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Cooperation in defence against a predator J. József Garay To cite this version: J. József Garay. Cooperation in defence against a predator. Journal of Theoretical Biology, Elsevier, 2009, 257 (1), pp.45. 10.1016/j.jtbi.2008.11.010. hal-00554534 HAL Id: hal-00554534 https://hal.archives-ouvertes.fr/hal-00554534 Submitted on 11 Jan 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Author’s Accepted Manuscript Cooperation in defence against a predator József Garay PII: S0022-5193(08)00603-6 DOI: doi:10.1016/j.jtbi.2008.11.010 Reference: YJTBI5368 To appear in: Journal of Theoretical Biology www.elsevier.com/locate/yjtbi Received date: 2 April 2008 Revised date: 14 November 2008 Accepted date: 14 November 2008 Cite this article as: József Garay, Cooperation in defence against a predator, Journal of Theoretical Biology (2008), doi:10.1016/j.jtbi.2008.11.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. 1 Cooperation in Defence against a Predator 2 3 József Garay 4 Research Group of Theoretical Biology and Ecology of Hungarian 5 Academy of Sciences and Department of Plant Taxonomy and Ecology, 6 L. Eötvös University, Pázmány P. sétány 1/C H-1117 Budapest, Hungary 7 8 Abstract 9 The origin and the evolutionary stability of cooperation between 10 unrelated individuals is one of the key problems of evolutionary biology. 11 In this paper, a cooperative defence game against a predator is introduced 12 which is based on Hamilton’s selfish herd theory and Eshel’s survival 13 game models. Cooperation is altruistic in the sense that the individual, 14 which is not the target of the predator, helps the members of the group 15 attacked by the predator and during defensive action the helper individual 16 may also die in any attack. In order to decrease the long term predation 17 risk, this individual has to carry out a high risk action. Here I show that 18 this kind of cooperative behaviour can evolve in small groups. The reason 19 for the emergence of cooperation is that if the predator does not kill a 20 mate of a cooperative individual, then the survival probability of the 21 cooperative individual will increase in two cases. If the mate is non- 22 cooperative, then – according to the dilution effect, the predator 23 confusionAccepted effect and the higher pred manuscriptator vigilance –, the survival 24 probability of the cooperative individual increases. The second case is 25 when the mate is cooperative, because a cooperative individual has a 26 further gain, the active help in defence during further predator attacks. 27 Thus, if an individual can increase the survival rate of its mates (no 28 matter whether the mate is cooperative or not), then its own predation risk 29 will decrease. 1 30 31 Key words: altruism, confusion effect, cooperation, dilution effect, ESS, 32 group augmentation, selfish herd, survival game, stakeholder altruism. 33 34 35 1. Introduction 36 37 A prerequisite to the emergence of cooperation is that individuals live 38 together. The „selfish herd” theory of Hamilton (1971) claims that the 39 evolutionary force behind the formation of a group is predation, since the 40 best shelter is in the centre of a group during each predator’s attack if 41 there is no refuge nearby. In other words, individuals reduce their 42 predation risk at the expense of others. Thus, it is natural to investigate 43 the emergence of altruistic defence against the predator in a “selfish 44 herd”. Clearly, altruistic defence against a predator increases the survival 45 probability of the attacked individual and decreases that of the helper 46 individual (Wilson 1975). For instance, alarm call is considered as one of 47 altruistic traits which benefit other group members at some cost to the 48 caller (Krebs & Davies 1978 p.142; Edmunds 1974, p. 254). Another 49 example is mobbing (Krebs & Davies 1978 pp 147; Edmunds 1974, p. 50 204) when the prey individuals must approach the predator within 51 dangerous distance. Moreover, active frightening behaviour is also well- 52 known whenAccepted the prey attack the predator manuscript (Edmunds 1974, p. 151). For 53 example, musk oxen (Ovibus moschatus) form defensive ring when 54 treated by a pack of wolves, and the cows and calves are in the centre of 55 the ring. Furthermore, elephants (Loxodonta africana) actively defend 56 themselves and their young against lions (Panthera leo) whereas buffalo 57 (Syncerus caffer) can kill lions (Hayward & Kerley 2005). 2 58 Eshel & Weinshall (1988) considered a sequence of survival games 59 with two players. The rules randomly changed from one encounter to 60 another. There were two types of games: the first one was a prisoner’s 61 dilemma game in which defection is the best strategy, while in the second 62 game cooperation is of immediate self reward. They showed that there are 63 cases when total cooperation is the best behaviour even in the prisoner’s 64 dilemma as well, since the opponent (if alive) will cooperate in the future. 1 65 Eshel & Shaked (2001) found that partnership may evolve in a random 66 supergame of survival. Supergame contains a continuum of events, in 67 each event each pair of individuals plays a prisoner dilemma game, in 68 which the pay-off matrix contains the survival probabilities which change 69 according to a time independent distribution. The survival rate of single 70 individuals may be higher than that of a non-fully cooperative pair. The 71 pay-off function is the life time of each individual. Eshel & Shaked 72 (2001) (using a little bit similar method to the well-known inclusive 73 fitness method), found that in a class of supergame cooperation can 74 evolve. 75 The aim of this paper is to “combine” Hamilton’s and Eshel et al.’s 76 ideas. For this purpose, the main assumptions of these theories must be 77 combined. In Hamilton’s scenario, the predator attack is asymmetric in 78 the sense that only one individual from a huge herd is attacked. There is 79 no such kind of asymmetry between two players in Eshel’s models. 80 HamiltonAccepted considered only one predator manuscript attack, while Eshel considered a 81 continuum of events. In the situation considered here, a predator can 82 attack only one individual at a time and each individual must survive an 83 average number of attacks. 1 Partnership is a relation between at least two individuals when it is their best interest to help each other since by doing so they increase the probability of being together in the future and they continue to help each other. 3 84 Let us consider a large enough asexual population, in which 85 unrelated individuals have no information on their mate’s behaviour 2 86 during random group formation. Every individual has the same ability to 87 defend itself against the predator, to recognize the stalking predator and 88 has the same payoff functions. They all have a genetically fixed (perhaps 3 4 89 mixed) behaviour phenotypes and no memory . Furthermore, in the 90 biological case considered here there is neither social nor dominant 5 91 structure. 92 Naturally, the size of the group determines defensive ability. If the 93 size of the group decreases then the defensive ability of the group also 94 decreases, thus the survival probability of any member of the group also 95 decreases (Wilson 1975). In this paper, the simplest possible case is 96 considered in which only two individuals constitute each randomly 97 formed group and when a member of a pair dies then new pair formation 98 from the survived members is not possible. 99 From a mathematical point of view, I use elementary stochastic 100 methods to calculate the average survival probabilities and then I consider 101 this survival rate as pay-offs. Thus, my approach is a deterministic one, 102 for instance, the stochastic fixation process of any strategy is not 103 investigated. 104 2 Under theseAccepted assumptions, kin selection (Hamilton manuscript 1964), group selection (Nowak 2006) and group formation on phenotypic similarity (Burtsev & Turchin 2006) cannot be used. 3 Since the genetically fixed behaviour is not conditional −i.e. an individual cannot change its behaviour according to the behaviour of others− neither direct (Brandt & Sigmund 2004) nor indirect reciprocity (Ohtsuki & Iwasa 2004; Nowak & Sigmund 2005) can be applied. 4 Thus, punishment is not possible in the present model (Fehr & Gachter 2002; Rockenbach & Milinski 2006). 5 The altruistic defence is not in order to allow an individual to stay in the group, and altruism is not considered as a result of the “pay-to-stay” hypothesis (Kokko & Johntone 2002). 4 105 2. One-attack game 106 Now let us describe an attack by a predator.
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