Anim. Behav., 1978, 26,959-963

Tree Frog Choruses :A Mixed Evolutionarily Stable Strategy?

Groups of males singing or calling in synchrony toattract females have been noted in many animal groups(see review by Alexander (1975) in Insects, Science andSociety, Pimentol, ed. New York : Academic Press) .Chorusing presumably evolved as a result of mateselection by females, but when a female approaches sucha group, how does she choose between its members?Whitney & Krebs (1975, Nature, Lond., 225, 325-326)demonstrated that, in Pacific tree frogs (Hyla regilla),females choose mates which call `most actively' (initiatecalling bouts and call loudest, longest and fastest). Theysuggest that when several males call at the same time,acoustical interference prevents the female from findingany of them, so that those which begin or end boutswould be easiest to locate .Whitney & Krebs point out a problem with this :

if males compete to call during periods when few othersare calling, why don't they develop other strategies ofcalling (i.e. cheat)? An obvious strategy would be to restduring bouts (when the other frogs interfere with oneanother) and start calling when the others stop . If asingle frog played this soloist strategy in a populationof chorusing individuals, it would clearly do very well,but what happens when more than one soloist is presentin a calling group is unclear . Each could interfere withthe others' attempts to call alone during the intervalsbetween bouts. Whitney & Krebs argue that the soloiststrategy would not be successful on two other grounds,suggesting that (1) a chorus of males attracts morefemales than the same number of males calling individu-ally, and (2) an individual should join in a chorus toprevent the others from attracting females, by acousticalinterference.

While these points may rule out the spread beyond avery low frequency of the simple soloist strategy outlinedabove, the arguments do not necessarily apply to themore complex cheating strategy we call solo/duet.Solo/duet males attempt to call between bouts, likesoloists, but also call if any frog is calling alone, toprevent him attracting a female.

For a behavioural strategy to persist it is not sufficientthat it be profitable. It must be resistant to invasion byother strategies as well. We were interested to discoverwhether a population of chorusing frogs could be invadedby solo/duet cheaters, and, if so, whether there was anevolutionarily stable mixture of the two (Maynard Smith& Price (1973) Nature, Lond., 246, 15-18) . Although2-player games can often be solved analytically, N-persongames such as this are more easily solved by computersimulation : note that our aim was not to simulate allthe details of chorusing behaviour of real frogs, butsimply to investigate the equilibrium frequencies of thetwo alternative strategies . We limited ourselves to onlyone simple strategy of cheating because we were interestedin the general principle of how well cheaters would dowithout devising a strategy so complex as to be un-realistic.

We divided our model population into `ponds' of sixmales. As long as some of the . frogs called in chorus,the number of females attracted to the pond was assumedto be constant . In each pond, males competed for thearriving females by calling, and reproduced in propor-tion to the amount of time they spent calling alone.

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Fig. 1 (A-C) . Simulated calling bouts. Solid lines indicateperiods of calling ; spaces indicate silent periods . In A,one frog is a solo/duetist, the other five are choristers .In B, three frogs play each strategy .

If a frog was not exhausted, the likehood of it startingor stopping calling was an exponential function of time .

P start = 1 - eK l - t, P stop = I - eK2 ' t .

The length of the refractory period following callingwas equal to a constant C plus the reciprocal of theprevious bout length. In simulations, K1, K2 and C, aswell as the starting frequencies of choristers and solo/duetists were altered independently . The model turnsout to be quite insensitive to the values of these para-meters. Lengths of calling and silent periods were altered,but the relative advantages of playing the two strategieswere not affected.

The simulation produces calling bouts closely resemb-ling those of tree frogs. A single solo/duetist in a popula-tion of choristers does very well (A), but, as predicted,the advantage is lost if there are several in the same pond(B).Note that we have assigned equal average calling

potential to all frogs, so there are no chorus leaders,that is, we are not modelling the behaviour of eachindividual frog. This unrealistically simple model issatisfactory for comparing the average success ofchoristers and solo/duetists, although it would be in-adequate for analysing the mating success of individualchoristers .

Figure 1C shows the proportion of choristers in thepopulation over a period of 35 generations . Repeatedsimulations showed that neither strategy was an ESS(each could be invaded by the other) . Regardless ofstarting frequencies, the population quickly reached astable proportion of about 2/3 choristers to 1/3 solo-duetists. It then oscillates around this equilibrium inresponse to random allocation of frogs to ponds in eachgeneration .

Offspring were allocated randomly to new ponds forthe next generation and the game was played again .Individuals were assigned one of the two strategies :chorister or solo/duetist . The strategies only differed in

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the rules governing when a frog called : all individualshad the same calling potential .

(1) Chorister : began calling if any other frog in thepond was calling and stayed in the chorus until exhausted .

We included exhaustion because there must be a costto calling or competition would lead to frogs callingcontinuously. We simulated this cost by making thelikelihood of stopping an exponential function of time(i .e. the longer a frog called, the more likely it was tostop). If the frog had called for a minimum length oftime, calling was followed by a refractory period duringwhich the frog could not begin calling again .

Whitney & Krebs observed that those frogs callinglongest were also the most likely to begin the next bout .To meet this criterion, we made the refractory periodequal to a small constant plus the reciprocal of theprevious bout length .After recovery from exhaustion, all individuals

initiated calling bouts with the same probability.(2) Solo/duetist : behaved the same as a chorister in

terms of starting bouts and exhaustion, but the likelihoodof joining or leaving a chorus depended upon the numberof individuals in it. Solo/duetists joined any frog callingalone, and were always silent when three or more frogswere calling. If two other frogs were calling, however,they joined with a probability of 0 . 5 . This was to mimic'mistakes' (when a frog could not be certain whethermore than one frog was calling).

Details and results of the model are shown in Fig . 1 .Repeated simulation showed that neither strategy is anESS (each can be invaded by the other) but that a mixedproportion of 2/3 choristers to 1/3 solo/duetists is stable .We do not know whether this proportion is a function ofpond size. These results suggest that it would pay somefrogs in the wild to cheat .

We can think of two explanations for why Whitney& Krebs did not observe any solo/duet callers . (1) Ourmodel may be too simple . For instance, making biggerchoruses more attractive would shift the ESS to a lowerfrequency of solo/duetists, although they would stilloccur. It may be that frogs in the wild are cheating, buttoo infrequently to be detected by Whitney & Krebs.(2) If males do not cheat, then females must rely on morethan simply the amount of solo calling to choose betweenthem. If females choose on the basis of calling abilitythen selection would favour reliance upon cues not easilyfaked . For example, females may be measuring the totalamount of calling by individual males. That is, they maybe perfectly capable of locating males during a chorus,but refrain from making a choice until the bout iscompleted.

We think that both (1) and (2) may be correct. Theextent to which they may alter the conclusions of Whitney& Krebs could beascertained by further field observations .We thank Nick Davies and Richard Dawkins for

helpful comments.BRIAN PARTRIDGE *JomN R . KREBS 4

*Dept. of Experimental Psychology,University of Oxford,South Parks Road, Oxford .

tEdward Grey Institute ofField Ornithology,

Dept. of Zoology,University of Oxford,South Parks Road, Oxford.(Received 8 November 1977 ; revised 14 February 1978 ;

MS. number : s-22)

Author's Reply to Partridge and Nunney Critique of'Tee-Generation Family Conflict'

Partridge & Nunney (1977) raise one specific and twogeneral points about my paper on three-generation familyconflict (Fagen 1976) . Each of these points reflects amajor controversial and unresolved issue in the study ofanimal behaviour . Are genes or individuals the units ofselection? Are sexual and agonistic behaviours the onlybehaviour patterns that clearly contribute to survivalor to reproductive success in humans? Does imperfectknowledge constrain animal behaviour and/or behavi-oural evolution?

I accept the specific criticism that the arrangement,mode of expression and resulting interaction of genescontrolling altruism and appearance should (in additionto cultural factors) theoretically determine whether ornot a grandparent will selectively favour those grand-children which in some sense resemble it more . W. D .Hamilton's 'gene' approach to the evolution of socialbehaviour (Dawkins 1976) offers simple, formal modelsrelevant to such analyses . I based most of my conclusionson mathematical formalisms that implicitly embodythis view, but my verbal argument on physical resemb-lance as a releaser for doting was inconsistent withHamilton's theory in its simplest version . Partridge &Nunney's analysis of the problem is the correct deductionfrom the 'gene' approach, which has, however, beencriticized (Gould 1977) . Such criticisms await the sup-port of a simple formal model reflecting these objections .'Gene' models have yet to be counterbalanced byalternative models starting from radically differentassumptions . Neither set of models should be confusedwith 'reality' (a word which, Vladimir Nabokov argued,should always appear within quotes) .Fundamental theoretical issues of developmental

behaviour genetics and of gene interactions in develop-ment arise from analyses of the behaviour of dotinggrandparents. Can mathematical models that neglectgene interactions have heuristic value in the study ofsocial behaviour? These existing models may profitablydirect attention to the neglected problem of co-operationand competition in grandparent-grandchild interactions,which could become the specific focus of a field study inelephants, chimpanzees, or other long-lived, group-forming animals such as humans .

The two general points raised by Partridge & Nunneyare : (1) what sorts of behaviours constitute appropriateexamples in human sociobiology and (2) the role ofimperfect information (imperfect knowledge) in socialtheory. Because human exploratory and playful manipu-lation of toys has been linked to development (White &Held 1967), to skill in tool construction and tool use(Sylva, Bruner & Genova 1976) and to other cognitiveskills (Dansky & Silverman 1973 ; Golomb & Cornelius1977; Hutt & Bhavnani 1972 ; Lieberman 1977 ; Sutton-Smith 1968 ; Sylva 1977), toys can contribute to humansurvival and reproductive success . Exploration and playare biologically important human behaviour patterns .(Incidentally, Partridge & Nunney incorrectly claim thatI cite 'the example of adult discipline of children over thesharing of toys' . The situation cited for illustrative pur-poses is three-generation conflict resulting from grand-parents' possible gifts of toys to grandchildren ; I didnot specifically invoke either 'discipline' or 'sharing') .

Imperfect information does not yet occupy the pivotalrole in social theory that it has long occupied in decisiontheory or in the theory of games . Unfortunately, evenwhen interests coincide, and even when self-deception