J. Social Biol. Struct. 1985 8,255 277 Towards a dynamicsof socialbehaviour: Strategicand geneticmodels for the evolution of animal conflicts Peter Schuster* and Karl Sigmundl rlnstitut für Theoretische Chemie uncl Strahlenchemie undllnstitut für Mathematik tler Universität Wien, A-1090 Wien and IIASA, Laxenburg' Austria persistenceof behavioural traits in animal societiesrather than optimality of reproduc- tive successis described in terms of the game-theoreticnotion of evolutionarily stable strategy.Game dynamics provides a suitable framework to accomodatefor the dynamic urp..i, of permanenceand uninvadability, to model the evolution of phenotypes with frequency äependent fitness and to relate the strategic models of sociobiology to the mechanismsol inheritance in population genetics. Introduction Classical ethology provided numerous investigations of animal populations exhibiting 'altruistic' social phenomena based on behaviour. In this non-antropomorphic context, an act performed by an amimal is called altruistic if it increasesthe fitnessof some other animul at the cost of its own. Such behaviour was often explained by notions such as 'benefit 'group selection' and of the species'(for example, seeWynne-Edwards, 1962)' But while there is no logical inconsistency in viewing groups or speciesas units of selection.it seemswell establishedthat Darwinian evolution acts much more commonly through selection of the level of individual organisms or eventually genes'The explan- ation of many altruistic traits in terms of individual selectionhas, therefore,to be viewed as a major successof sociobiology. Roughly speaking,there are three approacheswhich proved to be fruitful. They may be designatedas genetic.economic and strategic,and attributed to Hamilton (1964)' Trivers (1972) and Maynard Smith (1974), respectively,although the principles go back to Haldane and Fisher, and arguably to Darwin himself. (l) The geneticexplanation is basedon the notion of kin selection.A genecomplex programming altruistic acts which benefit relativesmay well spread,since it occurs,with a certain probability, within the relatives whose reproductive successis increased. (2) The economic explanation relies on the notion of reciprocal altruism. If the altruistic act is likely to be returned at some later occasion, then the behaviour may become established,especially so if there is individual recognition between members of the population preventing'cheaters'. 0l40 l750/85/030255+23 $03.00/0 .O 1985 Academic PressInc. (London) Limited 256 P. St'huster& K. Sismund (3) The strategicexplanation, finally, usesthe panoply of game theory to show how the fitnessof an individual may dependon what the other onesare doing. An act which in certainsituations would be calledaltruistic need not, in other situations,decrease the reproductivesuccess of the organismperforming it. In this paper.we shalldeal exclusivelywith the third approachstressing its dynamical aspects.It must, however, be stated at the outset that the game-dynamicapproach described here constitutes only a fraction of the game-theoretical analysis of animal behaviour. Our discussion is accordingly biased. The theory of games covers many applicationsin biologicalevolution. Some of them do not easilyfit into the mould of ordinary differentialequations. For a balancedsurvey of thesetopics. we may refer to a recentbook by Maynard Smith (1982). Our paper consistsof two parts. The first one discusses,within the static context of gametheory. the crucialnotion of evolutionarilystable strategy (ESS), which centreson aspectsof permanencerather than optimality. The secondpart usesgame dynamicsto obtain broadernotions of permanence,to model the evolution of behaviouralstrategies and to relate them to geneticmechanisms. Beforewe start our discussionof strategicmodels in sociobiologywe haveto comment on the evolutionaryaspects of this approach.The actual mathematicalmodels usedin sociobiologyas well as those of population geneticsdiscuss the spreadingof mutant genesin populations.Thecauses of mutationsare consideredas externalfactors and not as part of the systemunder consideratiom.A proper descriptionof the evolutionary process,however. is completeonly if it includesthe mechanismof mutant formation. At present.such complete theories of evolution exist for polynucleotidesin laboratory systems(for example,see Eigen & Schuster,1979; Biebricher et al.l982; Küppers, 1983) and eventually for simple viruses(Weissmann , 1914).The replication of bacteria follows a much more complicatedmechanism. Already ten enzymesor more are required for DNA copying. Although not all the mechanisticdetails of local mutations and DNA rearrangementsin bacteriaare known yet, the basicfeatures of bacterialevolution are establishedin principle and we can expect a complete molecular theory to become availablein the not too distant future. In caseof higher.multicellular organisms we encounternew sourcesof complications which obscurethe relation betweenmutations and their manifestationin phenotypic properties.The major problemconcerns the unfolding of the genotype:higher organisms obtain final form and shapethrough morphogenesis.This is a complicateddynamical processwhich is not completelyunderstood at presentand to which the genescontribute markers and other regulatory signalsfor intercellularcommunication apart from the ordinary dutiesin cell metabolismand cell division. Any completetheory of evolution of higher organismshence has to deal explicitelywith the influenceof mutations on morphogenesis,which is far outside the bounds of presentpossibilities. Behavioural traits are exceedinglycomplex phenotypicfeatures which apparentlycannot be traced 'mutant down to singlegenes. The notion of genes'leading to changesin behaviourhas to be understoodas a metaphor. Mutations actually operatewithin complicatednet- works of geneactions which are convertedinto the propertiesof the phenotypeby a highly complex transformationduring morphogenesis. Evolutionarily stable strategies 'evolutionarily The term stablestrategy' was introduced by Maynard Smith and Price (1913) to describecertain seneticallv determined traits of animal behaviour which are 257 Dynamics of sociul behaYiour evolutionarystability in uninvadableby mutants. The basic idea was to describesuch was subsequentlyused termsof the mathematicframework of gametheory. This method betweenspecies' and can to analysea wide varietyof biologicalconflicts. both within and be righily consideredto be a cornerstoneof theoreticalbiology. Game theorYand bblog.v first publishedin Ever sincevon Neumann and Morgenstern's(1953) classical treatise of all kinds of 1944.the methods of game theory have been applied to the modelling and plants' conflictsin human societies.The applicationsto conflictswithin animals using game however. date back to the last ten years.There were a few forerunners one describedhere' theoreticnotions in biology. but in a context quite differentfrom the 'games'were neglected With the benefitof hindsilht it seemsastonishing that biological military or economicones. Indeed. there are at least for so long in favour of political. 'natural' their two reasonswhich make human conflictsmore difficult to analyzethan counterpartsin biologY: sinceit is hard to evaluatemoney, (l ) The payoff in human systemsis often doubtful. In the evolutionarily social piestige,health and other factors on a singleutility scale. quantity. Although weightedcontests in biology, Darwinian fitnessis the ony relevant and empirical this notron presentsconsiderable di{iculties for theoretical analysis success' measurementit providesin principle,at least,a scalarestimate of reproductive 'rationality game theoristsat (2) The axiom' frequently setsempirical and theoretical such an assumption' odds. The investigationof animal behaviour is unhindered by the simplest Biologicalconflicts tend to be fairly straightforward.compared with all but lunacy' etc' h,.,manconflicts of any real interest.The possibilitiesfor conspiracy,fraud. are greatly reduced strategyIn It is interestingto note. however,that the notion of evolutionarily stable contestsis quite closeto that of Nash equilibrlum strategyfor noncooperative biological 'Characterizations gamesbetween rational players (cf. the sectionon of evolutionary is interpretedas !,ubl. r,.u,"gies(ESS)'below). In Parker& Hammerstein(1984), this 'quasi-ratioÄlity' of selection.In our view,the convergenceof the two conceptsis rather 'blind 'rational to adaptative due to the fäct that both selection'and decision' lead solutions.After all, rational behaviouritself is a product of selectron. Strategie s and Pheno t vPes entailsa shift The shift in the applicationof gametheory from human to animal conflicts 'strategy' 'payoff. 'Payoff, as we have in the meaningof the two basicnotions of and or-in subtler said.now correspondsto Darwinian fitness,i.e. the number of offspring generation' an situations-the number of copiesof genestransmitted to the next [For ( 'Strategy'.in the elaborateanalysis of the notion of fitnesswe refer to Dawkins l9S2)'] of moves' This context of chess.of war games,suggests a nicely calculatedsequence trait of fighting aspectof plotting and schemingwas replacedby the notion of an innate this viewpoint was biauioui (Maynard Smith & Price. 1973).It was soon realizedthat of any clash useful.not only in the analysisof aggressiveencounters. but in the modelling the length of the ol interests:in the parent offspringconflict, for example,concerning in parental weaningperiod. or in the