How important are partial preferences?

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computer simulation of group stability. Both have optima at a group size of 5. Note how the d W/dN changes on either side of the optimum. The stable group size for each function is indicated by the stars, the open star for the dashed line f(N), and the closed star for the solid line f(N).

each of five different f ( N ) . The f ( N ) were designed such that while Nopt remained constant at five the d W / d N at N Nopt. TWO extreme examples of the f ( N ) used are presented in Fig. 1. The simulations confirm that N~ can never be smaller than Nopt, a result derived analytically by Pulliam & Caraco (1984). Our simulations indicate, however, that Sibly's conclusion that Nopt < N~ is incorrect since in some cases N~= Nopt (Fig. 1). This occurs when the fitness of joining a group of optimal size is less than that of remaining alone (see Fig. 1). The generality of both Clark & Mangel's (1984) and Pulliam & Caraco's (1984) conclusion that individuals will more often be in groups larger than Nopt because Nopt is unstable resides in the generality o f f ( N ) for which Nopt < Ns. At present, there is no reason to believe that this type o f f ( N ) is more common than any other. In fact, when competition is important, we would expect f ( N ) to be of the shape that stabilizes Nopt. This is because competition could add costs to joining supra-optimal group sizes. These costs could be the result of harassment, aggression or even exclusion from the group. Spotted hyaenas for instance, establish dominance hierarchies that exclude lower-ranking individuals from having access to smaller carcasses (Tilson & Hamilton 1984). The costs of joining groups of supra-optimal size for lower-ranking individuals may be greater than the costs of joining groups of sub-optimal size. When lions hunt Thompson's gazelle in the Serengeti, Nopt (two lions) provides the maximum attainable harvesting rate (Caraco & Wolf 1975). Groups larger than the optimum could not provide the minimum daily food requirement of adult lions (Caraco & Wolf 1975). It is clear that lions joining a

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group in excess of Nopt would reduce their fitness considerably such that Nopt would be stable. Contrary to Sibly's conclusions therefore, one should not assume that Ns and Nobs are greater than Nopt before investigating the shape of the fitness functions. We wish to acknowledge the members of the McGill Graduate Seminar on Spatial Dispersion. Graham Bell, Don Kramer, Louis Lefebvre, Rob Peters, Marc Hauser and an anonymous referee provided constructive criticisms of an earlier version of the manuscript. We thank Sylvie Lanct6t and Roger Giraldeau for providing the computer. Financial support came from F,C,A.C. and N.S.E.R.C. scholarships to L.-A.G. and D.G. respectively. Luc-ALAIN GIRALDEAU* DARREN GILLIS Department of Biology, McGill University, 1205, Avenue docteur Penfield, Montreal, Canada H3A 1B1. * Present address: Department of Psychology, University of Toronto, Toronto, Ontario, Canada M5S 1AI. References

Caraco, T. & Wolf, L. L. 1975. Ecological determinants of group sizes of foraging lions. Am. Nat., 109, 343 352. Clark, C. W. & Mangel, M. 1984. Foraging and flocking strategies: information in an uncertain environment. Am. Nat., 123, 626-641. Pulliam, H. R. & Caraco, T. 1984. Living in groups: is there an optimal group size? In: Behavioural Ecology: An EvolutionaryApproach, 2nd edn (Ed. by J. R. Krebs & N. B. Davies), pp. 122-147. Sunderland, Mass.: Sinauer. Sibly, R. M. 1983. Optimal group size is unstable. Anim. Behav., 31, 947-948. Tilson, R. L. & Hamilton, W. J., lII. 1984. Social dominance and feeding patterns of spotted hyaena. Anim. Behav., 32, 715-724. (Received 25 July 1984; revised 17 September 1984; MS number: sc-214) H o w Important are Partial Preferences?

In the recent international conference on foraging behaviour, two speakers (G. H. Pyke & R. Gray) concluded that the existence of partial preferences is a major empirical failing of optimal diet theory (Pulliam 1974; Charnov 1976). I disagree, because under any reasonable statistical model of preference, it is practically impossible to observe anything but partial preferences, even if optimal diet theory is correct. Moreover, I argue that the

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Figure 1. A typical prediction of optimal diet theory: small mealworms should be ignored at encounter rates above 2*. (A) The supposed prediction of absolute preferences. At all encounter rates below 2, small worms are always taken, and at all encounter rates above 2* small worms are never taken. (B) The frequently observed pattern of a sigmoid dose-response relationship. This is the correct prediction of optimal diet theory when the observed threshold has some variance.

literature's pre-occupation with partial preferences (per se) has been counter-productive (see Krebs & McCleery 1984 for a list of proposed explanations). A typical prediction of optimal diet theory might be that 'above encounter rate 2., only large (more profitable) mealworms should be taken, and below 2., both small (less profitable) and large mealworms should be taken' (where 2* is the threshold encounter rate). 'Absolute' preferences would occur if at every encounter rate below 2,, all small mealworms encountered were eaten, and at every encounter rate above 2* none of the small mealworms encountered were taken (see Fig. 1A). Instead of this 'step function' experimentalists usually observe a smooth 'S-shaped' relationship (Fig. 1B) between encounter rate and the proportions of small mealworms taken. This is what is meant by 'partial preferences'. Suppose that 2. is the predicted time spent in a patch, instead of the predicted threshold. An experiment could be designed in which many patch-residence times (in identical patches) would be observed. The observed frequency distribution of patch-residence times (containing information about mean and variance) could then be compared with the predicted time. A statistical test of the

hypothesis that the mean patch-residence time is equal to the predicted patch-residence time could be performed, and statements about the significance of such a hypothesis could be made. Obviously, it would be unreasonable to expect that no variance would be observed around the predicted patch-residence time. The central problem is that a threshold (2,) cannot be directly observed. If the forager ignores a small mealworm at 2' then all that can be inferred is that 2. is less than 2'. According to the absolute preferences view, 2' must always be above 2., but if 2, is a random variable it is possible (in fact expected) that 2' will sometimes be above and sometimes be below the threshold 2,. Absolute preferences can only occur if the threshold (2*) has no variance. Observed thresholds will have some variance for a number of reasons: incomplete experimental control, within- and between-subject variability, and possible intrinsic variability. Suppose that an experiment was designed to test diet theory by controlling the encounter rate. Four levels of encounter rate are chosen 24 > 23 > 22 > 21. 2, (the predicted threshold) is between 22 and 23 (see Fig. 1). According to the 'absolute preferences' view of diet theory, small mealworms must be taken on 100~ of the encounters with small mealworms in treatments 21 and 22, and never taken in treatments 23 and 24. Suppose that the underlying distribution of the threshold is normal (with mean equal to 2* and unknown variance), and that a different realization of the threshold is 'drawn' from one encounter to the next (in the same way that a forager's patch residence time will vary from patch to patch even if the patches are identical). Because of the underlying stochasticity, 23 is usually above the threshold (more than 50~oof the time), but sometimes below it. Specifically, it will be above the threshold (predicting that small mealworms should be ignored) on ~[(23-2.)/cr] of the encounters. In an experiment like this, one can only observe the cumulative distribution function of the randomly-distributed threshold. The cumulative distribution function of the normal (q~) is sigmoid, and this is why a sigmoid dos~response relationship is not surprising, A number of statistical techniques can be used to estimate the mean and variance of a threshold. The most widely known technique is probit analysis (Finney 1962), which uses transformed response scores and weighted linear regression. This technique allows tests of the hypothesis: is the observed threshold significantly different from the predicted threshold? Finney discusses the problems of, and appropriate experimental designs for the study of these dose-response relationships. Although the conventional use of probit analysis is to estimate

Short Communications the toxicity of drugs and poisons, Finney points out that the technique was originally used for behavioural preference testing. I anticipate the following two objections. (1) It may be argued that the analysis I propose presupposes the existence of a threshold, which is one of the predictions of diet theory. This is true enough. However, this is the way statistical tests always work. Significance is a measure of error given that the underlying hypothesis is correct. The way around this difficulty is to test alternative hypotheses. A clear statement of the statistical implications of threshold predictions should help rather than hinder this process. One problem is that there is little observable difference between a graded sigmoid response to dose, and a randomly-distributed threshold. However, diet theory provides an a priori way to predict one of the dose-response curve's parameters (the mean; the variance must be estimated from the data), while the graded response view requires the estimation of two parameters from the data. (2) It may be argued that the presence of variance is an apology for optimal diet theory, and worse, that it hides many of the interesting phenomena which a science of foraging behaviour ought to address. Admitting that observed partial preferences are the result of variance enhances, rather than diminishes, attempts to understand them by providing a focus for further study. Five out of six of Krebs & McCleery's (1984) explanations of partial preferences can be viewed as components of variation (the sixth, simultaneous encounters, can be ruled out in any well-controlled experiment). The empirical support for diet theory would, of course, be most impressive if the variance about the threshold were small, but the variance must be measured before one can conclude that it is inordinately large. It is ironic that the literature's obsession with partial preference has prevented students of foraging from measuring how partial the preferences are. The simple existence of partial preferences is no more damning for optimal diet theory than the existence of variance is for the hypothesis that the sex ratio of mammals is one to one. Probit analysis can be used to test whether the data are systematically different from predictions. Optimal diet theory has not been allowed the luxury of such tests because of the pre-occupation with partial preferences. Using these techniques the phenomena which must be accounted for by the theoretical successors to optimal diet theory can be accurately and succinctly described. I thank A. A. E. J. Schluter, R. C. Ydenberg and L. M. Dill for their advice. This was written while I was a NATO postdoctoral fellow at the Depart-

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ment of Zoology, University of British Columbia. I am grateful to NATO and UBC for their support. D. W. STEPnENS Department of Biology, University o f Utah, Salt Lake City, Utah 84112, U.S.A.

References Charnov, E. L. 1976. Optimal foraging: the attack strategy ofa mantid. Am. Nat., ll0, 141 151. Finney, D. J. 1962. Probit Analysis. New York: Cambridge University Press. Krebs, J. R. & McCleery, R. H. 1984. Optimization in behavioural ecology, In: Behavioural Ecology: An Evolutionary Approach, 2nd edn (Ed. by J. R. Krebs & N. B. Davies), pp. 91--121.Sunderland, Massachusetts: Sinauer. Pulliam, H. R. 1974. On the theory of optimal diets. Am. Nat., 108, 59 74. (Received 12 July 1984; revised 26 September 1984," MS. number: AS-289)

Tactical Deception of Familiar Individuals in Baboons (Papio ursinus) In this paper we detail several samples of baboon behaviour, the significance of which is summarized in our title. 'Deception' is accepted as a well-established phenomenon in animals and plants (Wickler 1968; Hailman t977; Krebs and Dawkins 1984). By its very nature, it involves the use of an activity or body form which occurs also as an 'honest' counterpart, generally of higher frequency. However the relationships between honest and deceitful versions, and indeed between deceiver and victim, vary widely. The term 'tactical' is used here to emphasize a contrast between the very short-term changes occurring in a single animal between deceitful and honest versions of behaviour, and those involved in common types of deception described in the literature. Dawkins & Krebs (1978) note that 'interspecific deceit is so well known as to pass almost without comment' but that 'ethologists have failed to find many unequivocal cases of successful intraspecific deceit'. Mimicry and camouflage account for much of the former, where the deceptive and honest versions of anatomy and behaviour typically occur in different species and therefore cannot, of course, be flexibly alternated by one individual. Even in the less well documented intraspecific case, examples of honest and deceptive versions of behaviour tend to be the alternative strategies adopted by different