ARTICLE IN PRESSJournal of Theoretical Biology 232 (2005) 99–104www.elsevier.com/locate/yjtbiThe iterated continuous prisoner’s dilemma game cannot explain theevolution of interspecific mutualism in unstructured populationsIstván Scheuring Department of Plant Taxonomy and Ecology, Research Group of Ecology and Theoretical Biology, Hungarian Academy of Sciences and EötvösUniversity Budapest, Pázmány P. sétány 1/c, 1117 Budapest, HungaryReceived 14 May 2004; accepted 29 July 2004Available online 18 September 2004AbstractThe evolutionionary origin of inter- and intra-specific cooperation among non-related individuals has been a great challenge forbiologists for decades. Recently, the continuous prisoner’s dilemma game has been introduced to study this problem. In function ofprevious payoffs, individuals can change their cooperative investment iteratively in this model system. Killingback and Doebeli(Am. Nat. 160 (2002) 421–438) have shown analytically that intra-specific cooperation can emerge in this model system fromoriginally non-cooperating individuals living in a non-structured population. However, it is also known froman earlier numericalwork that inter-specific cooperation (mutualism) cannot evolve in a very similar model. The only difference here is that cooperationoccurs among individuals of different species. Based on the model framework used by Killingback and Doebeli (2002), this Noteproves analytically that mutualism indeed cannot emerge in this model system. Since numerical results have revealed that mutualismcan evolve in this model system if individuals interact in a spatially structured manner, our work emphasizes indirectly the role ofspatial structure of populations in the origin of mutualism.r 2004 Elsevier Ltd. All rights reserved.Keywords: Adaptive dynamics; Coevolution; Evolutionary stability; Reciprocal altruism; Variable investmentMutualistic interactions between members of differentspecies are widespread and play a central role inecosystems (Boucher et al., 1982; Bronstein, 2001a, b).However, the evolution of mutualism has been a greatchallenge for theoreticians for decades. How canmutualist individuals emerge in populations where allthe others are non-mutualists, and how can mutualistsprevent cheaters fromspreading in the population?The classical theoretical framework for studyingcooperation of unrelated individuals within a species isthe Prisoner’s Dilemma Game (Trivers, 1971), in whichpartners can choose either a defective (cheating) or acooperative strategy. If both partners defect, they get asmaller fitness than if both cooperate, but a defector hasan even higher fitness value if its opponent cooperates. Tel.: +36-1-209-0555x1706; fax: +36-1-381-2188.E-mail address: firstname.lastname@example.org (I. Scheuring).However, the cheated cooperator receives the smallestfitness if its opponent is a defector. It is easy to see thatdefection is the only evolutionary stable state in thismodel, and cooperators cannot spread in a defectingpopulation. On the other hand, defectors can invade anddestroy cooperation in a cooperative population (Trivers,1971; Axelrod and Hamilton, 1981). Cooperativestrategies emerge and are stable against the invasion ofdefective ones if individuals can interact with eachothers repeatedly (Axelrod and Hamilton, 1981; Nowakand Sigmund, 1992, 1993). The general conclusions ofintensive work in this field are that the successfulstrategies are either those that punish defector andreward cooperation in repeated encounters (Tit-for-Tat,Generous Tit-for-Tat) or those that retain their previoussuccessful strategy (Pavlov).Doebeli and Knowlton (1998) realized that there is noway for individuals to vary the degree of cooperation in0022-5193/$ - see front matter r 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.jtbi.2004.07.025
100ARTICLE IN PRESSI. Scheuring / Journal of Theoretical Biology 232 (2005) 99–104these model systems, which is an unrealistic assumptionin many biological situations. Therefore, they introducedan interspecific prisoner’s dilemma game wherethe payoffs can change according to the investmentsmade by the partners. The level of investment I h by ahost individual involves a cost CðI h Þ to the host andgives a benefit BðI h Þ to its symbiont partner. (Naturally,if the investment is zero then the cost and the benefit arezero too.) Thus if the symbiont partner invests I s thenthe payoff for the host is SðI h ; I s Þ¼BðI s Þ CðI h Þ andsimilarly, the payoff of the symbiont is SðI s ; I h Þ¼BðI h Þ CðI s Þ: (Species are called ‘‘host’’ and ‘‘symbiont’’only for convenience, but they have a completelysymmetrical role in the mutualistic interaction.) In thespecial case when one of them makes no investment tothe interaction we retrieve the classical prisoner’sdilemma game again (Table 1).Mutualistic connections were considered by Doebeliand Knowlton (1998) as a series of prisoner’s dilemmagames with variable investment, where the investment ofa partner depends on its payoff received in the previousround. Interaction is a general phrase covering differentkinds of exchange of commodities (Bronstein, 2001a, b).The investment decision is determined by two parameters:a; the initial offer, and b; the reward rate or therate of increase of investment depending on the payoff Sin the preceding round. So the investment of a host inround k þ 1isI ðkþ1Þh¼ a h þ b h SðI ðkÞh; I ðkÞs Þ; (1)where SðI ðkÞh; I ðkÞs Þ is the payoff of the host individualin round k, if it invested I ðkÞhand its symbiont opponentinvested I ðkÞs in the previous interaction. Toavoid negative investment I ðkþ1Þhis set to zero ifSðI ðkÞh; I ðkÞs Þo a h =b h : Similarly, the opponent’s investmentin the k þ 1-th round isI ðkþ1Þs¼ a s þ b s SðI ðkÞs; I ðkÞhÞ; (2)where a s and b s are initial offer and reward rate of theopponent. Since benefit generally increases less athigher investment (see e.g. Altmann, 1979; Schulmanand Rubenstein, 1983), Doebeli and Knowlton(1998) used a concave function for BðIÞ; namely BðIÞ ¼B 0 ½1 expð B 1 IÞŠ: They assumed that cost increasesTable 1Payoff matrix when the cooperative (C) strategy invests I40 and thedefective (D) does not invest into the interactionCC R ¼ BðIÞ CðIÞ S ¼ CðIÞD T ¼ BðIÞ P ¼ 0The matrix describes the classical prisoner’s dilemma game ifT4R4P4S and 2R4ðT þ SÞ; which is valid if BðIÞ4CðIÞ:Dlinearly, that is CðIÞ ¼C 0 I: The parameters B 0 ; B 1 andC 0 are positive constants describing the cost–benefitrelations for all individuals. It is worth investing in theinteraction only if B 0 4C 0 ; and therefore this relation isassumed in the following. There are a fixed number ofmutualistic interactions between two generations, andthe fitness of an individual is the sumof payoffscollected in each round.The partners, independent of which species theybelong to, have an initial phenotype ða; bÞ; so initiallyall of themhave the same fitness. However, slightlydifferent mutant phenotypes can emerge by chance inboth populations in every generation. If a mutant has ahigher fitness than the resident type, the latter isreplaced by the mutant phenotype. Invasion is thusassumed to imply fixation (Doebeli and Knowlton,1998). Mutualism emerges if the originally very small,but positive a and b phenotypic traits evolve towardshigher positive values.However, according to numerical simulations mutualismcannotevolve if the partners live in ‘‘well-mixed’’populations without spatial structure (Doebeli andKnowlton, 1998). Well-mixedness means that the probabilityof interactions between different phenotypes isequal to the product of their relative frequencies, and isassumed here for both intraspecific competition andinterspecific mutualism. This assumption makes themodel more tractable, but neglects the spatial structurepresent in most populations. It has been known forsome time that spatially structured evolutionary gamesbehave differently and in a more complex manner than‘‘well-mixed’’ models (Nowak and May, 1992). It is thusnot surprising that Doebeli and Knowlton (1998) placedthe individuals of interacting species on the grid pointsof separate lattices. An individual in the first populationcan interact with the individual on the same grid pointof the other lattice. The fitness of each individual isgiven by the sumof payoffs received in the interspecificinteractions, which determines the competitive successwith their local neighbors within the species. Thephenotype which has the highest fitness among theneighbors (including the focal individual itself) will enterthe next generation at the chosen site. It has been shownnumerically that mutualism can emerge in this spatiallyexplicit system, particularly if dispersion is absent orlimited, and some stochasticity is present in thecompetition or selection processes (Doebeli and Knowlton,1998). One of the crucial difference between the‘‘well-mixed’’ and lattice models that the successfulstrategies spread by growing patches in the latter case.Thus the similar (or identical) strategies interact witheach other with a high probability in the spatiallystructured models, while to meet a similar (e.g.cooperative) strategy depends on their relative frequenciesin the well-mixed systems. This ‘‘viscosity’’ ofspatial models enhance the benefit of cooperative
ARTICLE IN PRESSI. Scheuring / Journal of Theoretical Biology 232 (2005) 99–104 101strategies in a similar way as kin-selection acts on theevolution of cooperation in well-mixed models (vanBaalen and Rand, 1998).Variable investment is also a fruitful concept instudying cooperation among non-related individualsthat belong to the same species. The model of thisproblemis very similar to that introduced above, butnow the partners are members of the same population.Killingback et al. (1999) considered a population whereindividuals live on a two-dimensional lattice and play anon-iterated prisoner’s dilemma game with variableinvestment. Partners interact with their eight nearestneighbors. The fitness of each individual is given by thesumof payoffs received in the interactions with allneighbors. Similar to the two species case, the phenotypewhich has the highest fitness among the neighbors willenter the next generation. If phenotypes are allowed tochange their investment by mutations, the populationevolves to cooperation in this spatially structured modeldespite there being no iteration of investment (Killingbacket al., 1999). Furthermore, cooperation emergesand is stable against the invasion of defectors even in awell-mixed population (Killingback and Doebeli, 2002)assuming that investment can change according to theiteration process described above. The latter wasconfirmed by analytical results. Killingback and Doebeli(2002) have shown that if the initial reward rate bexceeds a critical level b c ¼ C 0 ð0Þ=½B 0 ð0Þ 2 C 0 ð0Þ 2 Š whereC 0 ð0Þ; B 0 ð0Þ means the derivative of cost and benefit atzero investment, then both a and b evolve towardshigher positive values. Evolution thus leads to cooperationin all cases, independently of the initial offerif the initial reward rate is high enough. The higher thebenefit is compared to the cost at low investment,the lower the threshold b c which must be exceeded.However, as we have stressed above, simulationssuggest that there is no way for interspecific mutualismin a well-mixed population to evolve, even if iteratedinvestment is built into the model. What is the differencebetween within species cooperation and interspecificmutualism, that can lead to such differentbehavior? The differences are intuitively clear: individualscompete with the con-specifics and cooperate withthe inter-specifics in the mutualistic case, while cooperationand selection occur within the same species inintraspecific cooperation. Based on the analysis ofKillingback and Doebeli (2002) we show analyticallythat this difference indeed implies that neither the smallinitial offer nor the reward rate can increase by variableinvestment, except at very special and biologicallyirrelevant combinations of parameters. Evolution ofmutualism is thus essentiallyimpossible in spatially nonstructuredversion of the Doebeli and Knowlton’s (1998)model.We are interested in whether the initially smalland positive parameters a i and b i (i ¼ h; s) can increasein the iterated investment game model of mutualism.For small values of these parameters the investment Idetermined by Eqs. (1), (2) are also small. Thereforeit is enough to consider the linear approximationsof CðIÞ and BðIÞ; that is CðIÞ ¼C 0 I and BðIÞ ¼B 0 I;where C 0 ¼ C 0 ð0Þ and B 0 ¼ B 0 ð0Þ are the derivativesof the functions at I ¼ 0 (Killingback and Doebeli,2002).Let us assume that a small e fraction of individualsmutate to a 0 i ; b0 i phenotypes in both species. UsingEqs. (1), (2) and the linear approximations of the costand benefit functions, we obtain a recursive equation forthe payoffs of resident and mutant phenotypes for bothspeciesS ðkþ1Þh¼ B 0 ½ð1 e h Þða s þ b s S ðkÞs Þþe h ða 0 s þ b0 s S0 sðkÞ ÞŠS 0 ðkþ1Þ h ¼ B 0 ½ð1S ðkþ1ÞC 0 ða h þ b h S ðkÞh Þ;e h Þða s þ b s S ðkÞs Þþe h ða 0 s þ b0 s S0 sðkÞ ÞŠC 0 ða 0 h þ b0 h S0 hðkÞ Þ;s ¼ B 0 ½ð1 e s Þða h þ b h S ðkÞS 0 ðkþ1Þ s ¼ B 0 ½ð1C 0 ða s þ b s S ðkÞs Þ;hÞþe sða 0 h þ b0 h S0 ðkÞ h ÞŠe s Þða h þ b h S ðkÞhÞþe sða 0 h þ b0 h S0 hðkÞ ÞŠC 0 ða 0 s þ b0 s S0 sðkÞ Þ: ð3ÞHere we assumed that an ða i ; b i ) strategy obtains anaverage benefit fromits mutualistic partner proportionalto the frequency of the strategies present in thatpopulation, that is, there is a ‘‘playing the field’’situation in well-mixed populations (Maynard Smith,1982). Killingback and Doebeli (2002) restricted theirattention to the infinitely large population limit (e i ¼ 0),thus their model is generalized here to the case e h ; e s a0:If there are high numbers of interactions between twogenerations (kb1), then S ðkÞias the S i ; S 0 iand S 0 ðkÞi can be consideredfixed points of Eqs. (3). Thus, the fitness of astrategy (a i ; b i ) is approximately kS i : Since the fitnessesof both the resident and the mutant strategies aremultiplied with the same constant k, it is enough toconsider the S i ; S 0 i fixed points in the following(Killingback and Doebeli, 2002). The fixed points canbe computed from the linear recursion system (3), butthey are too complex to display here (but see AppendixA). The relevant points here are that the fixed pointsexist, and they are asymptotically stable if b i and b 0 i aresufficiently small (Appendix B). To make the stabilityanalysis tractable we assume here and in the followingthat e h ¼ e s ¼ e:Since the initial offer (a i ) and reward rate (b i ) arecontinuous variables, it is convenient to useadaptive dynamics to investigate the evolution ofmutualism (Hofbauer and Sigmund, 1990; Geritz etal., 1998; Metz et al., 1992; Killingback and Doebeli,2002). According to this framework, the dynamics of
102ARTICLE IN PRESSI. Scheuring / Journal of Theoretical Biology 232 (2005) 99–104phenotypic traits are_a i ¼ @S0 i@a 0 i ða 0i ¼a i ;b 0 i iÞ;¼b_b i ¼ @S0 i ð4Þða 0i ¼a i ;b 0 i iÞ; ¼b@b 0 iwhere i denotes either the host (h) or the symbiont (s)species. Since our analysis is restricted to the invasion ofrare mutants we can assume that e51; thus we canconsider the linear approximation of S 0 i in e (seeAppendix A). Thus we obtain a relatively simple formfromEq. (4)_a i ¼_b i ¼1C 0 þ B2 0 b !j1 þ C 0 b i G e ;L ið1 þ C 0 b i ÞGC 0 þ B2 0 b jG e !; ð5Þwhere G ¼ 1 þ C 0 ðb s þ b h ÞþðC 2 0 B 2 0 Þb sb h ; L i ¼a i ½ C 0 þðB 2 0 C 2 0 Þb jŠþB 0 a j ; and i, j indexes denote hor s respectively. Mutualismemerges if the originallysmall a i and b i increase, that is if the right-hand sides of(5) are positive (Killingback and Doebeli, 2002).It can be seen fromEqs. (5) that the originally small a idecreases further at the infinite population size limit(e ¼ 0). The parameters b i can increase when L i =G isnegative. Since b i is small for both partners, G can beconsidered a positive number in this limit. However, ifb i o C 0¼ bB 2 0 C 2 c and a i is sufficiently smaller than a j then0L i can be negative. In this case b i increases, until itreaches a limit which is definitely smaller than b c :Consequently, the initial offer (a i ) decreases, the rewardrates (b i ) either increases or decreases initially, butcannot exceed the b c ; thus there is no way for evolutionof mutualism in the infinite population size limit.Now let us analyse the case when the population isfinite and mutants are rare (0oe51). Remember that bcould not be arbitrarily close to zero to increase even inthe single species cooperative model, so it is possible thatreasonable small thresholds exist here as well, abovewhich a i and b i will increase. Observe, that if G remainspositive, but it is close to zero then the expressions in theparentheses of Eqs. (5) can be positive. Since we focuson the emergence of mutualistic interactions from a nonmutualisticstate we assume that a i and b i are close tozero initially. Further, we assume that the initialpropensity to the mutualistic interaction are roughlythe same for both species (a h a s and b h b s ) and forthe phenotypic traits (a i b i ) too. Naturally, this is notsufficiently the case, but without these assumptions theanalysis become hopelessly complex. To make theestimation more tractable, let us assume in the futurethat initially a i ¼ b i ¼ b: The expressions in theparentheses of Eqs. (5) can be positive if 0oG51 whichrelations are valid if b is close to but still smaller thanb tr ¼ 1=ðB 0 C 0 Þ: (If b41=ðB 0 C 0 Þ then Go0; andthus a i will decrease if G is a large positive number thenthe second termcan be neglected in the parentheses ofEqs. (5), and this a i will decrease again). To satisfy thiscondition we assume thatb ¼ð1 dÞb tr ; where 0odo1measures the deviation from the threshold b tr : Using thisnotation and assuming that d 2 5d we conclude that G 2dB 0B 0 C 0in the d neighborhood of b tr : Substituting thesevalues of b and G into the first equation of Eqs. (5) andrearranging the expression in the parentheses, weconclude that a i increases ifdoB 0e¼ d tr : (6)B 0 e þ 2C 0The same condition for d guarantees that the expressionin the parentheses in the second equation of Eqs. (5) ispositive as well, but b _ i will be positive only if L i 40atthe same time. Substituting ð1 dÞb tr for b yields L i ¼1 d þ B 0þC 0B 0 C 0ð1 2dÞ; which is positive, at least if d isnot too close to one. We thus conclude that theparameters a i and b i will increase ifð1 dÞb tr obob tr : (7)The evolution of mutualism is possible if both b tr and d trare not very close to zero. In this case, the initially smalla i ; b i trait values will increase in a relatively wide intervalof a and b (see Eqs. (6) and (7)). The parameter b tr is nottoo small if B 0 is not much higher than C 0 : However,since e is a small number, thus d tr is close to zero in thiscase. That is, a i and b i can increase only in a restrictedinterval of b around a relatively high value (see Eq. (7)).According to Eq. (6) the parameter d tr can be relativelylarge if B 0 bC 0 ; but then b tr 51: Consequently, thisanalysis confirms the numerical results of Doebeli andKnowlton (Doebeli and Knowlton, 1998): that it ishighly improbable that mutualistic interaction wouldemerge in this model system. We emphasize here thatour analysis is restricted only to the large populationlimit (e51), which could not exclude the evolution ofmutualism in small populations. Similarly, our conclusiondoes not exclude the possibility of the emergence ofmutualism in cases when the initial a i and b i valuesdiffer to each other meaningfully (e.g. a h =a s 51;b h =b s 51; etc.), although these conditions seems to bebiologically less relevant. On the other side, this Letteremphasizes indirectly the important role of spatialstructures in the evolution of mutualism.AcknowledgementsThe author thank V. Mu¨ ller and T. Killingback andthe referees for helpful comments. The work issupported by the Hungarian National Science Foundation(OTKA, Grant no. T037726).
ARTICLE IN PRESSI. Scheuring / Journal of Theoretical Biology 232 (2005) 99–104 103Appendix A. The fixed pointIt is known that if A e ¼ A þ eB where e is a smallnumber then j l i ðAÞ l i ðA e ÞjpcondðSÞkBke (Kato,The fixed points of Eq. (3) are1980). The matrix S diagonalizes A; that is S 1 AS ¼1 C 0 ða i a 0 iS i ¼ÞþF diagðl i ðAÞÞ: condðSÞ ¼kS 1 kkSk is the condition numberof S: Since the eigenvectors of A are not closelyi;1 þ C 0 b i Y iparallel, the condition number of S cannot be high, andS 0 i ¼therefore the estimation is not ill-conditioned (Demmel,B 2 0 C 0ð1 eÞða 0 i a i Þb i ½eðb j b 0 j Þ b jð1 þ C 0 b 0 j ÞŠ þ F 1997). Using the euclidean normfor B it can be showni;thatY iðA:1Þ j l i ðAÞ l i ðA e Þjpffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwhere i denotes either the host or the symbiont. F i and pcondðSÞ 2 B0 max b 2 h þ b02 h;b 2 s þ b02 s e: ðB:1ÞY i areF i ¼ B 2 0 b iC 0 ð1 eÞða 0 i a i Þ½eðb j b 0 j Þ b j C 0 b j b 0 j Šþð1 þ C 0 b 0 j Þb jB 0 ðe 1Þ ½C 0 a j B 0 a i ðð1 eÞþea 0 i Šþð1 þ Thus the condition of j l i ðAÞ jo1 gives aC 0 b i Þ½B 0 a j ðe 1Þ Bea 0 j þ C 0a 0 i ÞŠð1 þ C 0b 0 j Þ;good estimation for j l i ðA e Þj to be smaller than one,Y i ¼ð1 þ C 0 b 0 i Þb iB 2 0 ð1 eÞ½eb0 j þ b jð1 eÞþC 0 b 0 j Šþ which guarantees the stability of the fixed points. Itð1 þ C 0 b i Þ½B 2 0 ð1 eÞeb jb 0 i ð1 þ C 0 b 0 j Þþð1 þ C follows fromsimple calculations that j l i ðAÞ jo1 is0b j ÞðB 2 0 e2 b 0 j b0 i ð1 þ C 0 b 0 j Þð1 þ C 0b 0 valid if b 0 hiÞÞŠ; where the i,j indexeso1=C 0; b 0 S o1=C 0; and 241 þ b h b s ðC 2 0B 2 0denote different species.Þ4ðb h þ b s ÞC 0 ; which is true if b i and b 0 i are smallenough.To determine the adaptive dynamics of a i and b iwe have to differentiate the fixed points within respectto these variables. This operation yields anothercomplex expression, which can be simplified by usingReferencesthat e51: Considering the linear approximation of thederivatives in the function of e we arrive at Eq. (4). The Altmann, S., 1979. Altruistic behavior: the fallacy of kin development.calculations have been performed by the software Anim. Behav. 27, 958–959.Mathematica 3.0.Axelrod, R., Hamilton, W.D., 1981. The evolution of cooperation.Science 211, 1390–1396.van Baalen, M., Rand, D.A., 1998. The unit of selection in viscouspopulations and the evolution of altruism. J. Theor. Biol. 193,Appendix B. The stability of the fixed point631–648.Boucher, D.H., James, S., Keeler, K.H., 1982. The ecology ofThe analysis of the stability of the fixed points in the mutualism. Ann. Rev. Ecol. System. 13, 315–347.Bronstein, J.L., 2001a. Mutualism. In: Fox, C., Fairbairn, D., Roff, D.two-species case is not as simple as in the one species(Eds.), Evolutionary Ecology, Perspectives and Synthesis. Oxfordcase (Killingback and Doebeli, 2002), but is still University Press, Oxford, pp. 315–330.tractable. A e ; the Jacobian matrix of (3) can be Bronstein, J.L., 2001b. The exploitation of mutualisms. Ecol. Lett. 4,considered as A þ eB; where277–278.01Demmel, J., 1997. Applied Numerical Linear Algebra. SIAM,C 0 b h 0 B 0 b s 0Philadelphia.0 C 0 b 0 h B 0 b s 0Doebeli, M., Knowlton, N., 1998. The evolution ofinterspecific mutualism. Proc. Natl Acad. Sci. USA 95,A ¼;B@ B 0 b h 0 C 0 b s 0 C8676–8680.AGeritz, A.A.H., Kisdi, E., Meszena, G., Metz, J.A.J., 1998. Evolutionarysingular strategies and the adaptive growth and branchingB 0 b h 0 0 C 0 b 0 s00 0 B 0 b s B 0 b 0 1of the evolutionary tree. Evol. Ecol. 12, 35–57.sHofbauer, J., Sigmund, K., 1990. Adaptive dynamics and evolutionary0 0 B 0 b s B 0 b 0 stability. Appl. Math. Lett. 3, 75–79.sB ¼:Kato, T., 1980. Perturbation Theory for Linear Operators. Springer,B@ B 0 b h B 0 b 0 h 0 0 CBerlin, Heidelberg, Tokyo.AKillingback, T., Doebeli, M., 2002. The continuousB 0 b h B 0 b 0 h 0 0prisonner’s dilemma and the evolution of cooperation throughreciprocal altruism with variable investment. Am. Nat. 160,To determine the eigenvalues of the Jacobian matrix is a 421–438.hopeless task, but since e51 it can be considered as a Killingback, T., Doebeli, M., Knowlton, N., 1999. Variableperturbation of matrix A: The eigenvalues of A are investment, the continuous prisonner’s dilemma, and the originqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC 0 b 0 h ; 1 2 ð C 0ðb h þ b s Þ 4ðB 2 0 C 2 0 ÞþC2 0 ðb h þ b s Þ 2of cooperation. Proc. Roy. Soc. London B, Biol. Sci. 266,Þ; 1723–1728.qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12 ð C 0ðb h þ b s Þþ 4ðB 2 0 C 2 0 ÞþC2 0 ðb h þ b s Þ 2 Þ; C 0 b 0 Maynard Smith, J., 1982. Evolution and the Theory of Games.s : Cambridge University Press, Cambridge.
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