dynamic model, apparent longevity, lifespan, wintering failure, honeybee, social insect, colony, extinction, overwintering, adult bees, capped brood, seasonal change, Field experiment, long-term
2 37 38 39 40 41 42 43 44 45 46 47 48 49
Summary The longevity of a honeybee colony is far more significant than the lifespan of an individual honeybee, a social insect. The longevity of a honeybee colony is integral to the fate of the colony. We have proposed a new mathematical model to estimate the apparent longevity defined in the upper limit of an integral equation. The apparent longevity can be determined only from the numbers of adult bees and capped brood. By applying the mathematical model to a honeybee colony in Japan, seasonal changes in apparent longevity were estimated in three long-term field experiments. Three apparent longevities showed very similar season-changes to one another, increasing from early autumn, reaching a maximum at the end of overwintering and falling approximately plumb down after overwintering. The influence of measurement errors in the numbers of adult bees and capped brood on the apparent longevity was investigated.
3 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
A lifespan of an animal, which is the period of time while an individual is alive, is an important index to evaluate individual activities. In the colony composed of eusocial insects such as honeybees (Apis mellifera) which exhibit age-polyethism, the lifespan of each individual cannot always give an assessment as to the activities of a colony but the longevity of colony could give it more appropriately. The longevity of a colony will have greater significance than the lifespan of each individual of the colony. The life of colony diversely depends on the inborn lifespan of an individual, the labor division distribution ratio of each honeybee performing a particular duty, the natural environment such as the weather, the amount of food, pests and pathogens, the environmental pollution due to pesticides and so on. The honeybee length of life has been observed or estimated before in the four seasons, which have a distinct bimodal distribution in temperature zones. According to previous papers, honeybees live for 2-4 weeks1 and 30-40 days2 in spring, for 1-2 weeks1, 25-30 days2 and 15-38 days3 in summer, for 2-4 weeks1 and 50-60 days2 in autumn, and for 150-200 days3, 253 days2, 270 days4, 304 days5 6-8 months6 and 150-200 days3 in winter, where it has been estimated that the difference of life length among seasons may come from the brood-rearing load imposed on honeybees1 and may mainly come from foraging and brood-rearing activity2. Incidentally, the lifetime of the queen seems to be three to four years (maximum observed nine years). The average length of life of worker bees in laboratory cages was observed to range from 30.5 to 45.5 days7. The study on the influence of altitude on the lifespan of the honeybee has found that the lifespans are 138 days at an altitude of 970 m and 73 days at an altitude of 200 m, respectively 8. Many papers have discussed what factors affect the length of life (lifespan, longevity, life expectancy) on a honeybee colony as follows: Proper nutrition may increase the length of life in a honeybee colony. Honeybees taking beebread or diets with date palm pollen (the best source for hypopharyngeal gland development) showed the longest fifty percent lethal time (LT50)9. The examination for the effect of various fat proteins on honeybee longevity have shown that honeybees fed diets of red gum pollen have the longest lifespan but those fed invert sugar have the shortest lifespan10. In the discussion on nutrition-related risks to honey bee colonies such as starvation, monoculture, genetically modified crops and pesticides in pollen and sugar, protein nutrient strongly affects brood production and larval starvation (alone and or in combination with other stresses) can weaken colonies11. And protein content in larval diet could positively affect worker longevity likely because of the increased antioxidant gene expression12. Through study on the effect of different dietary sugars and honey on longevity in hyper parasitoids, it has been found that reproductive success and longevity in honeybees varied significantly when fed different sugars13. It has been demonstrated that resveratrol significantly affects gustatory responsiveness and prolonged lifespan in wild-type honeybees under normal oxygen conditions, but the enhanced lifespan effect of resveratrol is abolished under hyperoxic conditions14. Comparisons among the nutritional effects of high protein feeds on honeybees in three steps of incubator, field and overwintering have shown that the longevity of caged honeybees in incubator is significantly affected by dietary treatments15. They concluded that soybean flour and bread yeast can be used as pollen supplements and substitute cakes. From the discussions on the ontogenic time and worker longevity in the Australian stingless, it has been inferred that extended longevity may result from evolutionary adaptations to the floral resource scarcity16. It has been demonstrated that pollen and royal jelly extend worker longevity17. It is reported that the load of work imposed on honeybees is deeply related to their length of life as follows: It has been developed that workloads imposed on worker bees beyond those normally encountered by them make their lifespans shorter18. The higher longevity of workers in winter seems to be probably connected with the interruption of the brood-rearing cycle1. Foraging and brood rearing activities are regarded as important factors affecting the lifespan2. Based on the hypothesis that the division of labor in foraging workers is a consequence of division of risk among foragers with differing life expectancies through field experiment, the validity of a dynamic programming model has been discussed19. A reverse relationship between the longevity of
4 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151
honeybees has been reported20. On the main driver of aging in long-lived winter honeybees, it has been concluded that all processes driving senescence in honeybees will have to be considered separately not only between queens and workers but also within the worker caste21. It has been inferred from their experiments that the mortality patterns in worker honeybees mainly influenced by the transition from in-hive workers to foragers, where the nearly tenfold lifespan differences between winter and summer bees defies explanation by the fact that the reduction of foraging activity and elimination of a number of external mortality factors increases the life expectancy of workers only by 4-5 days though it was experimentally confirmed that there was a higher mortality rate of foragers than non-foraging bees22. Furthermore, it was found that honeybees live longer in smaller than in larger colonies and it was concluded that it was caused by the primary effect of simultaneous consequences of foraging choice, overall workload and the rate of aging of worker honeybees23. The influence of a physical environment on the length of life in a honeybee colony is discussed. Comparing the longevity between field colonies and laboratory caged ones, there are seasonal inconsistencies between the two24. It has also been reported that eye color of honeybees can cause differences in longevity25. The influence of reproductive protein (vitellogenin) and brood (juvenile) pheromone on longevity has been discussed. The role of the vitellogenin in honeybee workers has been reviewed and a mathematical model has been proposed6. Vitellogenin gene activity protects worker bees from oxidative stress, where vitellogenin protein is synthesized at high levels in honeybee queens and is abundant in long-lived workers26. It has been demonstrated that brood rearing reduces worker vitellogenin stores and colony long-term survival27. It has been discussed how the manipulation of social environment in colonies with long-lived winter bees might alter the pace of functional senescence as well as of cellular senescence which is induced by such social intervention as changes in social task behaviors and social environment28. Extrinsic influences such as mites, pathogens and pesticides on the longevity have been examined. The examination of sub-lethal effects, including delayed larval development and adult emergence or shortened adult longevity on worker honey bees from pesticide residue exposure from contaminated brood comb, shows they can have indirect effects on the colony such as premature shifts in hive roles and foraging activity29. Nosema ceranae can infect honeybee larvae, reducing subsequent adult longevity30. Based on the information from observations of life length and probable factors influencing it as mentioned above, several mathematical models have been proposed in order to estimate seasonal changes in the length of life as follows: An attempt has been made to estimate the length of life in a honeybee colony assuming life expectancy as a function of both the future hatching rates and the net changes of the worker bee populations based on the number of worker-bees and open and sealed brood-cells observed at regular intervals31. A mathematical model with arbitrary parameters has been proposed under some assumptions to estimate the relationship between worker longevity and the intracolonial population dynamics of honeybees in temperature zones from early spring to late autumn32. And two mathematical models with arbitrary parameters have been proposed to estimate expected longevity of workers in colonies with age polyethism and that with caste polyethism under several assumptions while comparing that without each polyethism33, 34. Under several assumptions in order to offer a simple theoretical framework with which to explore how the dynamics of food flow through a colony might interact with population dynamics to determine colony growth, a mathematical model with arbitrary parameters has been proposed35, 36. Honeybee populations are predicted by using several parameters such as mortality within each bee caste under some assumptions37. These mathematical models might predict correctly the length of life and be useful and effective in beekeeping if the values of arbitrary parameters and the assumptions are close to the truth. The more parameters in a model, the more information can be obtained. The degree of accuracy in
5 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169
estimation deeply depends upon the veracity of each parameters value and the validity of each assumption. A single inaccurate parameter or assumption may lead to a large error in simulated results. It is difficult to correctly estimate the errors arising from the arbitrary parameters and the assumptions. The length of life in a honeybee colony is one of the most important indexes to assess the colony activity. Here we call the length of life in a honeybee colony the “apparent” longevity which reflects the influences of environment due to the weather and harmful pollutants such as pesticides, those of infections by pathogens, mites, wax-moth larvae and small hive-beetles, those of predation by birds, hornets and toads, and those of missing bees which are alive elsewhere. This work is focused on a proposal of a mathematical model to estimate only the apparent longevity, L(t), fairly accurately through all seasons. In order to estimate the apparent longevity as accurately as possible, the less parameters and the more plausible assumptions are better and it is desirable that the parameters can be determined from experimental data with high accuracy and the assumptions are closer to the truth. In this paper we propose a simple and realistic mathematical model whose parameters can be determined from the numbers of adult bees and capped brood obtained from the photos of combs and the inside of a beehive, which are comparatively easy to be accurately obtained from an experiment.
170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202
Examples of applying the model We obtained the numbers of adult bees and capped brood from three long-term field experiments that we conducted in 2011/2012 from July 2011 to April 201238, 2012/2013 from June 2012 to July 201339 and 2013/2014 from August 2013 to February 201440 in the same apiary in Shika, Japan as shown in Fig. 1 and Supplementary Table 1. To apply the proposed model to honeybee colonies, we made the following assumptions: (1) A pupa (brood) in a sealed (capped) cell is removed as soon as the pupa dies. (2) At the start of the experiment (when t = t0), the age distribution of the initial capped brood is uniform (i.e., the number of initial capped brood at each age (in days) is equal to one-twelfth the total initial number). (3) The eclosion rate, p(t), has a constant value of 0.941 throughout all seasons, although it may possibly be affected by other factors such as the ratio between the number of in-hive bees and foraging bees. (4) After eclosion, adult bees unexpectedly disappear (except in case of death through capture by birds, frogs, hornets, dragonflies, etc.) and cannot return to their beehive because of sudden changes in the weather and other factors such as in situ death due to pesticides and the onset of darkness. Here, apparent longevity includes all environmental factors, except the eclosion rate. From the abovementioned assumptions, we have estimated the apparent longevity of a honeybee colony using the numbers of adult bees and capped brood in Fig. 1 and Supplementary Table 2. Figure 2 shows the change in apparent longevity for each colony with days that have elapsed from June 28 in the 2011/201238, 2012/201339 and 2013/201440 experiments. We can perceive from Figs 1 and 2 that each estimated apparent longevity roughly has a similar change with the elapsed days regardless of different initial conditions and different numbers of adult bees and capped brood in each colony as follows: The apparent longevity begins to increase from late September and continues to increase until spring and after that it decreases suddenly in mid spring. That the longevity involving a sudden change was easily obtained from the measured numbers of adult bees and capped brood by solving a set of equations suggests from a mathematical viewpoint that both measured data were considerably accurate and little inconsistent with each other. In order to ensure the validity of the mathematical model, we tried to compare the previous reports2, 31. The previous results, however, seemed to be insufficient for the validity check (see Supplementary Figure 1). Examining the seasonal changes in atmospheric temperature in the Noto district around the experimental site (Weather archives in the Noto district of Ishikawa prefecture in Japan by the Kanazawa Local Meteorological Observatory), those during each experimental period seem to be approximately similar to each other. The similarity of the seasonal changes in apparent longevity of each colony may be due to that in atmospheric temperature during each experimental period. In
6 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237
other words, the apparent longevity in a colony will probably be very susceptible to the weather conditions. Strictly, the seasonal changes in apparent longevity, however, do not always coincide with those in atmospheric temperature. Especially, a drastic drop of apparent longevity in spring is distinct from a gentle rise in atmospheric temperature. This fact implies that the apparent longevity may come under the influence of not only atmospheric temperature but also other factors. Examining the seasonal changes in apparent longevity carefully, we can generally know the following findings: Almost every colony begins to increase in apparent longevity from late September even before it gets cold, then continues to increase through late April after overwintering and shortly afterward the apparent longevity decreases extremely rapidly. The extreme rapid decrease in apparent longevity in spring cannot be explained only from the environmental conditions which are weather conditions1, nutrition9, workloads such as foraging and brood-rearing2, 20 , extrinsic factors such as pathogens and pesticides29. Consequently, it seem plausible that the information on longevity may be genetically programmed to vary drastically as certain threshold values of environmental conditions such as atmospheric temperature are exceeded. On the other hand, a slight elevation of the apparent longevity shown in September may be caused by less foraging load in August when there are few flowers in bloom Why was the apparent longevity of the 2012/2013-B colony longer than that of the 2012/2013-A colony by about nine days after the drastic drop? Examining the records such as photographs and our research notes in the 2012/2013 experiment in further detail, we found that several queen cells were made on the combs only in the 2012/2013-B colony after overwintering (May 10, 2013), and afterwards the old queen in the 2012/2013-B colony was replaced by a new queen (May 26, 2013). From the above fact, we infer as follows: When the queen is lost, eggs which the old queen lays before she is replaced by a new one must survive longer because there is a time lag of oviposition between the old queen and a new one and the time lag may lead to a lack of nurse bees. Examining in more detail, the apparent longevity of the 2012/2013-B colony is maintained at an approximately constant value higher than that of the 2012/2013-A colony from early May to late June in 2013, and then it decreases to about the apparent longevity of the 2012/2013-A colony in early July. It may be inferred from this fact that the old queen continues to lay eggs with higher longevity untill she is replaced by a new queen. When queen cells begin to be made, the new queen begins to lay eggs with normal longevity. This may be genetically programmed. Comparing the apparent longevities of colonies which had succeeded in overwintering (2012/2013-A and B in Fig. 2), we found that the longer apparent longevity colony (2012/2013-A) during overwintering becomes smaller in size than the other after overwintering against expectations, judging from the numbers of adult bees and capped brood as shown in Fig. 1.
238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253
Effect of miscount errors in measurement When we obtain the numbers of adult bees and capped brood in a field experiment, the following errors sometimes creep in: (1) Errors arising from a counting method, where a weighing method is apt to have much more errors than a direct counting method: (2) Errors arising from the timing of measurement, where a measurement just after dawn before foraging bees leave from their beehive contains less errors than that in the daylight. We will discuss the effect of miscounting error on the apparent longevity and examine the possibilities that the apparent longevity can be used as a kind of consistent test to evaluate the soundness of experimental data in a field experiment. To take concrete examples, let us consider four cases where the number of adult bees is overestimated more than the original one of the 2013/2014-A colony by 20 % (CASE 1) and it is underestimated less than the original by 20 % (CASE 2); the number of capped brood is overestimated more than the original by 20 % (CASE 3) and it is under estimated less than the original by 20 % (CASE 4) for about one month from August 24, 2013 to September 27, 2013 in the 2013/2014 experiment. These estimated results of apparent longevity are shown in Table 1. From Table 1, we can find that the overestimations of adult bees and the underestimations of capped brood give longer apparent longevity than the original longevity or vice versa, and that a difference
7 254 255 256 257 258 259 260 261 262 263 264 265 266 267
in the number of adult bees from the original for a given miscounted period affects the apparent longevity only for the given miscounted period. The difference does not affect the apparent longevity for a correctly counted period when there is no difference in adult bees from the original. A difference in the number of capped brood from the original for a given period continues to affects the apparent longevity even over a long period when there is no difference in the number of capped brood from the original. Figure 3 shows the relative errors of apparent longevity in all cases. It can be seen from Fig. 3 that the maximum value of relative errors in CASE 1 is about plus 72 %, that in CASE 2 is about minus 23 %, that in CASE 3 is about minus 19 % and that in CASE 4 is about plus 80 %. The measurement errors in the number of capped brood can affect the apparent longevity over to a long period without the measurement errors though those of adult bees can affect the apparent longevity only for a period with the measurement errors.
268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293
Conclusion
294 295 296 297 298 299 300 301 302 303 304
Methods
A new mathematical model has been proposed, from which the apparent longevity of honeybee colony can be estimated only from the numbers of adult bees and capped brood. The apparent longevity proposed newly in this paper which is an indicator of overall longevity to characterize a complicated colony system, is likely to provide useful prior information not only on normal environmental conditions of a colony but also on anomalous ones. This mathematical model can be applied to the colonies of any other social insects having a clear life cycle. Using the mathematical model, we estimated the apparent longevity of a honeybee colony in long-term field experiments. As a result of the estimation, the following were made clear: (1) Similar seasonal changes in apparent longevity can be shown among the three long-term field experiments regardless of the different populations in colonies. (2) A drastic drop in apparent longevity in spring cannot be always explained by conventional theories and is thought to be probably ascribable to programmed code: (3) A slight elevation in apparent longevity in September seems to be caused by less workloads in summer: (4) A difference in apparent longevity of about 9 days after a drastic drop seems to be perhaps induced by the replacement of a queen under conditions of the existence of queen cells. (5) A longer apparent longevity colony during overwintering can lead to a smaller colony after overwintering. Measurement errors in the number of adult bees or capped brood seem to affect the apparent longevity. For instance, when either the number of adult bees is overestimated or that of capped brood is underestimated, relative errors in apparent longevity increase severalfold. Errors in the number of capped brood have a lasting effect on the apparent longevity compared to a period without measurement errors. The apparent longevity may be available for checking the consistency of field experimental data in a honeybee colony through the process by which a set of equations to estimate it is solved.
Mathematical model for bee population Many reports on the population dynamic model of honeybee colony consist of several parameters on environmental conditions. Simulation results depend on the accuracy of these parameters which is difficult to be accurately determined from the vagaries of a variety of environmental conditions. Therefore, we now focus our attention on one parameter model which can assess the anomalous change (bee behavior) in colony due to various environmental factors such as pesticide, temperature, humidity, food, bee-family structure (decay of polyethim) and so on, in order to estimate a long-term change in overall longevity of colony workers. Based on the numbers of adult bees and capped brood at regular intervals according to the similar
8 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334
procedure reported previously31, we attempt to resolve one of the remaining issues in the previous report31 such as median age of the colony, mortality rate, median life expectancy, etc. as follows: It is very important to know the general health status of a honeybee colony in order to predict any anomalous changes such as a massive death and a failure in wintering. General health status can be reflected in the average lifespan of the workers, which is obtained from the ratio of the total stationary population to the number of surviving bees immediately after eclosion. However, a considerable amount of time and energy is required to obtain temporal changes in average worker lifespan in a colony because following up the changes in age distribution of large numbers of individual adult bees is almost impossible. The information which can be easily obtained with high accuracy in a field experiment are the numbers of adult bees and capped brood. The two numbers are highly correlated with each other and also with the duration of life of a honeybee. The maximum duration of life does not always coincide with the longevity in a honeybee colony because the life of a missing honeybee which is alive out of a beehive cannot be considered as the duration of life. Here we will call the maximum duration of life obtained from the two measured numbers an “apparent longevity”. The sound estimation of apparent longevity can be performed by relying on the correct numbers of adult bees and capped brood. The inconsistency between the two measured numbers of adult bees and capped brood can be reflected as a singular apparent longevity which is estimated from the two measured numbers. That is, an apparent longevity (La) can be expected also as a reliability assessment of the two measured numbers in a field experiment, which will be able to evaluate the soundness of the two measured data obtained from a field experiment as is the case with the check of thermodynamic consistency in vapor-liquid equilibrium data sets42. If either or both of the two measured numbers which have a close connection with each other are inaccurate and inconsistent with each other, the apparent longevity cannot be obtained because the following equations cannot be solved. The change in La with the passage of time shows the dynamic activity of a honeybee colony which is influenced by the environment (weather, pesticide, enemies of honeybees, etc.). Here, we will introduce a dynamic parameter called “apparent longevity”, La, (L(t), defined in Eq. (1)), which characterizes the “border” age at which an average worker bee disappears from the colony:
335
a( ) =
336 337 338 339 340 341 342 343 344 345 346
, where a(t) and u(t) denote the number of adult bees and newly capped brood, respectively, per unit time t, p(t) is the eclosion rate of pupae capped at time t, and Tpupa denotes the average capped brood period, which is 12 days. It is noteworthy that if the colony is in a steady state, where the worker bees die at the same age, L(t) represents the lifespan of the adult worker bee and may therefore provide an estimation of the longevity of worker bees in a normal colony, where age distribution of dying bees is given by a narrow Gaussian. To obtain the apparent longevity, L(t), in Eq. (1), it is necessary to know a(t), u(t), and p(t). a(t) and p(t) may be obtained from our measurements, respectively. Since it is difficult to obtain u(t) directly, we estimate it from Eq. (2):
347 348 349 350 351 352 353 354
( )=
( )
( − ) ( − )
( − )[ ( − )] ⁄
(1)
,
(2)
where b(t) is the total number of capped brood at time t. Note that we obtain Eq. (2) assuming that a pupa is removed immediately when it is dead in a cell before eclosion. The combination of Eqs (1) and (2) is useful to characterize a colony by using its macroscopic quantities (total numbers of adults and capped brood), which are easily accessible. Though a(t) in Eq. (1) and b(t) in Eq. (2) are continuous, their measurements which can be obtained every measurement date are discrete in an actual experiment. Therefore, it is necessary to discretize Eqs.
9 355 356 357 358 359 360 361
(1) and (2) in order to obtain the apparent longevity L(t). That is, we obtain sets of data at several time points in an actual experiment rather than continuously. Their time-discretization can be achieved under the assumptions that the number of newly capped brood and the eclosion rate are constant between two successive measurements [i.e., u(t) and p(t) have the following functional forms]: ( )=
362
363
364
−
= ( < ) ℎ ( ≤ < ) ℎ ℎ ) ( ≤ < ℎ ℎ ⋮ ( ≤ < ) ℎ ( − 1) ℎ ℎ ℎ ⋮ ℎ ℎ ( ≤ < )
( )=
365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392
̅ ( < ) ̅ ( ≤ < ) ̅ ( ≤ < ) ⋮ ( ≤ < ) ⋮ ( ≤ <
(3)
(4)
)
where denotes the time at the kth measurement. The subscript k takes from 0 (at initial experiment) to N − 1 (at final measurement). Since ̅ is given by the eclosion rate reported previously41, there are N-variables to be determined in Eq. (2): , … . Using the measured number of capped brood at t = , b( ), we obtain the following system of linear equations: ( )=
(
−
)[ (
−
)] ⁄
for k = 0, …, N − 1
(5)
This system of linear equations is easily solved using standard Gaussian elimination, which yields a set of solutions, , … . Then, L(t0), …, L(tN) are found from the discretized version of Eq. (1) by using an iterative procedure, with the values of , … obtained from Eq. (5). Here we will explain the symbols used in this mathematical model below: Tegg is the egg and larval period (from the time the queen lays eggs till they become a capped brood) and Tegg = 9 days for a honeybee; Tpupa is the capped brood period, mainly pupal period (from the time a brood starts to be capped till the capped brood ecloses); Tpupa = 12 days for a honeybee; a(t) is the number of adult bees; v(t) is the number of eggs laid by a queen per day in an interval between the (k − 1)th measurement and the kth; b(t) is the total number of capped brood summed up the residual and newly capped brood; u(t) is the number of newly capped brood per day in an interval between the (k − 1)th measurement and the kth at the elapsed days of t, where u(t) = v(t − 9), because it takes 9 days for capping of the honeybee eggs; tk = time at the kth measurement, which is a discrete variable, t = t0, t1, ····, tN − 1, tN ; ūk is equal to u(tk) which is assumed to be constant between the (k − 1)th measurement and the kth, to obtain the value of u(t), we assume the following because of the fact that the capped brood hatches over 12 days for a honeybee; L(t) is the apparent longevity of a colony, which is a parameter to characterize the degree of longevity of that colony. As there are few lost bees in winter because of extremely low foraging activity, the apparent longevity is roughly equivalent to the longevity of oldest member of that group. On the other hand, it is not so always
10 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409
other than in winter; p(t) is the probability of the brood newly capped at time t eclosing and developing into imagos (adult bees); ̅ is p(tk) that is assumed to be constant between the (k − 1)th measurement and the kth, which is estimated from the eclosion rate curve of the capped brood. We assume the following assumptions to solve the above equations: An equal number of brood (larvae) are capped per day between a certain experiment and the next experiment. One-twelfth of all capped brood, b(k), which are measured in the kth experiment will turned into new adult bees per day between the kth experiment and k+1th experiment because of the fact that new adult bees emerge from capped brood after 12 days from the day when a brood is capped. When an interval between a certain experiment and the next one is longer than 12 days, the number of new adult bees emerging from capped brood remains the same to the next experiment. In this work Ruby programing language was used to solve the equations in the mathematical model mentioned above. Apparent longevity of a colony composed of social insects other than honeybees can be estimated using the mathematical model proposed in this paper when the life cycle of the insect is known (Tegg, Tpupa and p(t) can be obtained) and the numbers of imagos and pupae can be accurately measured at given time intervals.
410
References
411
1. Bodenheimer, F. S. Bee Populations II. Seasonal population-trends of the honey-bee. Q. Rev. Biol.
412 413 414 415 416 417
12(4), 406–425 (1937). 2. Fukuda, H. & Sekiguchi, K. Seasonal change of the honeybee worker longevity in Sapporo, north Japan, with notes on some factors affecting the life-span. Jpn. J. Ecol. 16(5), 206-212, https://doi.org/10.18960/seitai.16.5_206 (1966). 3. Remolina, S. C. & Hughes, K. A. Evolution and mechanisms of long life and high fertility in queen honey bees. Age (Dordr) 30(2-3), 177–185, doi: 10.1007/s11357-008-9061-4 (2008). 4. Sakagami, S. F. & Fukuda, H. Life tables for worker honeybees. Res. Popul. Ecol. 10(2), 127– 139, https://doi.org/10.1007/BF02510869 (1968). 5. Mattila, H. R., Harris, J. L. & Otis, G. W. Timing of production of winter bees in honey bee (Apis
418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439
mellifera) colonies. Insect. Soc. 48, 88–93, https://doi.org/10.1007/PL00001764 (2001). 6. Amdam, G.V. & Omholt, S. W. The Regulatory Anatomy of Honeybee Lifespan. J. Theor. Biol. 216(2), 209–228 (2002). 7. Kulinčević, J. & Rothenbuhler, W. C. Selection for length of life in the honeybee (Apis mellifera). Apidologie 13(4), 347-352, https://doi.org/10.1051/apido:19820402 (1982). 8. Mustafa, A. O. Effect of geographical and seasonal factors on honeybee longevity in the Erbil Governorate of Northern Iraq. AGRIS (International Information System for the Agricultural Science and Technology) RECORDS in Food and Agriculture organization of the United Nation, University Library University of the Philippines at Los Baños, 107-109 (2004). Available: http://agris.fao.org/agris-search/search.do?recordID=PH2007000820.
9. Alqami, A. S. Influence of some protein diets on the longevity and physiological conditions of honeybee Apis mellifera L. workers. J. Biol. Sci. 6(4), 734-737, https://scialert.net/abstract/?doi=jbs.2006.734.737 (2006). 10. Manning, R. The Effect of High and Low Fat Pollens on Honeybee Longevity. RIRDC Publication No 08/031 RIRDC Project No DAW-105A. (2008). Available: https://www.agrifutures.com.au/product/the-effects-of-high-and-low-fat-pollens-on-honeybee-lo ngevity/. 11. Brodschneider, R. & Crailsheim, K. Nutrition and health in honey bees. Apidologie 41(3), 278– 294, https://doi.org/10.1051/apido/2010012 (2010).
11 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462
12. Li, C., Xu, B., Wang, Y., Yang, Z. & Yang, W. Protein content in larval diet affects adult longevity and antioxidant gene expression in honey bee workers. Entomol. Exp. Appl. 151(1), 19–26, https://doi.org/10.1111/eea.12167 (2014). 13. Harvey, J. A., Cloutier, J., Visser, B., Ellers, J., Wäckers, F. L. & Gols, R. The effect of different dietary sugars and honey on longevity and fecundity in two hyperparasitoid wasps. J. Insect Physiol. 58(6), 816–823, doi: 10.1016/j.jinsphys.2012.03.002 (2012) 14. Rascón, B., Hubbard, B. P., Sinclair, D. A. & Amdam, G. V. The lifespan extension effects of resveratrol are conserved in the honey bee and may be driven by a mechanism related to caloric restriction. Aging (Albany NY) 4(7), 499-508 (2012). 15. Irandoust, H. & Ebadi, R. Nutritional Effects of High Protein Feeds on Growth, Development, Performance and Overwintering of Honey Bee (Apis mellifera L.) IJABBR 1(6), 601-613 (2013) 16. Halcroft, M., Haigh, A. M. & Spooner-Hart, R. Ontogenic time and worker longevity in the Australian stingless bee, Austroplebeia australis. Insect. Soc. 60(2), 259-264, https://doi.org/10.1007/s00040-013-0291-9 (2013). 17. Wang, H., Zhang, S.-W., Zeng, Z.-J. & Yan, W.-Y. Nutrition affects longevity and gene expression in honey bee (Apis mellifera) workers. Apidologie 45(5), 618–625, https://doi.org/10.1007/s13592-014-0276-3 (2014). 18. Wolf, T. J. & Schmid-Hempel, P. Extra loads and foraging life span in honeybee workers. J. Anim. Ecol. 58(3), 943-954, doi: 10.2307/5134 (1989). 19. Woyciechowski, M. & Kozlowski, J. Division of labor by division of risk according to worker life expectancy in the honey bee (Apis mellifera L.). Apidologie 29(1-2), 191-205, https://doi.org/10.1051/apido:19980111 (1998). 20. Neukirch, A. Dependence of the life span of the honeybee (Apis mellifera) upon flight
465
performance and energy consumption. J. Comp. Physiol. B 146(1), 35-40, https://doi.org/10.1007/BF00688714 (1982). 21. Aurori, C. M., Buttstedt, A., Dezmirean, D. S., Mărghitaş, L. A., Moritz, R. F. & Erler, S. What
466
is the main driver of ageing in long-lived winter honeybees: antioxidant enzymes, innate
467
immunity, or vitellogenin? J. Gerontol. A: Biol. Sci. Med. Sci. 69(6), 633-639, doi: 10.1093/gerona/glt134 (2014). 22. Rueppell, O., Bachelier, C., Fondrk, M. K. & Page, R. E., Jr. Regulation of life history determines lifespan of worker honey bees (Apis mellifera L.). Exp. Gerontol. 42(10), 1020–1032, doi: 10.1016/j.exger.2007.06.002 (2007). 23. Rueppell, O., Kaftanouglu, O. & Page, R. E., Jr. Honey bee (Apis mellifera) workers live longer in small than in large colonies. Exp. Gerontol. 44(6-7), 447–452, doi: 10.1016/j.exger.2009.04.003 (2009). 24. Rinderer, T. E., Danka, R. G. & Stelzer, J. A. Seasonal inconsistencies in the relationship
463 464
468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487
between honey bee longevity in field colonies and laboratory cages. J. Apicult. Res. 51(2), 218-219, https://doi.org/10.3896/IBRA.1.51.2.13 (2012). 25. Almeida, R. & Soares, A. E. E. Longevity of africanized worker honey bees (Apis mellifera) carrying eye color mutant alleles. Sci. Agric. 60(2), 277-281, http://dx.doi.org/10.1590/S0103-90162003000200011 (2003). 26. Seehuus, S.-C, Norberg, K., Gimsa, U., Krekling, T. & Amdam, G. V. Reproductive protein protects functionally sterile honey bee workers from oxidative stress. PNAS 103(4), 962-967, https://doi.org/10.1073/pnas.0502681103 (2006). 27. Smedal, B., Brynem, M., Kreibich, C. D. & Amdam, G. V. Brood pheromone suppresses physiology of extreme longevity in honeybees (Apis mellifera). J. Exp. Biol. 212(Pt23), 3795-3801, doi: 10.1242/jeb.035063 (2009). 28. Münch, D., Kreibich, C. D. & Amdam, G. V. Aging and its modulation in a long-lived worker
12 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522
caste of the honey bee. J. Exp. Biol. 216(Pt9), 1638-1649, doi: 10.1242/jeb.078915 (2013). 29. Wu, J. Y., Anelli, C. M. & Sheppard, W. S. Sub-Lethal Effects of Pesticide Residues in Brood Comb on Worker Honey Bee (Apis mellifera) Development and Longevity. PLoS ONE 6(2), e14720, https://doi.org/10.1371/journal.pone.0014720 (2011). 30. Eiri, D. M., Suwannapong, G., Endler, M. & Nieh, J. C. Nosema ceranae can infect honey bee larvae and reduces subsequent adult longevity. PLoS ONE 10(5), e0126330, https://doi.org/10.1371/journal.pone.0126330 (2015). 31. Bühlmann, G. Assessing population dynamics in a honeybee colony. M. D. Gesell. Allg. Ange. Entomol. 4(4-6), 312–316 (1985). 32. Omholt, S. W. Relationships between worker longevity and the intracolonial population dynamics of the honeybee. J. Theor. Biol. 130(3), 275-284, doi: 10.1016/S0022-5193(88)80030-4 (1988). 33. Tofilski, A. Influence of age polyethism on longevity of workers in social insects. Behav. Ecol. Sociobiol. 51(3), 234–237, https://doi.org/10.1007/s00265-001-0429-z (2002). 34. Tofilski, A. Influence of caste polyethism on longevity of workers in social insect colonies. J. Theor. Biol. 238(3), 527–531, https://doi.org/10.1016/j.jtbi.2005.06.008 (2006). 35. Khoury, D. S., Myerscough, M. R. & Barron, A. B. A quantitative model of honeybee colony population dynamics. PLoS ONE 6(4), e18491, https://doi.org/10.1371/journal.pone.0018491 (2011). 36. Khoury, D. S., Barron, A. B. & Myerscough, M. R. Modelling Food and Population Dynamics in Honey Bee Colonies. PLoS ONE 8(5), e59084, https://doi.org/10.1371/journal.pone.0059084 (2013). 37. Torres, D. J., Ricoy, U. M. & Roybal, S. Modelling Honey Bee Populations. PLoS ONE 10(7), e0130966, doi.10.1371/journal.pone.0130966 (2015). 38. Yamada, T., Yamada, K. & Yamada, Y. A clear difference in the impact on honeybee (Apis mellifera) colony between the two vehicles of sugar syrup and pollen paste. J. Biol. Ser. 1(3), 84-107, doi: 10.15413/jbs.2018.0400 (2018a). 39. Yamada, T., Yamada, Y. & Yamada, K. Difference between the impact of the neonicotinoid dinotefuran and organophosphate fenitrothion on a bee colony in a long-term field experiment: An evidence. J. Biol. Ser. 1(3), 108-137, doi: 10.15413/jbs.2018.0401 (2018b). 40. Yamada, T., Yamada, Y. & Yamada, K. Comparison of the influence of a pesticide at an environmentally realistic concentration level in Japan on a honeybee colony between neonicotinoids (dinotefuran, clothianidin) and organophosphates (fenitrothion, malathion). Journal of Biological Series, Article ID: JBS-18-402. doi: 10.15413/jbs.2018.0402. (in press). 41. Yang, E. C., Chang, H. C., Wu, W. Y. & Chen, Y. W. Impaired olfactory associative behavior of
524
honeybee workers due to contamination of imidacloprid in the larval stage. PLoS ONE 7(11), e49472, https://doi.org/10.1371/journal.pone.0049472 (2012).
525
42. Van Ness, H.C. Thermodynamics in the treatment of vapor/liquid equilibrium
523
526 527
(VLE) data. Pure Appl. Chem. https://doi.org/10.1351/pac199567060859 (1995).
67(6),
850-872,
13 528 529 530 531 532 533 534 535 536 537
Acknowledgments
538 539 540 541 542 543 544
Author contributions
545
Additional information
546
Supplementary Information accompanies this paper at
We would like to thank Dr. Ken Hashimoto, Yamada Bee Company, Inc., for his help, encouragement and guidance that has given us throughout this study. We are grateful to Mr. Christopher Ruth and Ms. Kimiko Mimura Ruth for their proofreading of the manuscript. This research was supported in part by Yamada Research Grant [grant number 74]. Here, we would also like to take this opportunity to thank all the individuals who assisted us in our research.
Yasuhiro Yamada (YY), Toshiro Yamada (TY), and Kazuko Yamada (KY) conceived and designed the study. YY proposed and made the mathematical model. YY made programming to estimate the apparent longevity and performed the calculation. TY and YY analyzed the data. TY and YY wrote the paper.
547 548
Competing financial interests: The authors declare no competing financial interests.
549 550
Disclosure statement: No potential conflict of interest was reported by the authors.
551 552
Reprints and permission information is available online at
553 554
Funding: Yamada Research Grant [grant number 74].
555 556 557
ORCID: https://orcid.org/0000-0001-7558-7617
14
Table 1 Estimated apparent longevity (L(t)) of 2013/2014-A in several cases where adult bees or capped brood is miscounted by plus or minus 20% from August 24 to September 27
Date
Calculated L(t) [day]
Elapsed day
Original
CASE 1
CASE 2
CASE 3
CASE 4
13-Aug-13
0
23.02
23.02
23.02
23.02
23.02
24-Aug-13
11
21.26
25.51
17.01
21.26
21.26
1-Sep-13
19
23.77
28.22
19.32
22.54
25.00
5-Sep-13
23
25.14
29.69
20.59
23.21
27.07
15-Sep-13
33
27.73
31.63
23.83
25.01
30.45
21-Sep-13
39
27.08
31.14
21.90
22.65
31.32
27-Sep-13
45
13.68
23.46
10.49
11.04
24.58
4-Oct-13
52
14.39
14.39
14.39
12.48
17.31
13-Oct-13
61
20.18
20.18
20.18
18.71
22.36
27-Oct-13
75
31.34
31.34
31.34
30.33
32.83
15-Nov-13
94
48.37
48.37
48.37
47.27
49.67
1-Dec-13
110
62.34
62.34
62.34
61.58
63.49
5-Jan-14
145
92.61
92.61
92.61
92.33
93.02
7-Feb-14
178
0.00
0.00
0.00
0.00
0.00
28-Feb-14
199
0.00
0.00
0.00
0.00
0.00
Adult bees 0%
Adult bees+20%
Adult bees -20%
Adult bees 0%
Adult bees 0%
Brood 0%
Brood 0%
Brood 0%
Brood +20%
Brood -20%
Change in number ranging from August 24th to September 27th
[Assumptions in Calculation] (1) Interval of Capped Broods = 12 days, (2) Emergence Probability (Eclosion rate) from Pupae = 90%
558
15 559 560
Figure Legends
561
Figure 1. Seasonal change in the numbers of adult bees (a) and capped brood (b) in three long-term field experiments.
562 563 564 565 566 567 568 569
The field experiments to obtain the numbers of adult bees capped brood were conducted as carefully as possible because their measurement accuracy could greatly affects the estimation accuracy of apparent longevity. Their numbers accurately counted with the help of an newly-developed image processing software while being directly corrected on enlarged photographs of all combs and the inside of a hive because their numbers obtained from photographs could be counted again at any time when in doubt. An experiment is conducted just after dawn before foraging bees go out anywhere.
570 571 572 573 574 575 576 577
Figure 2. Seasonal change in apparent longevity at a general view (a) and that at an enlarged view (b). Figure 3. Relative errors of apparent longevity to the original due to measurement errors in four cases.
35000
Number of Adult Bees
30000 25000
2011/2012-A Adult 2012/2013-A Adult 2012/2013-B Adult 2013/2014-A Adult 2013/2014-B Adult
a
20000 15000 10000 5000 0 21-May 10-Jul 29-Aug 18-Oct
Number of Capped Brood
16000 14000 12000
7-Dec
26-Jan 17-Mar 6-May 25-Jun 14-Aug
2011/2012-A Brood 2012/2013-A Brood 2012/2013-B Brood 2013/2014-A Brood 2013/2014-B Brood
3-Oct
b
10000 8000 6000 4000 2000 0 21-May 10-Jul 29-Aug 18-Oct
7-Dec
26-Jan 17-Mar 6-May 25-Jun 14-Aug
3-Oct
Apparent Longevity, L(t)
[days]
200 180
2012/2013-A
160
2012/2013-B
140
a
2011/2012-A
2013/2014-A 2013/2014-B
120 100 80 60 40
General View
20 0 20-Jun
20-Aug
20-Oct
20-Dec
20-Feb
20-Apr
20-Jun
50
b
Apparent Longevity, L(t)
[days]
45
40
35
30
25
20
Enlarged View
15
10
5
0 20-Jun
20-Aug
20-Oct
20-Dec
20-Feb
20-Apr
20-Jun
Relative erros of apparent longevity to the original [%]
90
70
Miscounted Period
Relative errors of L(t) to the original[%] CASE 1 Relative errors of L(t) to the original[%] CASE 2 Relative errors of L(t) to the original[%] CASE 3 Relative errors of L(t) to the original[%] CASE 4
50
30
10
-10
-30
A mathematical model to estimate the seasonal change in apparent longevity of bee colony Yasuhiro Yamada1†, Toshiro Yamada2††*, Kazuko Yamada2††
1
Department of Applied Physics, Graduate School of Engineering, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-8656, Japan
2
Graduate School of Natural Science & Technology, Kanazawa University, Kakuma-machi, Kanazawa 920-1192, Japan
†Present address: Department of Physics, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan †† Present address: 2-10-15, Teraji, Kanazawa, Ishikawa, 921-8178, Japan
*Corresponding author. Email:
[email protected]. Present situation: professor emeritus of Kanazawa University
Supplementary Table 1. The numbers of adult bees and capped brood in 2011/2012, 2012/2013 and 2013/2014 experiments. The detailed experimental methods are described in our previous papers 38-40.
Date 9-Jul-11 16-Jul-11 22-Jul-11 29-Jul-11 6-Aug-11 12-Aug-11 18-Aug-11 26-Aug-11 10-Sep-11 17-Sep-11 24-Sep-11 29-Sep-11 7-Oct-11 21-Oct-11 30-Oct-11 4-Nov-11 18-Nov-11 26-Nov-11 3-Dec-11 17-Dec-11 16-Feb-12 2-Apr-12
Elapsed Days 0 7 13 20 28 34 40 48 63 70 77 82 90 104 113 118 132 140 148 162 223 269
2011/2012 EXP-2011-A Adult Brood 3392 6827 6399 7737 7893 7675 8873 9327 9249 9762 11252 10736 12015 11253 10958 10654 11303 12390 12109 11811 10514 9622
5819 3644 4692 4861 7179 8390 6125 5797 8803 8327 7034 5810 6100 4100 4300 3900 3430 1750 2038 212 0 1873
Date
Elapsed Days
2012/2013 EXP-2012-A Adult Brood
EXP-2012-B Adult Brood
28-Jun-12 0 7136 4746 5832 08-Jul-12 10 9621 5806 8917 15-Jul-12 17 8695 10215 8265 21-Jul-12 23 9647 10254 9665 22-Jul-12 24 10136 10210 9558 27-Jul-12 29 10633 10617 10770 28-Jul-12 30 10391 10858 10901 03-Aug-12 36 12083 10000 11939 04-Aug-12 37 12389 9687 12041 08-Aug-12 41 14065 7154 12978 16-Aug-12 49 13371 6111 12207 25-Aug-12 58 11961 6014 10997 06-Sep-12 70 11165 8783 11582 15-Sep-12 79 11980 5531 11825 21-Sep-12 85 12166 6086 11025 05-Oct-12 99 10715 7615 10510 19-Oct-12 113 11726 7280 10038 25-Nov-12 150 13255 36 12477 13-Dec-12 168 12858 0 13316 01-Feb-13 218 9306 0 9421 01-Mar-13 246 7464 17 8426 09-Mar-13 254 7512 497 8017 17-Mar-13 262 6862 1691 7372 23-Mar-13 268 7312 2431 7416 29-Mar-13 274 7720 4097 8018 07-Apr-13 283 9518 7326 9833 13-Apr-13 289 12523 10166 12594 19-Apr-13 295 15677 11324 16221 26-Apr-13 302 20574 9725 20412 03-May-13 309 23935 5808 24100 10-May-13 316 23629 4551 27670 17-May-13 323 16210 6036 30325 26-May-13 332 9981 9778 17904 02-Jun-13 339 9289 7396 26814 14-Jun-13 351 6135 4875 16284 21-Jun-13 358 5590 3089 11501 28-Jun-13 365 4210 1522 10454 08-Jul-13 375 1962 978 11162 26-Jul-13 393 0 0 4817 Note: EXP-2012-A became extinct on July 26 in 2013.
5094 4710 11143 11301 10967 10329 10581 10025 10269 8472 5977 6684 8126 7135 9202 5679 6628 2937 305 0 0 660 2419 3624 5165 8414 11211 12975 14287 14521 13380 12864 10157 3454 1600 7725 10165 7534 4498
Date
Elapsed day
2013/2014 EXP-2013-A Adult Brood
EXP-2013-B Adult Brood
13-Aug-13 0 7579 4167 5953 24-Aug-13 11 7000 3363 6041 1-Sep-13 19 7324 1825 5565 5-Sep-13 23 7487 2261 5549 15-Sep-13 33 6415 5462 4027 21-Sep-13 39 6697 4683 3829 27-Sep-13 45 5711 2538 3889 4-Oct-13 52 5048 649 3824 13-Oct-13 61 4210 151 4322 27-Oct-13 75 3152 314 3371 15-Nov-13 94 2714 50 3181 1-Dec-13 110 2189 3 2904 5-Jan-14 145 1100 0 2218 7-Feb-14 178 0 314 1799 28-Feb-14 199 0 0 1594 Note: EXP-2013-A became extinct on February 7 in 2014.
4683 2146 558 578 2665 3198 2882 2142 113 0 0 0 0 0 0
Supplementary Table 2. Estimated apparent longevity, L(t), i n 2011/2012, 2012/2013and 2013/2014 experiments. Date 9-Jul-11 16-Jul-11 22-Jul-11 29-Jul-11 6-Aug-11 12-Aug-11 18-Aug-11 26-Aug-11 10-Sep-11 17-Sep-11 24-Sep-11 29-Sep-11 7-Oct-11 21-Oct-11 30-Oct-11 4-Nov-11 18-Nov-11 26-Nov-11 3-Dec-11 17-Dec-11 16-Feb-12 2-Apr-12
Elapsed Days
0 7 13 20 28 34 40 48 63 70 77 82 90 104 113 118 132 140 147 161 222 268
2011/2012 EXP-2011-A L(t) [days] 7.38 14.85 14.59 21.27 21.89 22.26 17.63 15.17 18.82 15.75 16.44 16.37 20.65 23.79 26.73 27.44 33.75 39.66 43.64 50.76 107.14 133.10
Date 28-Jun-12 08-Jul-12 15-Jul-12 21-Jul-12 22-Jul-12 27-Jul-12 28-Jul-12 03-Aug-12 04-Aug-12 08-Aug-12 16-Aug-12 25-Aug-12 06-Sep-12 15-Sep-12 21-Sep-12 05-Oct-12 19-Oct-12 25-Nov-12 13-Dec-12 01-Feb-13 01-Mar-13 09-Mar-13 17-Mar-13 23-Mar-13 29-Mar-13 07-Apr-13 13-Apr-13 19-Apr-13 26-Apr-13 03-May-13 10-May-13 17-May-13 26-May-13 02-Jun-13 14-Jun-13 21-Jun-13 28-Jun-13 08-Jul-13 26-Jul-13
Elapsed Days
0 10 17 23 24 29 30 36 37 41 49 58 70 79 85 99 113 150 168 218 246 254 262 268 274 283 289 295 302 309 316 323 332 339 351 358 365 375 393
2012/2013 EXP-2012-A L(t) [days] 19.03 25.65 21.94 21.53 21.12 14.51 13.79 15.41 14.88 16.23 18.39 19.88 23.00 21.31 21.00 21.14 19.59 47.53 64.81 108.91 133.75 141.82 148.28 154.08 159.04 166.80 172.59 177.64 181.88 185.32 34.93 26.01 22.57 16.76 11.18 14.36 12.24 13.31 0.00
EXP-2012-B L(t) [days] 14.49 22.15 21.07 23.06 20.76 13.24 13.15 13.52 13.70 14.86 15.60 18.37 21.68 20.47 18.04 16.13 19.03 37.74 51.75 93.76 118.44 124.67 128.57 131.34 133.66 136.13 138.63 141.88 38.93 30.47 28.37 28.76 17.50 27.14 26.34 27.64 31.03 21.93 10.72
Note: EXP-2012-A became extinct on July 26 in 2013.
Date 13-Aug-13 24-Aug-13 1-Sep-13 5-Sep-13 15-Sep-13 21-Sep-13 27-Sep-13 4-Oct-13 13-Oct-13 27-Oct-13 15-Nov-13 1-Dec-13 5-Jan-14 7-Feb-14 28-Feb-14
Elapsed day
0 11 19 23 33 39 45 52 61 75 94 110 145 178 199
2013/2014 EXP-2013-A L(t) [days] 23.02 21.26 23.77 25.14 27.73 27.08 13.68 14.39 20.18 31.34 48.37 62.34 92.61 0.00
EXP-2013-B L(t) [days] 16.09 16.32 19.22 21.56 26.86 29.13 29.88 16.20 20.63 29.89 48.19 63.18 95.68 127.15 147.03
Note: EXP-2013-A became extinct on February 7 in 2014.
Supplementary Figure 1. Comparison of apparent longevity between previous works (Bühlmann 31, Fukuda & Sekiguchi2) and this work. The mean longevity reported by Fukuda and Sekiguchi (1966) was plotted as apparent longevity as it was. The apparent longevities of Bühlmann (1985) and this work was calculated from the numbers of adult bees and capped (sealed) brood by using our mathematical model on the assumption of the eclosion rate = 0.9. The apparent longevities at the colony extinction (L(t) = 0) were removed when being plotted, which were on July 26 in 2012/2013-A and on February 28 in 2013/2014-A. All data of apparent longevity were plotted from the end of June. From the figure, we can find that there are following clear distinctions between the previous apparent longevities reported by Fukuda and Sekiguchi (1966) and Bühlmann (1985) and that in this work in which the apparent longevities show very similar season-changes to one another, increasing from early autumn, reaching a maximum at the end of overwintering and falling approximately plumb down after overwintering: (1) The apparent longevity of Fukuda and Sekiguchi (1966) rapidly increases from the end of August and already reaches a maximum (about 140 days) at the beginning of October. Thereafter it begins to decrease gradually and it decreases again in nearly the same decrement from the beginning of April after there is a lack of data from the end of October to the end of March. (2) The apparent longevity of Bühlmann (1985) also rapidly increases from the end of August and reaches a maximum (about 245 days) at the end of February. After it reaches a maximum, it begins to decrease with time from the beginning of March to early May. Thereafter it reaches a certain value (10 – 20 days). That is, after it reaches a maximum, it does not show a sudden drop like our results but do a comparatively gradual decrease during and just after overwintering.
Apparent longevity, L(t) [day]
300
Bühlmann 1979
Bühlmann 1980
Bühlmann 1981
Fukuda A-Mean
Fukuda B-Mean
This work 2011/2012-A
This work 2012/2013-A
This work 2012/2013-B
This work 2013/2014-A
This work 2013/2014-B
250
200
150
100
50
0
15-May
15-Jun
16-Jul
16-Aug
16-Sep
17-Oct
17-Nov
18-Dec
18-Jan
18-Feb
21-Mar
21-Apr
22-May
22-Jun
23-Jul
23-Aug
