- Email: [email protected]
Butterfly contests: contradictory but not paradoxical
ANIMAL BEHAVIOUR, 2000, 59, F1–F3 Article No. anbe.1999.1305, available online at http://www.idealibrary.com on
FORUM Butterfly contests: contradictory but not paradoxical SCOTT A. FIELD* & IAN C. W. HARDY†
*Department of Entomology, Hebrew University of Jerusalem †Department of Animal Biology and Ecology, University of Granada, Spain (Received 20 January 1999; initial acceptance 30 July 1999; final acceptance 17 September 1999; MS. number: AF-4)
‘common sense’ and the majority of empirical studies, small H. sara males had an advantage; it appeared that ‘intruders quickly recognized small residents as unfavourable adversaries’ (page 537) and abandoned their attempt to acquire the territory. The authors suggest that ‘the success of small males may represent a paradoxical strategy (sensu Maynard Smith & Parker 1976), which promotes victory despite lower resource holding power’ (page 537). Here we point out that this suggestion runs contrary to the established definition of a paradoxical strategy for this game, and that the evolution of a paradoxical strategy in H. sara is in any case quite unlikely on theoretical grounds. We begin by briefly recapping the theoretical development of the concept of paradoxical strategies, and after reconsidering Hernandez & Benson’s (1998) conclusions, highlight a seeming ambiguity in the definition of paradoxical ESSs. We conclude with a comment on the prospects for actually demonstrating their existence in nature. The concept of a paradoxical strategy ‘evolved’ out of early explorations of two models of contest behaviour, the ‘hawk–dove’ game (Maynard Smith & Price 1973) and the ‘war of attrition’ (Maynard Smith 1974). Various authors later analysed ‘asymmetric’ contests in which one of the contestants was a better fighter or held a positional advantage (higher resource-holding power, RHP), and/or had more to gain than the other (higher resource value, RV or V) (e.g. Maynard Smith & Parker 1976; Hammerstein 1981; Parker & Rubenstein 1981; Hammerstein & Parker 1982). As common sense would suggest, in these games where the asymmetries directly affected the chances of winning or the payoff, ESSs were found in which the individual in the ‘favoured role’ (i.e. the one with a higher RHP or RV, or higher ratio of resource value to cost accrual, V/K), won the contest. Unexpectedly, however, they also discovered that under certain conditions (admittedly evolutionarily restrictive) there existed ‘paradoxical’ ESSs in which the contestant
aradoxes conflict with preconceived notions of what is reasonable or possible. Hence they are usually identified when empirical observation conflicts with established theory, although in some instances it is theoreticians who uncover the possibility of a paradox first. Such was the case in behavioural ecology with the so-called ‘paradoxical ESS’ (evolutionarily stable strategy), an evolved behavioural convention dictating that an individual of lower fighting ability, and/or with less to gain from winning a fight, obtains access to a disputed resource at the expense of a more able or motivated opponent (Maynard Smith & Parker 1976). In hindsight, the behaviour of a social spider reported by Burgess (1976), in which resident males routinely flee from a hiding place when disturbed by an intruder, seemed to provide an example (Maynard Smith 1982, page 96). However, important variables require empirical estimation before this interpretation can be fully accepted (Mesterton-Gibbons & Adams 1998). Further empirical examples of paradoxical outcomes have remained exceedingly sparse. In a recent issue of Animal Behaviour, Hernandez & Benson (1998) report what they speculate is a paradoxical strategy for resolving territorial contests between males in the butterfly Heliconius sara (Nymphalidae). Males have two alternative mating tactics: perching next to a pupa from which a female is about to emerge (here larger males may have an advantage) and defence of a mating territory. The latter involves the resident male engaging intruders in a short circling contest flight, with the loser being escorted out of the territory. In accord with most animal contests, Hernandez & Benson (1998) found that the resident male won the majority (in this case all) of the territory encounters. However, in contrast to both
P
Correspondence: I. C. W. Hardy, Ecology Centre, The Science Complex, University of Sunderland, Sunderland SR1 3SD, U.K. (email: [email protected]). S. A. Field is at the Department of Applied and Molecular Ecology, University of Adelaide, Waite Campus, Glen Osmond 5064, South Australia, Australia. 0003–3472/00/010F01+03 $35.00/0
F1
2000 The Association for the Study of Animal Behaviour
F2
ANIMAL BEHAVIOUR, 59, 1
in the disfavoured role won. Subsequent work also showed paradoxical ESSs to exist in the ‘sequential assessment game’ (Leimar & Enquist 1984; Enquist & Leimar 1987) and the ‘iterated hawk–dove game’ (MestertonGibbons 1992). We now turn to Hernandez & Benson’s (1998) argument, put forward in detail on page 538. They infer that the butterflies are engaging in a display in which victory goes to the more persistent, and therefore phrase their ideas in terms of the war of attrition (WOA). In this game, the common-sense strategy ‘for the individual with the lower resource value/rate of cost accrual (V/K) would be to retreat’. They argue that ‘size should not greatly influence the fitness rewards of territorial defence’, meaning that V should be approximately equal for large and small males. However, the cost K should be lower for large males because ‘holding territories may be physiologically less costly for larger individuals’. But a counterbalance is provided by reproductive opportunities lost during a contest, which should be greater for large males, making K higher (and hence V/K lower) for them even if, as one might expect, ‘smaller adversaries run a greater risk of serious injury in escalated combats’. In summary, the authors imply that an ESS is operating in which smaller males win because their ratio of resource value to rate of cost accrual is generally higher. The flaw in the suggestion that this result is paradoxical should now be clear: the statement above is the very definition of the common-sense strategy for this game. For the strategy to be paradoxical, victorious males would need to be in the ‘disfavoured’ role. Thus, leaving aside the fact that the parameters critical to verifying Hernandez & Benson’s (1998) argument are yet to be estimated, the suggestion that these territorial contests could exemplify a paradoxical strategy is clearly untenable. Furthermore, it is dubious whether a paradoxical strategy for contest resolution in H. sara is even theoretically possible. Hammerstein & Parker (1982) have shown that for the continuous WOA, paradoxical solutions do not exist. Only if individuals are forced at some point to commit themselves in advance to a reasonably costly discrete period of display (a ‘round’) during which they cannot retreat, and do so at a stage when they have not yet accrued much cost, can paradoxical ESSs arise (Hammerstein & Parker 1982). This situation of a discrete WOA is sometimes quite probable in nature, for example, where giving up in the middle of a round may expose a player to some uncontrollable risk of injury. However, from the description of H. sara contests given by Hernandez & Benson (1998, page 534), it does not appear to be the case here. The error we have highlighted seems to have arisen from two sources: confusion between the everyday and technical meaning of ‘paradoxical’, and a misunderstanding of the relationship between paradoxical strategies and ‘contradictory asymmetries’ in RV and RHP. Parker & Rubenstein (1981) point out that when an animal has high RHP, it will usually have low RV: better fighters have easier access to resources, so each particular unit of resource is worth relatively less to them. They call
this inverse correlation a ‘contradictory interaction’. Hernandez & Benson (1998) correctly point out that a contradictory interaction between RV and RHP appears to be operating in H. sara, but incorrectly equate this fact with the strategy being paradoxical. True, males with lower RHP win, but this is not enough to make the strategy paradoxical, at least if their hypothesis that another component of K, viz. lost reproductive opportunities, has a stronger opposing effect, is true. The critical ratio V/K is still hypothesized to be higher for the (smaller) winning males, which makes the strategy common-sense by definition. In general, paradoxical strategies do not ‘result from a contradictory asymmetry’ as Hernandez & Benson (1998) state in the Introduction (page 534); they can arise quite independently of whether RV and RHP are inversely correlated. Nevertheless, the notion of contradictory asymmetries does raise an interesting issue for the definition of a paradoxical strategy, and one which seems to us insufficiently clear from previous work. If Parker & Rubenstein (1981) are correct, then in an asymmetric contest, players will usually differ in both RHP and RV, and in opposite directions. In models like the sequential assessment game (e.g. Leimar & Enquist 1984), in which the favoured role is defined by separate consideration of RV and RHP (unlike the WOA, in which elements of RV and RHP are combined in the role-determining ratio V/K), it is therefore possible to argue that a common-sense strategy ‘higher RV wins’ is also simultaneously paradoxical, because it is a case of ‘lower RHP wins’ (the same applies for the reverse case). The question, then, is how to define the disfavoured role in such cases. We suggest it might help avoid confusion if the term ‘paradoxical’ was reserved for those cases in which it is known either that a contradictory asymmetry is not present, or that either RHP or RV is of overriding importance in resolving contests, and is lower for the winner. In practice, such knowledge can be obtained experimentally, by manipulating RHP and RV independently, thus enabling their respective effects on contest outcome to be estimated (e.g. Keeley & Grant 1993). From this, it seems that demonstration of paradoxical strategies in nature might be restricted to cases in which either a contradictory asymmetry can be ruled out, or in which sufficient ecological data are at hand such that the individual in the favoured role can be determined unambiguously. Theorists have long argued that paradoxical strategies can evolve only under a much more restricted set of conditions than common-sense strategies, and are unlikely to be found in nature. The points we raise above are perhaps further cause for pessimism about being able to confirm their existence empirically. Nevertheless, knowing that some types of fights and specific ecological settings are particularly favourable (Maynard Smith & Parker 1976; Hammerstein & Parker 1982; Mesterton-Gibbons 1992), the quest to discover paradoxical behaviour remains a tantalizing goal. For the time being, however, we are no closer to seeing the evolutionary paradoxes suggested by the mathematics of theoretical models being played out for real in nature.
FORUM
We thank Peter Hammerstein, Michael MestertonGibbons, Woodruff Benson and two anonymous referees for comments on the manuscript. S.A.F. was funded by a Golda Meir Postdoctoral Fellowship from the Hebrew University of Jerusalem and I.C.W.H. by a TMR established researcher grant (ERBFMBICT 983172) from the European Commission. References Burgess, J. W. 1976. Social spiders. Scientific American, March, 100–106. Enquist, M. & Leimar, O. 1987. Evolution of fighting behaviour: the effect of variation in resource value. Journal of Theoretical Biology, 127, 187–205. Hammerstein, P. 1981. The role of asymmetries in animal contests. Animal Behaviour, 29, 193–205. Hammerstein, P. & Parker, G. A. 1982. The asymmetric war of attrition. Journal of Theoretical Biology, 96, 647–682. Hernandez, M. I. M. & Benson, W. W. 1998. Small-male advantage in the territorial tropical butterfly Heliconius sara (Nymphalide): a paradoxical strategy? Animal Behaviour, 56, 533–540.
Keeley, E. R. & Grant, J. 1993. Asymmetries in the expected value of food do not predict the outcome of contests between convict cichlids. Animal Behaviour, 45, 1035–1037. Leimar, O. & Enquist, M. 1984. Effects of asymmetries in owner–intruder conflicts. Journal of Theoretical Biology, 111, 475– 491. Maynard Smith, J. 1974. The theory of games and the evolution of animal conflicts. Journal of Theoretical Biology, 47, 209–221. Maynard Smith, J. 1982. Evolution and the Theory of Games. Cambridge: Cambridge University Press. Maynard Smith, J. & Parker, G. A. 1976. The logic of asymmetric contests. Animal Behaviour, 24, 159–175. Maynard Smith, J. & Price, G. R. 1973. The logic of animal conflict. Nature, 246, 15–18. Mesterton-Gibbons, M. 1992. Ecotypic variation in the asymmetric hawk–dove game: when is Bourgeois an evolutionarily stable strategy? Evolutionary Ecology, 6, 198–222. Mesterton-Gibbons, M. & Adams, E. S. 1998. Animal contests as evolutionary games. American Scientist, 86, 334–341. Parker, G. A. & Rubenstein, D. I. 1981. Role assessment, reserve strategy, and acquisition of information in asymmetric animal conflicts. Animal Behaviour, 29, 221–240.
F3
FORUM Butterfly contests: contradictory but not paradoxical SCOTT A. FIELD* & IAN C. W. HARDY†
*Department of Entomology, Hebrew University of Jerusalem †Department of Animal Biology and Ecology, University of Granada, Spain (Received 20 January 1999; initial acceptance 30 July 1999; final acceptance 17 September 1999; MS. number: AF-4)
‘common sense’ and the majority of empirical studies, small H. sara males had an advantage; it appeared that ‘intruders quickly recognized small residents as unfavourable adversaries’ (page 537) and abandoned their attempt to acquire the territory. The authors suggest that ‘the success of small males may represent a paradoxical strategy (sensu Maynard Smith & Parker 1976), which promotes victory despite lower resource holding power’ (page 537). Here we point out that this suggestion runs contrary to the established definition of a paradoxical strategy for this game, and that the evolution of a paradoxical strategy in H. sara is in any case quite unlikely on theoretical grounds. We begin by briefly recapping the theoretical development of the concept of paradoxical strategies, and after reconsidering Hernandez & Benson’s (1998) conclusions, highlight a seeming ambiguity in the definition of paradoxical ESSs. We conclude with a comment on the prospects for actually demonstrating their existence in nature. The concept of a paradoxical strategy ‘evolved’ out of early explorations of two models of contest behaviour, the ‘hawk–dove’ game (Maynard Smith & Price 1973) and the ‘war of attrition’ (Maynard Smith 1974). Various authors later analysed ‘asymmetric’ contests in which one of the contestants was a better fighter or held a positional advantage (higher resource-holding power, RHP), and/or had more to gain than the other (higher resource value, RV or V) (e.g. Maynard Smith & Parker 1976; Hammerstein 1981; Parker & Rubenstein 1981; Hammerstein & Parker 1982). As common sense would suggest, in these games where the asymmetries directly affected the chances of winning or the payoff, ESSs were found in which the individual in the ‘favoured role’ (i.e. the one with a higher RHP or RV, or higher ratio of resource value to cost accrual, V/K), won the contest. Unexpectedly, however, they also discovered that under certain conditions (admittedly evolutionarily restrictive) there existed ‘paradoxical’ ESSs in which the contestant
aradoxes conflict with preconceived notions of what is reasonable or possible. Hence they are usually identified when empirical observation conflicts with established theory, although in some instances it is theoreticians who uncover the possibility of a paradox first. Such was the case in behavioural ecology with the so-called ‘paradoxical ESS’ (evolutionarily stable strategy), an evolved behavioural convention dictating that an individual of lower fighting ability, and/or with less to gain from winning a fight, obtains access to a disputed resource at the expense of a more able or motivated opponent (Maynard Smith & Parker 1976). In hindsight, the behaviour of a social spider reported by Burgess (1976), in which resident males routinely flee from a hiding place when disturbed by an intruder, seemed to provide an example (Maynard Smith 1982, page 96). However, important variables require empirical estimation before this interpretation can be fully accepted (Mesterton-Gibbons & Adams 1998). Further empirical examples of paradoxical outcomes have remained exceedingly sparse. In a recent issue of Animal Behaviour, Hernandez & Benson (1998) report what they speculate is a paradoxical strategy for resolving territorial contests between males in the butterfly Heliconius sara (Nymphalidae). Males have two alternative mating tactics: perching next to a pupa from which a female is about to emerge (here larger males may have an advantage) and defence of a mating territory. The latter involves the resident male engaging intruders in a short circling contest flight, with the loser being escorted out of the territory. In accord with most animal contests, Hernandez & Benson (1998) found that the resident male won the majority (in this case all) of the territory encounters. However, in contrast to both
P
Correspondence: I. C. W. Hardy, Ecology Centre, The Science Complex, University of Sunderland, Sunderland SR1 3SD, U.K. (email: [email protected]). S. A. Field is at the Department of Applied and Molecular Ecology, University of Adelaide, Waite Campus, Glen Osmond 5064, South Australia, Australia. 0003–3472/00/010F01+03 $35.00/0
F1
2000 The Association for the Study of Animal Behaviour
F2
ANIMAL BEHAVIOUR, 59, 1
in the disfavoured role won. Subsequent work also showed paradoxical ESSs to exist in the ‘sequential assessment game’ (Leimar & Enquist 1984; Enquist & Leimar 1987) and the ‘iterated hawk–dove game’ (MestertonGibbons 1992). We now turn to Hernandez & Benson’s (1998) argument, put forward in detail on page 538. They infer that the butterflies are engaging in a display in which victory goes to the more persistent, and therefore phrase their ideas in terms of the war of attrition (WOA). In this game, the common-sense strategy ‘for the individual with the lower resource value/rate of cost accrual (V/K) would be to retreat’. They argue that ‘size should not greatly influence the fitness rewards of territorial defence’, meaning that V should be approximately equal for large and small males. However, the cost K should be lower for large males because ‘holding territories may be physiologically less costly for larger individuals’. But a counterbalance is provided by reproductive opportunities lost during a contest, which should be greater for large males, making K higher (and hence V/K lower) for them even if, as one might expect, ‘smaller adversaries run a greater risk of serious injury in escalated combats’. In summary, the authors imply that an ESS is operating in which smaller males win because their ratio of resource value to rate of cost accrual is generally higher. The flaw in the suggestion that this result is paradoxical should now be clear: the statement above is the very definition of the common-sense strategy for this game. For the strategy to be paradoxical, victorious males would need to be in the ‘disfavoured’ role. Thus, leaving aside the fact that the parameters critical to verifying Hernandez & Benson’s (1998) argument are yet to be estimated, the suggestion that these territorial contests could exemplify a paradoxical strategy is clearly untenable. Furthermore, it is dubious whether a paradoxical strategy for contest resolution in H. sara is even theoretically possible. Hammerstein & Parker (1982) have shown that for the continuous WOA, paradoxical solutions do not exist. Only if individuals are forced at some point to commit themselves in advance to a reasonably costly discrete period of display (a ‘round’) during which they cannot retreat, and do so at a stage when they have not yet accrued much cost, can paradoxical ESSs arise (Hammerstein & Parker 1982). This situation of a discrete WOA is sometimes quite probable in nature, for example, where giving up in the middle of a round may expose a player to some uncontrollable risk of injury. However, from the description of H. sara contests given by Hernandez & Benson (1998, page 534), it does not appear to be the case here. The error we have highlighted seems to have arisen from two sources: confusion between the everyday and technical meaning of ‘paradoxical’, and a misunderstanding of the relationship between paradoxical strategies and ‘contradictory asymmetries’ in RV and RHP. Parker & Rubenstein (1981) point out that when an animal has high RHP, it will usually have low RV: better fighters have easier access to resources, so each particular unit of resource is worth relatively less to them. They call
this inverse correlation a ‘contradictory interaction’. Hernandez & Benson (1998) correctly point out that a contradictory interaction between RV and RHP appears to be operating in H. sara, but incorrectly equate this fact with the strategy being paradoxical. True, males with lower RHP win, but this is not enough to make the strategy paradoxical, at least if their hypothesis that another component of K, viz. lost reproductive opportunities, has a stronger opposing effect, is true. The critical ratio V/K is still hypothesized to be higher for the (smaller) winning males, which makes the strategy common-sense by definition. In general, paradoxical strategies do not ‘result from a contradictory asymmetry’ as Hernandez & Benson (1998) state in the Introduction (page 534); they can arise quite independently of whether RV and RHP are inversely correlated. Nevertheless, the notion of contradictory asymmetries does raise an interesting issue for the definition of a paradoxical strategy, and one which seems to us insufficiently clear from previous work. If Parker & Rubenstein (1981) are correct, then in an asymmetric contest, players will usually differ in both RHP and RV, and in opposite directions. In models like the sequential assessment game (e.g. Leimar & Enquist 1984), in which the favoured role is defined by separate consideration of RV and RHP (unlike the WOA, in which elements of RV and RHP are combined in the role-determining ratio V/K), it is therefore possible to argue that a common-sense strategy ‘higher RV wins’ is also simultaneously paradoxical, because it is a case of ‘lower RHP wins’ (the same applies for the reverse case). The question, then, is how to define the disfavoured role in such cases. We suggest it might help avoid confusion if the term ‘paradoxical’ was reserved for those cases in which it is known either that a contradictory asymmetry is not present, or that either RHP or RV is of overriding importance in resolving contests, and is lower for the winner. In practice, such knowledge can be obtained experimentally, by manipulating RHP and RV independently, thus enabling their respective effects on contest outcome to be estimated (e.g. Keeley & Grant 1993). From this, it seems that demonstration of paradoxical strategies in nature might be restricted to cases in which either a contradictory asymmetry can be ruled out, or in which sufficient ecological data are at hand such that the individual in the favoured role can be determined unambiguously. Theorists have long argued that paradoxical strategies can evolve only under a much more restricted set of conditions than common-sense strategies, and are unlikely to be found in nature. The points we raise above are perhaps further cause for pessimism about being able to confirm their existence empirically. Nevertheless, knowing that some types of fights and specific ecological settings are particularly favourable (Maynard Smith & Parker 1976; Hammerstein & Parker 1982; Mesterton-Gibbons 1992), the quest to discover paradoxical behaviour remains a tantalizing goal. For the time being, however, we are no closer to seeing the evolutionary paradoxes suggested by the mathematics of theoretical models being played out for real in nature.
FORUM
We thank Peter Hammerstein, Michael MestertonGibbons, Woodruff Benson and two anonymous referees for comments on the manuscript. S.A.F. was funded by a Golda Meir Postdoctoral Fellowship from the Hebrew University of Jerusalem and I.C.W.H. by a TMR established researcher grant (ERBFMBICT 983172) from the European Commission. References Burgess, J. W. 1976. Social spiders. Scientific American, March, 100–106. Enquist, M. & Leimar, O. 1987. Evolution of fighting behaviour: the effect of variation in resource value. Journal of Theoretical Biology, 127, 187–205. Hammerstein, P. 1981. The role of asymmetries in animal contests. Animal Behaviour, 29, 193–205. Hammerstein, P. & Parker, G. A. 1982. The asymmetric war of attrition. Journal of Theoretical Biology, 96, 647–682. Hernandez, M. I. M. & Benson, W. W. 1998. Small-male advantage in the territorial tropical butterfly Heliconius sara (Nymphalide): a paradoxical strategy? Animal Behaviour, 56, 533–540.
Keeley, E. R. & Grant, J. 1993. Asymmetries in the expected value of food do not predict the outcome of contests between convict cichlids. Animal Behaviour, 45, 1035–1037. Leimar, O. & Enquist, M. 1984. Effects of asymmetries in owner–intruder conflicts. Journal of Theoretical Biology, 111, 475– 491. Maynard Smith, J. 1974. The theory of games and the evolution of animal conflicts. Journal of Theoretical Biology, 47, 209–221. Maynard Smith, J. 1982. Evolution and the Theory of Games. Cambridge: Cambridge University Press. Maynard Smith, J. & Parker, G. A. 1976. The logic of asymmetric contests. Animal Behaviour, 24, 159–175. Maynard Smith, J. & Price, G. R. 1973. The logic of animal conflict. Nature, 246, 15–18. Mesterton-Gibbons, M. 1992. Ecotypic variation in the asymmetric hawk–dove game: when is Bourgeois an evolutionarily stable strategy? Evolutionary Ecology, 6, 198–222. Mesterton-Gibbons, M. & Adams, E. S. 1998. Animal contests as evolutionary games. American Scientist, 86, 334–341. Parker, G. A. & Rubenstein, D. I. 1981. Role assessment, reserve strategy, and acquisition of information in asymmetric animal conflicts. Animal Behaviour, 29, 221–240.
F3