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Conflict resolution Evolutionary stable strategies (ESS) –Game theory –Assumptions Evolution of display –Hawks and doves –Arbitrary asymmetry

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Assessment in red deer

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What is an ESS? Strategy = the behavioral response of an individual ESS = a strategy which if adopted by all members of a population cannot be invaded by any alternative strategy The ESS is found using game theory. Game theory is needed when the fitness consequences of a behavior depend on what others are doing, i.e. is frequency dependent

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ESS assumptions Infinite population Asexual (haploid) reproduction All strategies are specified Either pairwise contests occur or one individual competes against a group

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Evolution of display: Hawks vs Doves Possible behaviors: –Display –Fight but risk injury –Retreat Possible strategies: –Hawk: fight until injured or opponent retreats –Dove: display initially but retreat if opponent attacks

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Payoff matrix Payoffs to Actor after confronting opponent: V = value of resource being contested C = cost of fighting due to injury Opponent:HawkDove Actor:Hawk Dove (V-C)/2

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Payoff matrix Payoffs to Actor after confronting opponent: V = value of resource being contested C = cost of fighting due to injury Opponent:HawkDove Actor:Hawk Dove (V-C)/2 V

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Payoff matrix Payoffs to Actor after confronting opponent: V = value of resource being contested C = cost of fighting due to injury Opponent:HawkDove Actor:Hawk Dove (V-C)/2 V 0

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Payoff matrix Payoffs to Actor after confronting opponent: V = value of resource being contested C = cost of fighting due to injury Opponent:HawkDove Actor:Hawk Dove (V-C)/2 V 0V/2

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Pure ESS Resource > cost; V = 2; C = 1 Opponent:HawkDove Actor:Hawk Dove 1/2 2 0 1 1/2 > 0, so Hawks resist invasion by doves 2 > 1, so Hawks can invade doves ESS = all Hawks => pure ESS

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Mixed ESS Resource < cost; V = 1; C = 2 Opponent:HawkDove Actor:Hawk Dove -1/2 1 0 1/2 -1/2 < 0, so Doves can invade Hawks 1 > 1/2, so Hawks can invade doves ESS = mix of Hawks and Doves => mixed ESS If the frequency of Hawks is p, and Doves is 1-p and at the ESS the fitness of Hawks = the fitness of Doves, then (-1/2)p + (1-p) = (0)p + (1/2)(1-p) 1 - 3p/2= 1/2 - p/2 1/2 = p

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Mixed ESS mechanisms Stable strategy set in which a single individual sometimes performs one strategy and sometimes another with probability p Stable polymorphic state in which a fraction, p, of the population adopts one strategy while the remainder, 1-p, adopts the other Note that highest payoff is not the ESS

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ESS Solutions Opponent:AB Actor:A B � A is a pure ESS Opponent:AB Actor:A B � Stable mixed ESS Opponent:AB Actor:A B � �� � Unstable mixed ESS � = highest payoff in column

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Opponent:HawkDoveBourgeois Actor:Hawk Dove Bourgeois Uncorrelated asymmetry Opponents differ, but not with regard to fighting ability Example: hawk - dove - bourgeois –Bourgeois strategy: if owner play hawk, if intruder play dove –Assume that owner and intruder are equally frequent and get equal payoffs (V-C)/2 V 0 V/2

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Opponent:HawkDoveBourgeois Actor:Hawk Dove Bourgeois Uncorrelated asymmetry Opponents differ, but not with regard to fighting ability Example: hawk - dove - bourgeois –Bourgeois strategy: if owner play hawk, if intruder play dove –Assume that owner and intruder are equally frequent and get equal payoffs (V-C)/2 V 0.5H:H + 0.5H:D 0 V/2 0.5D:H + 0.5D:D 0.5H:H + 0.5H:D + 0.5H:D + 0.5D:H 0.5D:D 0.5D:H

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Opponent:HawkDoveBourgeois Actor:Hawk Dove Bourgeois Uncorrelated asymmetry Opponents differ, but not with regard to fighting ability Example: hawk - dove - bourgeois –Bourgeois strategy: if owner play hawk, if intruder play dove –Assume that owner and intruder are equally frequent and get equal payoffs: (V-C)/2 V3V/4-C/4 0 V/2 V/4 (V-C)/43V/4 V/2

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Hawk-Dove-Bourgeois Therefore, arbitrary asymmetries should resolve conflicts Opponent:HawkDoveBourgeois Actor:Hawk Dove Bourgeois 1/2 2 5/4 0 1 1/2 1/4 3/2 1 If V > C (V = 2, C = 1), then H is pure ESS Opponent:HawkDoveBourgeois Actor:Hawk Dove Bourgeois -1/2 1 1/4 0 1/2 1/4 -1/4 3/4 1/2 If V < C (V = 1, C = 2), then B is pure ESS

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Residency in speckled wood butterflies Note, however, that this effect has been found to be due to the body temperature of the resident (Stutt and Wilmer 1998)

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Finding the ESS by simulation If you have a Mac computer, you can download the game theory Simulation from Keith Goodnight at http://gsoft.smu.edu/GSoft.html

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