Proc. Natl. Acad. Sci. USA Vol. 88, pp. 36-38, January 1991 Ecology Cooperative foraging, productivity, and the central limit theorem (group size/sampling theory/progressive provisioning/sociality/Polstes) JOHN W. WENZEL AND JOHN PICKERING Department of Entomology, University of Georgia, Athens, GA 30602 Communicated by Charles D. Michener, September 28, 1990 (received for review May 24, 1990)

ABSTRACT The central limit theorem is applied to group In accord with the predictions of CLT, both empirical and foraging to show an automatic and universal benefit to group theoretical literature on flocking birds find that flocking can living. This may explain the paradoxical inverse correlation decrease variation in foraging success: ". . . the primary between group size and per capita brood production in prim- advantage of flocking for low-ranking individuals was the itively eusocial insects and why only one of the five major reduced chance of obtaining no food at all" (10). "If indi- lineages of social insects contains species that revert to solitary viduals prefer to avoid variation in foraging time. . . flocking habit. should often be favored" (11). Of course, the reward for forming groups and the exact form of the curves represented A fundamental paradox among group-living animals is that as in Fig. 1 will vary according to the details of the species' groups contain more individuals, they generally appear to ecology and society, but we have illustrated the conventional have lower per capita productivity (1). If animals produce and well-understood situation where normality holds. In offspring more efficiently when alone or in small groups, then reality, an animal that has already collected a surplus is less why should they form larger groups? Proposed explanations likely to continue hunting, while the lower limits are trun- for group living include observer bias (1), kin selection (2), cated at the value of zero success, likely less than the full variation in physical fitness (3, 4), protection from mutual hypothetical range illustrated here. Other factors, such as enemies (5), and recovery from catastrophe (6), while others trading information about successful hunting sites or non- believe that "[t]here is no automatic or universal benefit from independence ofefforts, may increase the advantage ofliving group living" (7). The predictability with which groups attain in groups. the mean per capita productivity is an important and over- Production schedules in social groups may be based on the looked issue. Small samples (small groups) are more likely to variance rather than the mean. If a colony has enough brood show large deviations from the expected mean and are less to consume occasional surplus, then the production schedule predictable than are large samples (large groups), according for the number of brood per female will approximate the to the central limit theorem (CLT) (8). Here we apply CLT to upper curve of Fig. 1, a strategy called "no wasted food." social grouping among foragers sampling from the environ- Alternatively, if the cost of replacing offspring is more than ment. that of replacing wasted food, animals may follow the other strategy by tracking points in the lower halfofthe figure, "no Except for halictid bees and stingless bees, social insects wasted brood." Although the 95% interval chosen for illus- provision brood progressively like most birds and mammals, tration here is extreme, recent models (12) show that "bet- supplying offspring with food repeatedly throughout the hedging" by substantial overproduction of offspring is ap- developmental period. In times of plenty there may be too propriate when the cost of abortion is low and environmental few larvae to eat all available food, while there may be too variation is high. Higher per capita number of brood main- many larvae in times of dearth. Unpredictable supply is most tained by smaller groups (1) may be an optimistic response to problematic if food cannot be stored, as in predatory wasps. their decreased predictability of resources. The cost of am- If each forager's success is considered as an independent bition will include energetic input toward eggs and larvae that datum sampled from a normal distribution of values for daily will be aborted before maturity (12) or longer larval devel- food intake (with environmental and spatial variation pro- opment time due to periods of dearth. viding random deviations from the expected rate of prey If groups do respond optimistically to the expected varia- capture), then the expected daily intake for groups of differ- tion, then one easily measured prediction ofCLT is that small ent sizes may be regarded as a Student's t distribution. Fig. groups should exhibit both the highest productivity and the 1 demonstrates how groups enjoy more dependable intake on highest abortion rate. A study ofPolistes annularis in Kansas a given day. Foragers can reduce the range of deviations (13) shows the production schedule of this population exem- away from expectation by forming cooperative groups that plified the relationship Michener (1) has discussed; during the pool resources. The 95% confidence interval for a lone foundress period (prior to emergence of workers), there was forager's effort may include values up to 12 SD units away a negative correlation between group size and number ofcells from expectation. If two foragers work separately and then per female (N) (by linear regression r = -0.82, P < 0.01, N pool their resources (or if one searches twice as many places = 28, pooled over years). In 1985, abortion rates were low while her partner stays home), they will depart from expec- with a total of 15 events on 13 nests during the foundress tation by <4 units 95% of the time. By the more modest 80% period, in no clear pattern. But in 1986, rates were three times interval, a pair offoragers displays about half the variation of as high, with 53 events on 15 nests. Ofthese colonies, one pair loners. If avoidance of negative deviations is important, then of females failed entirely after 34 days. An additional nest of this reduction in variance constitutes the automatic and one female failed after 7 days (not included in the analysis universal benefit to group living that Alexander (7) repudi- because of the short period of observation). Kolmogorov- ated. Smirnov analysis shows abortion was associated with the smaller groups (Fig. 2). As predicted by CLT and optimistic bet and not other smaller The publication costs of this article were defrayed in part by page charge hedging (12), by proposals (1-6), payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact. Abbreviation: CLT, central limit theorem. 36 Downloaded by guest on September 30, 2021 Ecology: Wenzel and Pickering Proc. Natl. Acad. Sci. USA 88 (1991) 37 with respect to group size in mature nests measured through- +12 - T 1~~~~95% limits | | ~~~O80% limits out the same season (June to August). From the total data set of 540 cells on six mature nests (with adult cohorts of 4, 8, 8, CD 14, 22, and 69 females), the largest nest is the only one where 2 -0 +6- development times appear narrowly and normally distributed \No Wasted Food + 0 about the mean value (27.2 0.2 days). All others show +4 - A- 0 Of greater variation and bimodality due in part to egg eating and ( 0 ~~~~~__ . . c LL +2- .0 0 extended larval development. Kolmogorov-Smimov analy- 0 >0 0 sis shows the difference is significant and profound; on the 0 0 x _ ° 0 0 0 0 nest of 69 females, 90% of the larvae pupated in 32 days or c > -2 0 0 c less, whereas only half the larvae on a nest of 14 females did 0 so (Fig. 3; see ref. 14 for methods). Large colonies (6 or more co)'-0 -4 No Wasted Brood foundresses) of the Asian subtropical wasp Ropalidia fasci- -6 ata show a similar, significant reduction in larval develop- cn ment time (15). CLT appears to predict the relative magnitude of abortion -12 frequency and brood development times for groups of dif- ferent size in both temperate and of U tropical species primi- . . . tively eusocial wasps. Thus, while larger groups have fewer 1 2 3 4 5 6 7 8 20 60 brood cells per capita, they invest less in brood lost by Group Size abortion and likely suffer less jeopardy of catastrophic nest loss or adult mortality (16) prior to offspring maturity. This FIG. 1. Confidence limits (80% and 95%) for Student's t distri- contributes to what has been called "homeostasis" in large bution (9) for sample means based on samples ofdifferent size, where groups (17, 18). Their advantage over small groups will a lone female represents n = 1, a pair is n = 2, etc. increase further if they can recruit foragers to good sites or if they build up large stockpiles of food, as do ants, honey groups paid a higher price in a year with relatively high brood bees, and stingless bees. Of course, the magnitude of the abortion, despite their higher productivity in general (mea- effects and the size ofgroups that are considered "small" or sured as number of cells per capita in the foundress period). "large" will vary with each population's ecological milieu. Brood development time provides a separate test of the Unlike progressive provisioners, halictid bees provision importance of CLT. A large group should provide more cells with a mass of food, usually collected in one day. The stable food intake, producing a more narrow and uniform cell is then sealed and the larvae complete development. The distribution of larval development time than that found in a development and success of sealed larvae is not threatened small group. Minimum development time of larvae is likely by the female's inability to find adequate pollen and nectar for fixed physiologically, but long times may be due either to subsequent cells. Thus, halictid species rely less on temporal retarded development or to undetected replacement of continuity than do progressive provisioners. Neither do aborted eggs (or small larvae) by new eggs. Polistes cana- halictids store large quantities of food, like the mass- densis in Panama demonstrates a bias in development times provisioning meliponines and other social bees, which would increase the inherent stability ofa group. Accordingly, and in 2 1 sharp contrast to the strict reliance on social groups among - 1. 100 1 -w -0 .-O 0- 2 0110 000 c - 20 - 0 90 4 AO* / 0 - MGA a- 80 ~0 0 10 0U) 0r 0c) 70 / 9- co 0 - 0 0 0 60 - / 40 50 60 .00 a)0 0 50 I E 40- - 23.3%1 z (0 40 - -- - Expected I 30- 30 Observed c MG3 20 Numerals Indicate 20 no. nests per point c) 10 10 L 0%A I w.11 2 1 2 3 4 5 6 7 8 9 . 111,1 Number of Females 20 so 40 5g0 60 Days to Pupation FIG. 2. Abortion events in P. annularis by colony size prior to emergence of workers, based on location of larvae recorded at least FIG. 3. Days elapsed from oviposition to pupation on P. cana- every 4 days. Kolmogorov-Smirnov analysis shows a significant densis nest MGA (14 females) and MG3 (69 females). Other nests excess of abortion in small groups when tested against the assump- with 4, 8, 8, and 22 females showed a broad range similar to that tion that abortion is equally/ likely on nests of all sizes (D,,m = 23%, illustrated for the nest of 14 females, but with fewer observations per versus the critical value of 22% at P = 0.05, N = 15). Only colonies nest. Kolmogorov-Smirnov analysis shows the illustrated distribu- that were observed regularly for at least 30 days prior to worker tions are significantly different (Dmax = 38%, versus the critical value emergence are illustrated. of 20% at P = 0.001, N1 = 239, N2 = 149). Downloaded by guest on September 30, 2021 38 Ecology: Wenzel and Pickering Proc. Natl. Acad Sci. USA 88 (1991) the termites [>100 genera, 2000 species (19)], paper wasps 9. Rohlf, F. J. & Sokal, R. R. (1981) Statistical Tables (Freeman, [Vespinae and Polistinae, >30 genera, 800 species (ref. 20, San Francisco). but cf. ref. 21)], ants [>200 genera, 10,000 species (22)], and 10. Baker, M. C., Belcher, C. S., Deutsch, L. C., Sherman, G. L., Thompson, D. B. (1981) Anim. Behav. 29, 137-142. social apid bees [Apini, Bombini, Meliponini, >18 genera, 11. Caraco, T. (1981) Ecology 62, 527-531. 700 species (Charles D. Michener, personal communica- 12. Kozlowski, J. & Stearns, S. C. (1989) Evolution 43,1369-1377. tion)], many species in the 80 genera of the tribe Halictini 13. Wenzel, J. W. (1989) J. Insect Behav. 2, 679-699. display flexibility between social and solitary habit (23-29) 14. Pickering, J. (1980) Dissertation (Harvard Univ., Cambridge, depending on environmental factors. MA). The CLT appears to be broadly applicable to cooperative 15. Kojima, J. (1989) Insectes Soc. 36, 197-218. in social animals, and its application to social insects 16. Queller, D. C. (1989) Proc. Natl. Acad. Sci. USA 86, 3224- foraging 3226. may provide an explanation for Michener's paradox of pro- 17. Emerson, A. E. (1956) Ecology 37, 248-258. ductivity and offers clues about the cause of exceptional 18. Jeanne, R. L. (1991) in The Biology ofSocial Wasps, eds. Ross, reversal to solitary habit in halictids. K. G. and Matthews, R. W. (Cornell Univ. Press, Ithaca, NY), pp. 191-231. We thank J. M. Carpenter, W. W. Hargrove, R. L. Jeanne, C. D. 19. Wilson, E. 0. (1971) The Insect Societies (Harvard Univ. Michener, S. Pearson, J. Shykoff, and W. T. Wcislo, as well as two Press, Cambridge, MA). anonymous reviewers for comments. 20. Carpenter, J. M. (1982) Syst. Entomol. 7, 11-38. 21. Wenzel, J. W. (1987) J. Kans. Entomol. Soc. 60, 549-556. 1. Michener, C. D. (1964) Insectes Soc. 11, 317-341. 22. H1lldobler, B. & Wilson, E. 0. (1990) Ants (Harvard Univ. 2. Hamilton, W. D. (1964) J. Theor. Biol. 7, 1-52. Press, Cambridge, MA). 3. West Eberhard, M. J. (1978) J. Kans. Entomol. Soc. 51, 23. 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