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1 City Researc Online City, University of London Institutional Repository Citation: Broom, M., Luter, R. M., Ruxton, G. D. & Ryctar, J. (008). A game-teoretic model of kleptoparasitic beavior in polymorpic populations. Journal of Teoretical Biology, 55(1), pp doi: /j.jtbi Tis is te accepted version of te paper. Tis version of te publication may differ from te final publised version. Permanent repository link: ttp://openaccess.city.ac.uk/978/ Link to publised version: ttp://dx.doi.org/ /j.jtbi Copyrignd reuse: City Researc Online aims to make researc outputs of City, University of London available to a wider audience. Copyrignd Moral Rigts remain wit te autor(s) and/or copyrigt olders. URLs from City Researc Online may be freely distributed and linked to. City Researc Online: ttp://openaccess.city.ac.uk/ publications@city.ac.uk

2 A game-teoretic model of kleptoparasitic beavior in polymorpic populations Mark Broom a, Roger M. Luter a Graeme D. Ruxton b Jan Ryctář c a Department of Matematics, University of Sussex, Brigton BN1 9RF, UK b Institute of Biomedical and Life Sciences, University of Glasgow, Glasgow G1 8QQ c Department of Matematics and Statistics, University of Nort Carolina Greensboro, Greensboro, NC 740, USA. Abstract Kleptoparasitism, te stealing of food by one animal from anoter, is a widespread biological penomenon. In tis paper we build upon earlier models to investigate a population of conspecifics involved in foraging and, potentially, kleptoparasitism. We assume tat te population is composed of four types of individuals, according to teir strategic coices wen faced wit an opportunity to steal and to resisn attack. Te fitness of eac type of individual depends upon various natural parameters, for example food density, te andling time of a food item and te probability of mounting a successful attack against resistance, as well as te coices tat tey make. We find te Evolutionarily Stable Strategies (ESSs) for all parameter combinations and sow tat tere are six possible ESSs, four pure and two mixtures of two strategies, tat can occur. We sow tat tere is always at least one ESS, and sometimes two or tree. We furter investigate te influence of te different parameters on wen eac type of solution occurs. Key words: Kleptoparasitism, ESS, Game teory, Strategy 1. Introduction In tis paper we sall investigate a model of kleptoparasitism, te stealing by one animal of food tat as been caugt by anoter (Rotscild and Clay 195). Te most Corresponding autor. addresses: M.Broom@sussex.ac.uk (Mark Broom), R.Luter@sussex.ac.uk (Roger M. Luter), G.Ruxton@bio.gla.ac.uk (Graeme D. Ruxton), ryctar@uncg.edu (Jan Ryctář). 1 Te researc was initiated wen Mark Broom visited UNCG. Te visit was supported by te UNCG New Faculty Grant eld by Jan Ryctář. Te contribution of Roger Luter was carried out wile e was a D.Pil student supported by te EPSRC. Preprint submitted to Elsevier 31 July 008

3 common observations of kleptoparasitism ave been amongst birds. An extensive review (Brockmann and Barnard 1979) gives a list of observations of kleptoparasitism by birds. It is noted tere tat tis beavior is muc more common in some orders of birds tan oters, and it is especially prevalenmongst sea-birds. Observations include te sigting of gulls attacking eac oter for food (Steele and Hockey 1995), oystercatcers feeding on cockles (Triplet el 1999) and skuas attacking albatrosses and giant-petrels (Spear el 1999). It sould be noted tat kleptoparasitism as been observed in many oter types of animals as well, including insects (Jeanne 197), fis (Grimm and Klinge 1996) and mammals (Kruuk 197). Tere is now a substantial literature of works using game teoretic models to investigate kleptoparasitic beavior in nature (eg Barnard and Sibly 1981; Stillmann el 1997; Broom and Ruxton 1998; Ruxton and Broom 1999; Broom and Ruxton 003). In one of te most recent papers (Broom el 004), te generality of te original model of Broom and Ruxton (1998) was expanded in two key ways: allowing flexibility in te likeliood tan attacker will be able to successfully steal a prey item from a andler, and allowing attacked individuals te flexibility to surrender items witou time-consuming contest. Tis sowed tat (depending on te values given to ecological variables) tree different types of ESSs were possible: one were individuals bot attacked oters for food items and resisted attacks from oters (Hawk), one were individuals attacked but did not resist (Marauder), and one were individuals did nottack, but would resist if temselves attacked (Retaliator). Furter, in some circumstances, more tan one of tese ESS s was possible as alternates, depending on te istory of te system as well as its current parameter values. Te Marauder ESS is particularly interesting ecologically, giving an economic explanation for one individual to surrender a valuable food item witou figt to anoter individual in te absence of dominance ierarcies or intrinsic asymmetries in competitive abilities between individuals.. Te model, beavioral stages and strategies Te basic structure of our model follows tat of Broom and Ruxton (1998). Individuals forage for food, and can be in one of four beavioral stages. Tey are eiter a searcer (looking for food), a andler (preparing to consume food it as found) an attacker (trying to steal te food item from a andler it as found) or a resister (trying to resist te attack of anoter). We assume tat individuals take an exponential time to andle a food item, and tat te food is consumed in no time at te end of tis period. Note tat Broom and Ruxton (003) considered andling times wic were constannd different food types, were some types were consumed instantaneously at te end of te andling period as in tis paper, but oters were consumed continuously. One consequence of tis is tat te time spent andling a food item was of great importance, and tis led to rater different beavior to te oter models. In particular noll kleptoparasitic coices were identical; tere was a tresold were opportunities were only taken if sufficiently little andling ad been done (continuous consumption) or if sufficiently long andling time ad elapsed (instantaneous consumption). We consider a polymorpic population consisting of te four different bird types introduced in Broom el (004), (see also Broom and Ryctář, 007; Luter al, 007). A bird s type is determined by its strategy, i.e. by te reaction of an individual to an

4 encounter wit anoter, were one of te two birds is andling food, and te oter as an opportunity to try to steal it. Te four strategies are: Hawk Dove Retaliator Marauder always attack, always resist wen attacked never attack, never resist wen attacked never attack, always resist wen attacked always attack, never resist wen attacked Eac bird is initially searcing for food items. We assume tat Doves and Retaliators find food at rate ν f f (were f is te density of food) as opposed to Hawks and Marauders wo find food at rate ν g f. Wen food is acquired te individual becomes a andler, te andling time of te food item following an exponential distribution wit mean t, after wic it is instantaneously consumed. Tus te food item can be tougt of as an animal tat as to be removed from a sell, but tat te time it takes to extract te animal is unpredictable, and variable from item to item. Hawks and Marauders are searcing for andlers as well (at rate ν ). Hawks and Marauders may tus ave to divide teir attention between te two searces and it is possible tat because of tis tat ν g < ν f. We sall in any case consider ν g and ν f to be potentially different, and will also consider te case were ν g ν f. Wen Marauder or Hawk find a andler, tey attack and try to steal its food. If tey encounter Dove or Marauder, te andler surrenders te food item. If tey encounter Retaliator or Hawk, te andler resists te attack and tus bot te searcer and te andler engage in a figt. Tere are potentially many different types of costs of te figt (injury, energy loss, time loss). In our model we assume tat te only cost is te time spent in te contest. Te figts take a random time wit exponential distribution wit mean ta. Te attacker wins te figt wit probability α (0, 1). Let P be te total density of te population and H d, D d, M d, R d be te densities of Hawks, Doves, Marauders and Retaliators, respectively. Every individual goes troug a searcing and andling period, Hawks and Marauders may be involved in figts as attackers and Hawks and Retaliators may be involved in figts as resisters. Let H s, H, H a, H r denote te densities of Hawks in searcing, andling, attacking and resisting stages. Te corresponding notation is used for oter types, see Table 1. We assume te total density of te population is constannd denote by P s, P, P a, P r te densities of te population involved in searcing, andling, attacking and resisting. Tus P = H d + D d + M d + R d (.1) P s = H s + D s + M s + R s (.) P = H + D + M + R (.3) P a = H a + M a (.4) P r = H r + R r (.5) We also assume tat te fitness of an individual is proportional to its uptake rate, te inverse of te total consumption time (i.e. finding te food item and eating it, including all possible interruptions by oters). Te following equation can be found in Broom and Ryctář (007) and determines te total consumption time T for Hawks in terms of its inverse andling ratio H d /H (a similar equation olds for oter strategies). 3

5 Parameter P P s, P, P a, P r meaning density of te population density of searcers, andlers, attackers and resisters D d, R d, H d, M d density of Doves, Retaliators, Hawks and Marauders D s, R s, H s, M s density of searcing Doves, Retaliators, Hawks and Marauders D, R, H, M density of andling Doves, Retaliators, Hawks and Marauders H a, M a R r, H r r f ν f ν g ν t α density of attacking Hawks and Marauders density of resisting Retaliators and Hawks andling ratio H /P in a population of Hawks only density of food items area Doves and Retaliators can searc for food per unit time area Hawks and Marauders can searc for food per unit time area Hawks and Marauders can searc for andlers per unit time expected time to consume a food item (if undisturbed) expected duration of a figt contest over food probability tat te attacker wins te figt Table 1 Te model parameters and notation. T = t Hd H. (.6) Te uptake rate is tus directly proportional to te proportion of time tat eac type of individual spends in te andling stage (see Broom and Ruxton, 1998, amongst oter papers). Tis means tat, te sorter te consumption time, te iger te fitness. We sall consider eac of te strategies in turn and evaluate its uptake rate in te mixed population. We sall investigate wic mixtures are evolutionarily stable. A mixture is evolutionarily stable if a) all birds present in te mixture ave equal fitness (i.e. teir inverse andling ratio is te same); b) all birds not present in te mixture would ave smaller fitness if tey were only present in very small numbers (wit a density approacing 0); and c) increasing te proportion of one strategy by a small amount lowers its fitness relative to te oter strategy, wereas decreasing te proportion increases its fitness relative to te oter strategy. 3. Equilibrium equations for te dynamical system In tis section we will consider eac strategy and evaluate its inverse andling ratio, from wic its consumption time and ence uptake rate can be found. Transitions between te states follow te scematic description given in Figure 1, wic ten translates into te differential equations (3.1), (3.), (3.5), (3.6), (3.7), (3.10), (3.11), (3.1), (3.15), (3.16), (3.17), (3.18) described and developed later in tis section. 4

6 Fig. 1. Pase diagrams for a) Doves, b) Retaliators, c) Marauders, d) Hawks. We sall assume tat te populations converge to equilibrium exponentially fast, and we tus concentrate on tese equilibrium values only. For te model of Broom and Ruxton (1998), tis fast convergence was sown in Luter and Broom (004). Te proof of Luter and Broom (004) also works for te special case of a single strategy population for eac of te four strategies of our model. We believe tat tis result olds more generally for our situation wit multiple strategies, and tis as certainly proved te case in simulations. Te parameter values tat we ave cosen are plausible for real populations wen te time units are minutes; a sample set of simulated solutions in Figure sow convergence witin ten minutes from a population initially composed of searcers, as is reasonable at te start of a foraging period (e.g. at te start of a new day) Doves Doves can go troug searcing and andling stages only. If a Dove is searcing, it can become a andler if it finds a food item (wit te rate ν f f). If te Dove is a andler, it can become a searcer if it finises andling (wit rate t 1 ) or is found by a searcing Hawk or Marauder (wit te rate ν (H s + M s )). Tis provides te following set of equations. 5

7 Fig.. Fast convergence of solutions of equations (3.1), (3.), (3.5), (3.6), (3.7), (3.10), (3.11), (3.1), (3.15), (3.16), (3.17), (3.18). Te parameter values are t = 1, ν = 1, α = 0.5, ν f f = 1, ν gf = 0.8. All individuals start originally as searcers wit D s = R s = M s = H s = 10. Black lines are for = 0.3, gray line for = 1, ligt gray line for =. d dt D s = t 1 D + ν (H s + M s )D ν f fd s (3.1) d dt D = t 1 D ν (H s + M s )D + ν f fd s (3.) D d = D s + D (3.3) In te equilibrium, bot sides of te equations (3.1) and (3.) are equal 0 wic togeter wit (3.3) provides D d = D t ν f f + ν (H s + M s ) (3.4) ν f f 3.. Retaliators Retaliators can go troug searcing, andling and resisting stages. If a Retaliator is searcing, it can become a andler if it finds food (wit rate ν f f). If te Retaliator is andling, it can become a searcer if it finises andling (wit rate t 1 ) a resister if it is found by a searcing Hawk or Marauder (wit rate ν (H s + M s )). If te Retaliator is resisting, it can become 6

8 a andler if it wins te figt (wit rate (1 α)( ta ) 1 ), a searcer if it loses te figt (wit rate α( ta ) 1 ). Tis provides te following set of equations (already assumed to be in te equilibrium conditions). 0 = d dt R s = ν f fr s + t 1 R + αt 1 a R r (3.5) 0 = d dt R = t 1 R ν (H s + M s )R + ν f fr s +(1 α)t 1 a R r (3.6) 0 = d dt R r = t 1 a R r + ν (H s + M s )R (3.7) R d = R s + R + R r (3.8) By (3.8), (3.5) and (3.7), R d = (t 1 R ν f f + αν (H s + M s ) ) + ν (H s + M s ) (3.9) 3.3. Marauders Marauders can go troug searcing, andling and attacking stages. If a Marauder is searcing, it can become a andler if it finds food (wit te rate ν g f) or a andling Dove or Marauder (wit te rate ν (D + M )), an attacker if it finds a andling Hawk or Retaliator (wit te rate ν (H + R )). If te Marauder is andling, it can become a searcer if it finises andling (wit rate t 1 ) or is found by a searcing Hawk or Marauder (wit rate ν (H s + M s )), If te Marauder is attacking, it can become a andler if it wins te figt (wit rate α( ta ) 1 ), a searcer if it loses te figt (wit rate (1 α)( ta ) 1 ). Tis provides te following set of equations. 0 = d dt M s = ν P M s ν g fm s + ( t 1 0 = d dt M = (t 1 + ν (H s + M s ) ) M +(1 α)t 1 a M a (3.10) + ν (H s + M s ))M + (ν g f + ν (D + M )) M s +αt 1 a M a (3.11) 0 = d dt M a = t 1 a M a + ν (H + R )M s (3.1) M d = M s + M + M a (3.13) By (3.1), M a M = M a M s Ms M = ν (H + R ) Ms M 7

9 and by (3.10) M s t 1 = + ν (H s + M s ) M ν P + ν g f (1 α)ν (H + R ). Tus, M d t 1 = ν ( (H s + M s ) M ν P + ν g f (1 α)ν (H + R ) 1 + ν (H + R ) t ) a. (3.14) 3.4. Hawks Hawks can go troug four different stages - searcing, andling, attacking, and resisting. If a Hawk is searcing, it can become a andler if it finds food (wit rate ν g f) or a andling Dove or Marauder (wit rate ν (D + M )), an attacker if it finds a andling Hawk or Retaliator (wit rate ν (H + R )). If te Hawk is andling, it can become a searcer if it finises andling (wit rate t 1 ), a resister if it is found by a searcing Hawk or Marauder (wit rate ν (H s + M s )). If te Hawk is attacking, it can become a andler if it wins te figt (wit rate α( ta ) 1 ), a searcer if it loses te figt (wit rate (1 α)( ta ) 1 ). If te Hawk is resisting, it can become a searcer if it loses te figt (wit rate α( ta ) 1 ), a andler if it wins te figt (wit rate (1 α)( ta ) 1 ). It gives te following set of equations. 0 = d dt H s = ν P H s ν g fh s + t 1 H + (1 α)t 1 a H a +αt 1 a H r (3.15) 0 = d dt H = ν (H s + M s )H t 1 H + (ν g f + ν (M + D )) H s +αt 1 a H a + (1 α)t 1 a H r (3.16) 0 = d dt H a = t 1 a H a + ν (H + R )H s (3.17) 0 = d dt H r = t 1 a H r + ν (H s + M s )H (3.18) H d = H s + H + H a + H r (3.19) By (3.18) by (3.17) and by (3.16) H r H = ν (H s + M s ) ; H a H = H a H s Hs H = ν (H + R ) Hs H, H s H = + αν M s. (3.0) ν g f + ν (D + M ) + αν R t 1 8

10 Tus, by (3.19), H d H = 1 + t 1 + αν M s ν g f + ν (D + M ) + αν R +ν (H s + M s ) ( 1 + ν (H + R ) t ) a (3.1) Note tat by using equation (3.15) rater tan (3.16) in te above process we get te following expression equivalent to (3.1) H d t 1 = αν ( (H s + M s ) H ν g f + ν (D + M ) + αν (H + R ) 1 + ν (H + R ) t ) a +ν (H s + M s ) 4. General comparisons (3.) We can use te inverse andling ratios from te previous sections to establis some conditions wen a given strategy as (or as not) an advantage againsnoter strategy in a general population mixture. Since all stable population mixtures require te total consumption times of all strategies involved in te mixture to be identical, tis in turn will sow wen te mixture can be invaded by te strategy in question. We consider pairs of strategies in turn, comparing te inverse andling ratio from te previous section. In eac case we give te condition for te first strategy to be better tan te second one (te condition for te second strategy to be better is just te reverse of tis condition). By (3.4) and (3.9), Dove beats Retaliator if 1 α < ν f f (4.1) (unless H s + M s = 0 wen bot always do equally well). By (3.4) and (3.14), Dove beats Marauder if ν f f ν g f ν > D + M + (H + R ) ( α ν f f t ) a. (4.) By (3.14) and (3.), Marauder beats Hawk if ν ( gf (1 α) + ν D + M + (H + R )(α 1) ) > 0 (4.3) (unless H s + M s = 0 wen bot always do equally well). Using (3.9) and (3.1), Retaliator beats Hawk under exactly te same conditions as Dove beats Marauder and in any population mixture Hawks ave iger payoffs tan Retaliators if and only if Marauders ave a iger payoff tan Doves. Expressed in formulae tis is H > R M > D H d R d M d D d For Hawk to be an ESS it must ave a iger payoff tan Retaliator, Dove or Marauder i.e. ( H R > max, D, M ) H d R d D d M d 9

11 Clearly tis requires H > R H d R d wic in turn implies M > D M d D d i.e. ( H R > max, D, M ) H ( R > max, M ) H d R d D d M d H d R d M d Hawk is tus an ESS if and only if it cannot be invaded by Marauder or Retaliator, and consideration of invasion by Dove is superfluous. Similarly wenever Dove can be invaded by Hawk it can be invaded by one of te oters. Te same argument works for Marauder and Retaliator, so tat wenever Retaliator is invaded by Marauder, or Marauder is invaded by Retaliator, te strategy will also be invaded by eiter Hawk or Dove. It is tus never necessary to consider invasion by te opposite strategy (were bot attacking and resisting beavior differ). 5. Evolutionarily stable mixtures We sall now consider te various possible population mixtures in turn. Tere are 15 possibilities to consider (eac of Hawk, Marauder, Retaliator and Dove can eiter be included in te population or not, excluding te case were none are present). Some of tese results can be obtained by revisiting te paper Luter el (007). Tat paper considered two populations of individuals from te same species, one of wic was a pure forager, and te oter foraged bulso indulged in kleptoparasitism (at some cost to its foraging ability). Te potential kleptoparasite could attack or not (altoug it could not tell wic type its potential victim was) and eiter type could coose to resist any attack or not. Under certain circumstances evolution eliminated one of te groups to leave a single population, in oters a mixture of te two groups survived. Tis actually incorporates our situation wenever tere are exactly two different strategies in a mixture one of wic is a kleptoparasite (Marauder or Hawk) and one of wic is not (Dove or Retaliator), or if tere is jus single strategy. Tus from te paper we can directly obtain some of te following results (for ν f > ν g ) Pure ESS s In tis section we will ask wen a population consisting of a single strategy can or cannot be invaded Dove as an ESS Strictly speaking, Dove is never an ESS, because it can always be invaded by Retaliator (by drift). Once we allow a constant presence of a small amount of Hawks and/or Marauders, we get tat Retaliator cannot invade Dove as long as (4.1) olds: 1 α < ν f f 10

12 (i.e. if tere is enoug food or figts are too long). By (4.), Dove can not be invaded by Marauder if Since, by (3.4), for D d P, we get tat Dove is an ESS if D < ν f f ν g f ν. D = t ν f f t ν f f + 1 P, 1 α < ν f f and P < ν f f ν g f tν f f + 1. (5.1) ν t ν f f In particular, pure Dove can be an ESS only if ν g f < ν f f, i.e. Doves find food faster tan t Hawks or Marauders. And even ten, only if ν P ν f f t ν f f+1 is smaller tan ν f f ν g f, i.e. te rate tat Hawks and/or Marauders find andling Doves is smaller tan te Dove s advantage in finding food. Anoter interpretation is tat tis only occurs in sufficiently low density populations (suc populations do not give kleptoparasitic invaders a cance of finding a andler very fast) Retaliator as an ESS Similarly to te Dove situation, pure Retaliator can never be an ESS because it can always be invaded by Doves by drift. Neverteless, wen we allow a small presence of Hawks and/or Marauders, we get (as above) tat Retaliator cannot be invaded by Doves if and only if te condition (4.1) does not old, i.e. ν f f < 1 α. Wen R d P we get from (3.14) and (3.1) tat M d M = H d H. As discussed in section 4, Retaliator beats Hawk if and only if Dove beats Marauder. Tus, by (4.), one needs or Since R = R ( α ν f f t ν f f t ν f f+1p, Retaliator is an ESS if ) < ν f f ν g f ν. ν f f < 1 α, ν f f < α, P < ν f f ν g f ν 1 α ν f f ta tν f f + 1 t ν f f ν f f < 1 α, ν f f > α, P > ν gf ν f f 1 ν ν f f ta α tν f f + 1 t ν f f (5.) (5.3) Te first condition says tat if figts are sort or tere is not enoug food, ten Retaliator is an ESS in a low density population only (sort figts are good for Marauders or Hawks; not enoug food increases te searcing time, i.e. it, relatively, decreases te figt time.) Moreover, Retaliator is an ESS only if ν f f > ν g f, i.e. if kleptoparasitic birds ave a disadvantage. Te second condition is possible only for α < 0.5, i.e. wen resisters ave an advantage against teir attackers. Retaliator is ten an ESS in sufficiently ig density populations and wen te figts do not take too muc time. Long figts would mean tat Doves do 11

13 better. A ig density population is needed because kleptoparasites are forced to engage in figts and tus lose time wile te substantial remainder of te Retaliator population can arvest te food undisturbed. In tis sense, te denser te population, te better for te Retaliators Marauder as an ESS In te population of Marauders only (M d = P ), tere are no figts, so M a = 0 and consequently te equation (3.10) reduces to Tus, and M = P 0 = d dt M s = ν g fm s + t 1 M. M s = 1 M t ν g f, M d = M t ν g f t ν g f 1 + t ν g f, M 1 s = P 1 + t ν g f. By (4.), Marauder cannot be invaded by Doves if M > ν f f ν g f ν, i.e. if P > ν f f ν g f 1 + t ν g f. (5.4) ν t ν g f Marauders cannot be invaded by Hawks if, by (4.3), ν M > 1 α ν gf, i.e. if P > ( 1 α t ) a ν ν gf tν g f + 1 (5.5) t ν g f We do not need to consider te invasion by Retaliators since if Marauder is invaded by Retaliator it will also be invaded eiter by Dove or Hawk as discussed in Section 4. Conditions (5.4) and (5.5) togeter give or P > ( max { (1 α) Tus, pure Marauder is an ESS if eiter ν f f } ), ν f f ν g f tν g f + 1 ν t ν g f ν f f > 1 α, and P > ν f f ν g f 1 + t ν g f ν t ν g f ( < 1 α, and P > 1 α t ) a ν ν gf (5.6) (5.7) tν g f + 1. (5.8) t ν g f Te distinction between te two conditions lies in te relationsip between ν f f ta and 1 α. If ν f f ta > 1 α, it means a) figts are long and b) it is not beneficial to resist. Tus, Marauders cannot be invaded by Hawks nor by Retaliators. Marauders cannot be invaded by Doves if te population is dense enoug (no matter ow big te disadvantage 1

14 of kleptoparasites) because in te population were everybody steals, Doves lose teir food very fast but do nocquire it back by stealing, as Marauders do. If ν f f ta < 1 α, resisting te attack is beneficial. Yet, wen te population is dense enoug, ten Hawks cannot invade Marauder (tey would be doomed to resist too muc) and Retaliators cannot invade Marauders for te same reason Hawk as an ESS Wen Hawk is te only type in te population, by (3.17) and (3.18), H a = ν H sh = H r and, by (3.16), H = ν g ft H s. By (3.19), we can evaluate H in te population of Hawks only. As done already in Ruxton and Moody (1997), H = P r, were r, te andling ratio, is te positive root of te quadratic equation Tis yields r ν P + r (1 + ν g ft ) ν g ft = 0. (5.9) H = r P = 1 + ν gft ν ( ν t P ν g f (1 + ν g ft ) ), (5.10) wic can be later used for determining conditions on P from conditions on H. Retaliator cannot invade Hawk only if Dove cannot invade Marauder, i.e., by (4.) if ( ν f f ν g f < H α ν f f t ) a, ν i.e. if ν f f < α and P r > ν f f ν g f ν or ν f f > α and P r < ν gf ν f f ν 1 α ta ν f f, 1 ν f f α. By (4.3), Hawk cannot be invaded by Marauder if i.e. if ν gf (1 α) < H ν (1 α), α < 1 and P r > ν gf (1 α)/ ν or α > 1 and P r < ν gf (1 α)/ ν 1 1 α, 1 1 α. By results of Section 4, if Doves invade Hawks ten eiter Marauders or Retaliators invade Hawks so te above are te only conditions we need. 13

15 5.. Mixed ESS s We reiterate tat in te current work a mixed ESS (or a mixture) refers to a population of different individuals wo eac play a unique pure strategy, and not to individuals wo are capable of playing more tan one strategy Marauders and Doves A mixture of Marauders and Doves occurs if Doves can invade Marauders, Marauders can invade Doves and neiter Hawks or Retaliators can invade. From te general invasion conditions, since Doves and Marauders do equally well, Retaliators and Hawks do equally well and so it is enoug to consider te invasion of Doves by Retaliators only. Tis is equivalent to finding wen it is optimal to not resisttacks in suc a mixture. Te inequalities (4.1) and (4.) (te second one must old for M d P and must not old for D d P ) yield te conditions ν f f > 1 α, (ν f f ν g f)(ν f ft + 1) ν f ft ν < P < (ν f f ν g f)(ν g ft + 1) ν g ft ν (5.11) Notice tat tis is possible only if ν f f > ν g f, i.e. if kleptoparasites ave a disadvantage. A mixture of Dove and Marauder is tus only possible if kleptoparasites forage less efficiently tan non-kleptoparasites, figts are sufficiently long, and te population density is an intermediate level. Te conditions above are bot sufficiennd necessary for a stable mixture to occur. Indeed, wen tere is an equilibrium between Marauders and Doves, te condition (4.) provides D + M = ν f f ν g f ν and it follows tat ν f f > ν g f. Adding equations (3.1) and (3.10) and adding D s + M s to te result yields after rearranging t ν g f 1 + t ν g f P < D + M < P t ν f f 1 + t ν f f, (5.1) wic is exactly te condition we introduced above. Notice tat te left and side of te inequality (5.1) is P in a pure Marauder population, wile te rigt and side of (5.1) is P in a pure Dove population Hawks and Retaliators A mixture of Hawk and Retaliator occurs if Retaliator can invade Hawk, Hawk can invade Retaliator and in suc a mixture it is optimal to resisttacks, wic prevents te invasion of Doves or Marauders. As in Section 4, invasion of Dove or Marauder does not occur if and only if Dove does not invade Retaliator, i.e. if, by (4.1), 1 α > ν f f ta. Since Retaliator invades Hawk (and vice versa) if and only if Dove invades Marauder (and vice versa), te condition (4.) yields P ν f ft 1 + ν f ft ( α ν f f ) > ν f f ν g f ν 14 > P r ( α ν f f t ) a.

16 t ν f f Since R = P 1+t ν f f in a Retaliator only population, H = P r in a Hawk only population and r < ν f ft 1+ν f ft, te above inequalities are possible only for α > ν f f ta and ν f > ν g, wic yields te conditions ν f f < (1 α), 1 + ν f ft 1 ( ) ν f f ν g f < P < 1 1 ( ) ν f f ν g f ν f ft α νf f ta ν r α νf f ta ν A mixture of Hawk and Retaliator is tus only possible if kleptoparasites forage less efficiently tan non-kleptoparasites, figts are not too long, and te population density is an intermediate level. Note te similarity to te conditions for te Dove and Marauder mixture, te major difference being tat for te Hawk-Retaliator mixture, figts must be sort Mixtures involving Dove and Retaliator In tis section we consider all of te possible mixtures wic include te strategies Dove and Retaliator, possibly wit oter strategies as well. Tese mixtures are 1) Dove and Retaliator, ) Marauder, Dove and Retaliator, 3) Hawk, Dove and Retaliator, 4) Marauder, Hawk, Dove and Retaliator. = 1 α, or By (4.1), Doves and Retaliators ave te same payoff only if a) ν f f ta b) tere are no Hawks or Marauders in te population. Te condition ν f f ta = 1 α occurs only a precise coincidence of ecological parameters, wic as zero probability for any real situation, and so is a degenerate case. Tus, tere are no mixed equilibria in mixtures ), 3) and 4). If tere are no Hawks or Marauders in te population, Doves and Retaliators ave equal payoffs ere irrespective of te proportion of Doves, and so te population is subject to genetic drift, and tere is no unique equilibrium. Note tat if we allowed te repeated attempted invasion by mutants of oter strategies, ten eiter Dove or Retaliator would ave a fitness advantage and so a pure solution would result (tis is discussed in te sections for pure Dove and pure Retaliator) Hawks and Doves A mixture of Hawk and Dove individuals only is not possible (because of invasion by one or oter of Marauder and Retaliator). Tis is an extension of te result from Luter el (007). Tis result follows directly from te comparison conditions in Section 4. Indeed, if Marauder does not do better tan Dove, ten Hawk does no better tan Retaliator. So te only possibility to ave a Dove and Hawk mixture is wen all types do equally well (and tis situation is not possible as discussed in 5..3) Marauders and Retaliators Similarly to te Hawk - Dove mixture, a mixture of Marauder and Retaliator individuals only is not possible, due to invasion by eiter Dove or Hawk. Tis resulgain 15

17 follows from te general comparisons from Section 4 in te same manner and is again an extension of one from Luter el (007) Mixtures involving Hawk and Marauder Te remaining tree mixtures are 1) Marauder and Hawk, ) Marauder, Hawk and Dove, 3) Marauder, Hawk and Retaliator, wic all include Marauder and Hawk. If tree strategies ave te same payoff, te fourt one needs to ave te same payoff as follows from te general invasion conditions in Section 4. Tis means tat neiter a mixture of Hawks, Marauders and Doves, nor a mixture of Hawks, Marauders and Retaliators can be an ESS. Finally we must consider Hawk and Marauder only. Combining several earlier equations ((3.10)-(3.11)+(3.15)-(3.16)) gives t ν g fp s = P (5.13) Equation (3.18), combined wit te facts tat P a = P r, and H r = P r gives P a + P r = ν H P s Tus we can express P = P a + P r + P s + P in terms of P and H giving P = t ν g f ν H P t ν g f (5.14) We sow tat tere is no stable mixture. To ave suc a mixture, we need H /H d = M /M d, wic is, by (4.3), equivalent to (1 α) ν gf ν (P H (1 α)) = 0 (5.15) Te left and side of (5.15) reaces its minimum value in an all Marauder population since P attains te maximum wen tere are no figts, and H is equal to 0. Tus if tere are any roots to (5.15), pure Marauder muslso be an ESS. Tis in turn means tas we decrease te proportion of Marauders, M d, from P, te first root of (5.15) we reac (i.e. te largest root of (5.15)) is unstable. Tus, if tere is a stable root of (5.15), tere must be at least two roots. By (5.14), increasing P corresponds to decreasing H. Hence, tere can only be at most one value of P wic satisfies (5.14) and (5.15). Tus, for tere to be a stable root of (5.15), tere must be (at least) two population mixtures wic yield te same P. By (5.14), H is te same in bot mixtures as well. Tis determines M and P s, since M = P H and, by (5.13), P s = P /(t ν g f). From (3.0) we ave H s = t 1 + αν M s. H ν g f + ν M For two different population mixtures wit te same H and P to satisfy te above, bot H s and M s are must be larger in one of te populations tan te oter, wic contradicts te statement tat P s is constant. Tus no suc mixtures are possible, and tere is no stable Hawk-Marauder mixture. 16

18 Dove Retaliator Marauder Hawk mixture of Marauders and Doves in any pattern for relatively small P and for ν f f > max{ν gf, (1 α) } only if (1 α) > min{ν gf, α } ν f f sould be below (1 α) in any pattern, usually for ig P in all patterns except for (1 α) ν f f sould be no more tan (1 α) in any pattern for relatively medium P and for ν f f > max{ν gf, (1 α) } mixture of only if ν gf < min{ (1 α), α } Hawks and Retaliators and for ν f f between tem Table Possible ESS and teir occurrence. 6. Patterns of ESSs < ν gf < α ; In tis section we consider te various combinations of ESS types tat can occur for te same parameters. Tis is important, as if tere is more tan a single ESS a particular combination of parameters, ten identical populations in different locations can display completely different beavior due to istorical conditions or cance, rater tan any differences in inerent caracteristics. We use te term Patterns of ESSs for tis idea, after te work of Cannings and Vickers (1988); Vickers and Cannings (1988) (see also for example Broom, Cannings and Vickers, 1994). Te Pattern of a game is te collection of supports (te set of pure strategies wic occur wit non-zero probability) of te ESSs of te game. For example te pattern (M, HR) in our game refers to te case were tere is a pure Marauder ESS as well as a mixture of Hawk and Retaliator for te same parameter values (and so te same game), and no oter ESSs. Note tat in te study of patterns of ESSs, only te strategies involved in an ESS, rater tan teir frequencies, are considered. Tere are six possible ESS s, four of tem are te strategies Dove, Retaliator, Marauder and Hawk, two of tem are mixtures of strategies: a mixture of Marauders and Doves and a mixture of Hawks and Retaliators. No matter wat te parameters, tere is always at least one ESS. Table sows for wat parameter values te ESSs occur. For te six ESSs, eac can occur as te unique ESS, but it is also possible to ave two or tree ESSs for particular sets of parameter values (tere cannot be four). Te possible combinations of two ESSs are Hawk or Marauder, Retaliator or Marauder, and Marauder or a mixture of Hawks and Retaliators. Te possible combinations of tree ESSs are Hawk or Marauder or Retaliator. In total, tere are ten distinct ESS combinations over te range of all parameters (six unique ESSs and four cases of multiple ESSs). Tere are essentially six parameters of our model. Te population density, P, te speed kleptoparasites and non-kleptoparasites searc for food, ν g f and ν f f, te duration of figts, ta, attacker s cances of winning te contest, α, and te time needed for andling te food, t. We will fix (arbitrarily) all parameters but ν f f and P and will investigate ow te ESS outcome canges depending on tem. Tere are tree natural breakpoints for ν f f: 17

19 Hawk or Marauder Retaliator or Marauder Hawk or Retaliator Marauder or a mixture of Hawks and Retaliators Marauder or Hawk or Retaliators in all orderings but (1 α) in all orderings but (1 α) only if α < ν gf < (1 α) only if ν gf < min{ (1 α), α } only if α < min{ν gf, (1 α) } < ν gf < α < min{ν gf, α } Table 3 Multiple ESSs and teir occurrence in relation to te six size orderings of ν gf, α and (1 α) (i) (1 α), (ii) α, and (iii) ν g f Tere are six possible size orderings of tese breakpoints (excluding equalities), and te pattern of ESSs tat we get depends to a certain extent on wic ordering occurs, but te essential structure does not differ unless te ordering does. Table 3 sows for wic of tese size orderings te multiple ESSs occur. Tere is a total of eleven different patterns of ESSs. Figure 3 sows nine of tese patterns for te ordering ν g f < α < (1 α). An alternative structure wic again as nine patterns, including te two patterns of multiple ESSs Hawk or Retaliator, or Hawk or Retaliator or Marauder, not in Figure 3, is sown in Figure 4 for te ordering α < ν g f < (1 α). Te most interesting ESS is pure Marauder. It can occur as a unique ESS for some parameter values, or at te same time as any of pure Hawk, pure Retaliator, a mixture of Hawk and Retaliator, or pure Hawk and pure Retaliator. It cannot occur togeter wit Dove or te Marauder-Dove mixture. Te pure Marauder ESS occurs always for ig population densities. In te population of all Marauders, tere are no figts and tus te total time for consumption of a food item equals (ν g f) 1 + t (wic means te sum of time needed to find a food item plus te time needed to andle te item), altoug in reality, te food item is often acquired by stealing from a andler. Wen te density of te population is ig, Marauders are uninvadable because any intruder would eiter be spending too muc time figting (if it would resist wit non zero probability) or it would be losing food items wile andling but nocquiring te food by stealing efficiently enoug (if not stealing wit probability one). Te Dove ESS occurs for any of te ordering of parameters above as long as ν f f is large enoug and P is not too large. Having ig ν f f means tat food is easy to find and tus it is not beneficial to resist. In relatively small density populations te encounters between individuals are not too frequennd terefore te opportunity to attack is not wort te associated loss of foraging efficiency (ν g f < ν f f), so tat Marauders cannot invade. A mixture of Marauders and Doves also occurs as an ESS for all orderings, as long as ν f f is large and te population is of intermediate density. Tis mixture occurs as a transition between pure Dove and pure Marauder for large ν f f. Te actual stable mixture varies from almosll Dove near te lower boundary to te region, increasing to almost all Marauder near te upper boundary. Hawk is an ESS only for relatively small values of ν f f. Small ν f f means tat food is scarce and tus valuable. It is tus beneficial to figt for ind to resist if attacked. 18

20 Fig. 3. Patterns of ESSs for te ordering ν gf < α < (1 α). Legend: H = Hawk, M = Marauder, R = Retaliator, ( D = ) Dove, XY = mixed ESS of X and Y, X + Y = ESSs of eiter X or Y ; (1 α) P M = ν gf tν ν gf+1 t ν ν. For tis particular picture: νgf = 0.3, α = 0.4, t gf = ν = = 1, ν f f ranges from 0 to 1.5, P ranges from 0 to 9. Hawk can be an ESS for low density populations as well as for ig density ones. Te only ordering wen Hawk is non ESS is wen (1 α) < ν g f < α wen Marauder is te only ESS for small ν f f. Tis is natural since te conditions imply a) α > 1/, i.e. it is beneficial to attack since te cances of success are ig and b) a food item is not valuable, since ν g f is relatively ig and tus it is not beneficial to resisnd lose time by figting. Retaliator is an ESS for medium values of ν f f. Te value ν f f sould never exceed (1 α) wic is te point were resisting stops being beneficial due to te ig cost of time spent figting compared to te cances of winning te figt. On te oter and, ν f f sould be ig enoug in order to assure tat food is not too scarce and tus it is not beneficial to look for andlers. Te mixture of Hawks and Retaliators occurs in a region between pure Hawks and pure Retaliator if ν g f < min{ (1 α), α }. In tis setting, wen ν f f is small, it is beneficial to figt for food and Hawk is an ESS. For larger ν f f it is not beneficial to fignd Retaliator is an ESS. If ν f f takes a value somewere in between, Hawk can invade Retaliator and vice versa and te strategies coexist. For larger population densities, te same pattern occurs, but Marauder is a second ESS alternative. If α < min{ν g f, (1 α) }, ten tere can be tree different ESS - Hawk or Marauder or Retaliator for te same set of parameter values. 19

21 Fig. 4. Patterns of ESSs for te ordering α < ν gf < (1 α). Legend: H = Hawk, M = Marauder, R = Retaliator, ( D = ) Dove, XY = mixed ESS of X and Y, X + Y = ESSs of eiter X or Y ; (1 α) P M = ν gf tν ν gf+1 t ν ν. For tis particular picture: α = 0.5, νgf = t gf = ν = = 1, ν f f ranges from 0 to, P ranges from 0 to. 7. Discussion In tis paper we ave developed te game-teoretic model of kleptoparasitism introduced in Broom and Ruxton (1998). Individuals use one of te four different strategies first described in Broom el (004), depending upon weter tey will attack conspecifics for food, and weter tey are prepared to resisny suc attack. We ave also maintained te simplification tall birds are equally able figters (so tat te probability of te attacker winning is independent of te types attacking and resisting). Our model allows for a difference in te foraging rates of attackers and non-attackers (peraps because non-attackers can concentrate on searcing for food rater tan food and conspecifics). Te model, in common wit tat of Broom and Ruxton (1998) and most subsequent models, assumes tat contest times are exponentially distributed and not under te control of te participants. Note tat in Ruxton and Broom (1999), owever, te lengt of te contest was under te control of te individuals, wo played a classical war of attrition to decide te winner. From tis model, an exponential contest lengt occurs naturally from optimal play witin te contest for food. Te same resullso occurs in te important earlier foraging models of Hines (1976, 1977) were individuals foraged 0

22 and also entered contests over food, since tese papers again used te war of attrition as te basis for contests, introducing it to a more complex foraging setting for te first time. As in our model, te aim was to find an evolutionarily stable foraging strategy were an individual s fitness was given by its foraging rate. Tere were two key differences between te Hines (1976, 1977) models and ours. Firstly tere was effectively no andling time for a food item, so an individual tat was successful in a conteslways consumed te item, and did not ave to face subsequent callenges. Secondly te proportion of food items contested was independent of te strategies played by individuals; in our model te cosen strategies directly affect te likeliood of any subsequent item being competed for, and also wic type of opponent would be faced. Tis effect is central to tis and earlier models, and is te cause of teir complexity, wit individual contests generally being very simple. In contrast te individual contests of Hines (1976, 1977) were more complex, and are te central feature of tese papers. We ave found general expressions for te inverse andling ratio, and ence te uptake rate and te fitness, of eac strategy in a general population mixture of different strategies, and ten found conditions for wic eac of te strategies or mixtures of strategies are ESSs. Weter a strategy is an ESS or not of course depends upon te values of te population parameters (see Table 1). All four strategies can be ESSs and two of te possible mixtures of strategies can also be an ESS. Tere are tus six ESSs in total, eac of wic can be te unique ESS; indeed tere is always at least one ESS. It is possible tat tere are two ESSs, and one case of tree ESSs (but never more tan tree). Wen tere are multiple ESSs, in all but one case one of tese is te Marauder ESS, wic can occur in conjunction wit Hawk, Retaliator, a mixed ESS containing bot Hawk and Retaliator, or wit bot pures Retaliator and Hawk. Te multiple ESS case witout Marauder includes pure Hawk and pure Retaliator. Tis gives eleven possible ESS patterns in total, wic are best summarised by Figures 3 and 4. In Section 5 we describe te conditions wen eac ESS occurs in more detail. In particular Hawk occurs wen te food gatering ability of foragers talso attack, or general food availability, is poor and te population density is not too large (recalling tat interactions appen more frequently te denser te population). Marauders generally trive wen te population density is large and food availability is also large. Retaliators do better wen food levels are intermediate and te population density is not large, and Doves do better wen food is plentiful and te population density is not too large. All of tese results are intuitively reasonable. In eac of te situations were tere is more tan one ESS, wic is more likely to occur? Te only case were two ESSs oter tan Marauder occurred simultaneously was for Hawk and Retaliator (eiter as two pure ESSs or tree pure ESSs including Marauder), wic was in te case wen te Hawk s foraging rate was greater tan te Retaliator s. Tis case is peraps unrealistic because a Hawk must split its attention between searcing for food and for kleptoparasitic opportunities, wereas a Retaliator can dedicate itself to searcing for food, and tus tis combination is unlikely to occur. Te Marauder and Retaliator case is similar to te owner intruder game of Maynard Smit and Parker (1976) were tere is a symmetrical game (in te sense of payoffs for escalating or not) but wit te individuals in different roles. Te Marauder ESS is in some ways a paradoxical strategy, in a similar way to te anti-bourgeois strategy X, were te owner always gives way to te intruder. Te autors argued tat it would be muc less likely to occur in practice tan Bourgeois, altoug tey identified one case 1

23 were it did occur. Tus peraps wen multiple ESSs are available, Marauder migt not be observed. If strategies wic involve resistance are stable, ten non-resistance may never evolve. Tis would yield a natural single solution for realistic parameter values. Tis does owever suppose tat te individuals would naturally resist, and only not resist if tis was subsequently beneficial. Even if tis is true, and it migt not be, if parameters cange troug time, tey may well move to a set of values were Marauder is te unique ESS, and ten move to one were two ESSs are possible, but te population starts in te still stable Marauder strategy. Te situation is in fact rater different in te case of Marauder and Retaliator being ESSs, to wen Marauder and Hawk are ESSs. For te Hawk and Marauder case, Marauders do better in populations wit iger food availability, Hawks do better wen tis availability is lower. However wen all individuals play Hawk time is wasted in figting and te consumption rate goes down, effectively mirroring lower food availability. Tere are tus parameter values wen Marauders do better in a population of Marauders and Hawks do better in a population of Hawks. In tis paper we ave considered te four most basic types of attitude to kleptoparasitic opportunity and treat; namely te four combinations of attack or nottack, wit resist or not resisn attack. How do tese strategies relate to te beavior of real birds? Most birds effectively play eiter te Dove or Retaliator strategy, and exist in a population were no food stealing takes place. If a population as te strategy Hawk tere will be widespread kleptoparasitism wit visible contests, and it is tis strategy wic yields te most obvious kleptoparasitic beavior. Our teoretical model makes predictions about wen eac of te basic types of beavior or mixtures of tese basic types sould occur, and we ope tat te model will be compared to real populations to test te accuracy of its predictions. Tese four different tactics all seem plausible in terms of underlying biology. A real population tat played Marauder would be caracterised by frequent very sort contests. Beavior of tis type occurs for various wading birds (see e.g. Stillman el, 1997); often only some birds will attack, and so tis example is more caracteristic of our mixed populations in suc cases (note tat tis could also be linked to dominance relationsips between te birds). Te Marauder tactic may also equate to te concept of stealt kelptoparasitism introduced by Giraldeau and Caraco (000). Here kleptoaparasitic acts are not immediately detected by te victim, or te victim is not in a position to react immediately, or feature great speed by te tief. Giraldeau and Caraco discuss observations of kelp gulls (Larus domenicanus) stealing from eac oter reported by Hockey el. (1989) as an example of stealt kleptoparasitism. Te gulls are feeding on bivalue sellfis. Tey sometimes break te protective sell of teir prey by dropping te bivalve from a great eigt onto rocks. Tis may give oter gulls standing on te rocks te opportunity to rus in and steal te newly-exposed prey before te focal gull can swoop down from ig above. In suc cases, te victim does not seem generally to pursue te tief, peraps because it would take a long time to close te eigt differential between tem, during wic time te tief will ave made good teir escape or even consumed te prey. Individuals play bot roles, sometimes being aggressive tieves and sometimes passive victims, exactly as in te Marauder strategy. A furter example of suc stealt kleptoparasitism occurs in some species of birds and mammals tat orde food. If one individual discovers te unguarded cace of anoter ten it may invest significant energy in exposing and exploiting tat cace, but wen te victim returns tey may not invest

24 in pursuit of te tief, wo may by ten be a long time gone and consequently ard to find, Giraldeau and Carcaco discuss several studies tat observe tis beavior. Te oter tree tactics can be seen to more easily reflect conventional aggressive contests. Again in gulls, Hesp & Barnard (1989) discuss differences between immature and mature black-eaded gulls, Larus ridindus. Younger gulls tended bot to avoid attacking and to sow little resistance to attack (like te Dove strategy), wereas adults sowed strong aggressiveness wen in eiter position (like te Hawk strategy). Interestingly, aggressiveness seemed to develop first in terms of resistance to attack rater tan in self-induced attacks, tus leading to individuals progressing troug teir lives from te Dove strategy, troug Retaliator to te Hawk strategy. Furter, generally aggressiveness can be related to ig-value ard-to-find items (Morand-Ferron el. 007) and so asymmetry between individuals in teir ability to find prey or teir current need to eat can lead to asymmetry in teir evaluation of te value of te contested prey item, leading to differential investmennd so Marauder or Retaliator strategies. Suc asymmetry may also be introduced by te spatial interaction of te two participants. It may simply be tat if te tief can swoop on teir victim in fligt from above and use gravity to accelerate away, ten pursuit is futile and we see a Marauder-like situations, wereas te same victim may defend teir prize muc more vigorously against te same tief if bot meet on te ground (leading to a Hawk-type tactic involving te same participants). References Barnard C.J., Sibly, R.M Producers and scroungers: A general model and its application to captive flocks of ouse sparrows. Animal Beaviour 9, Brockmann H.J., Barnard C.J Kleptoparasitism in birds. Animal Beaviour 7, Broom M., Cannings C. and Vickers,G.T. (1994) Sequential metods for generating patterns of ESS s. Journal of Matematical Biology 3, Broom M., Ruxton G.D Evolutionarily Stable Stealing: Game teory applied to kleptoparasitism. Beavioral Ecology 9, Broom M., Ruxton G.D Evolutionarily stable kleptoparasitism : consequences of different prey types. Beavioral Ecology 14, Broom, M., Luter,R.M., Ruxton,G.D Resistance is useless? - extensions to te game teory of kleptoparasitism. Bulletin of Matematical Biology 66, Broom, M., Ryctář,J Te evolution of a kleptoparasitic system under adaptive dynamics. Journal of Matematical Biology 54, Cannings, C., Vickers, G.T Patterns of ESS s. Journal of Teoretical Biology 13, Dukas, R. 00. Beavioural and ecological consequences of limited attention. Pilosopical Transactions of te Royal Society of London B 357, Dukas, R., Ellner, S Information Processing and prey detection. Ecology 74, Dukas, R., Kamil, A.C Limited Attention: te constraint underlying searc images. Beavioral Ecology 1, Furness, R.W Kleptoparasitism in seabirds. In Seabirds: Feeding ecology and role in marine ecosystems (Croxall JP, ed ) CUP. 3