A simple model of group selection that cannot be analyzed with inclusive fitness

Transcription

1 A smple model of group selecton that cannot be analyzed wth nclusve ftness

2 1.0 Hamlton s rule (bology) Folk theorem (economcs)

3 Ant-group selecton team Pro-group selecton team Jerry Coyne Rchard Dawkns (c) Andy Gardner Alan Grafen Laurent Keller Laurent Lehmann Steven Pnker Davd Queller Francos Rousset Stuart West Geoff Wld Letca Avles Rob Boyd Samuel Bowles Lee Dugatkn Herbert Gnts Charles Goodnght Jon Hadt Pete Rcherson Arne Traulsen DS Wlson (c) EO Wlson

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5 Two dfferent ssues 1) Has group selecton shaped (human, cooperatve) behavour? 2) Is group selecton equvalent wth nclusve ftness?

6 'Group selecton', even n the rare cases where t s not actually wrong, s a cumbersome, tme-wastng, dstractng mpedment to what would otherwse be a clear and straghtforward understandng of what s gong on n natural selecton. Rchard Dawkns (2012)

7 Inclusve ftness theory, summarsed n Hamlton s rule, s a domnant explanaton for the evoluton of socal behavour. A parallel thread of evolutonary theory holds that selecton between groups s also a canddate explanaton for socal evoluton. The mathematcal equvalence of these two approaches has long been known. Marshall (2011)

8 No group selecton model has ever been constructed where the same result cannot be found wth kn selecton theory West, Grffn & Gardner (2007)

9 Inclusve ftness models and group selecton models are extremely smlar to each other. Ther only fundamental dfference s n how they choose to decompose ftness. Other dfferences are trval matters of the form of presentaton. Queller (1992)

10 Mathematcal gene-selectonst (nclusve ftness) models can be translated nto multlevel selecton models and vce versa. One can travel back and forth between these theores wth the pont of entry chosen accordng to the problem beng addressed. Hölldobler and Wlson (2009)

11 The Prce formulaton convnced Hamlton that kn selecton was group selecton. Wade et al. (2010)

12 Hamlton (1975)

13 Inclusve ftness / group selecton Hamlton s mssng lnk, 2007 Hamlton s rule 1964 Prce equaton 1970, 1972 Karln & Matess 1983, 1984 On the use of the Prce equaton, 2005 Unto Others 1998 GS IF 2009 NTW Wllams 1966 Hamlton 1975 Queller 1992 Traulsen & Nowak 2006 Nal n the coffn of group selecton 2009

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15 Shsh Luo Burt Smon

16 Indvdual reproducton Group reproducton a b ntensty 1 ntensty 1 ntensty 1

17 a Indvdual reproducton b Group reproducton a b

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19 The PDE that descrbes the dynamcs loss n ndvdual reproducton gan n collectve reproducton wave movement to the left ncrease reproducton rate -groups ncrease (unform) death rate

20 Change n frequency of cooperators at large n the mddle large f groups heterogeneous

21 Change n frequency of cooperators at

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23 Of course, t s now generally understood that the correct defnton of relatedness s that whch makes nclusve ftness theory work. Marshall / Grafen

24 A rule s not a rule f t changes from case to case. Van Veelen, 2012

25 Go Procrustes, go!

26 Change n frequency of cooperators at

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30 Replcator dynamcs Standard: 2 players, random matchng Generalzaton: n players, assortatve matchng 1 1 ( ) ( ) ( ) ( ) 4 8

31 Replcator dynamcs 2 players f f f players f f f f

32 It s the equal gans from swtchng, st! 1 _ + 1 1

33 Queller (1985) but then Prce-less + altrusm selected + altrusm selected aganst bstablty + + coexstence

34 f 1 1 f 1 1 _ + f0 1 f 2 1 f 0 1 f 2 1 Hamlton Queller

35 Replcator dynamcs 1) the populaton structure mples a constant r, and 2) the game satsfes generalzed equal gans from swtchng, p t e e 1 Kt K t0 t K t0 t t where e K r b - c

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39 A lst of numbers Left hand sde = Rght hand sde

40 If you want to wn a game, you should score [at least] one goal more that your opponent Johan Crujff

41 The frequency has gone up because the frequency has gone up the Prce Equaton

42 Is there such a thng as Prce s theorem?

43 Theorem 1 (bology): If the left hand sde n the Prce equaton s computed as suggested n Prce (1970) and the rght hand sde as well, then they are equal.

44 Theorem 1 (football): If team A scores more goals than team B, then team A wns.

45 Theorem 2 (football): If team A and B have equally able players, and nteractons occur accordng to Assumpton 1,, Assumpton N, and team A plays and B plays 4-4-2, then team A s more lkely to wn than team B.

46 Theorem 2 (bology): If the ftness of an ndvdual depends on ts own and the other ndvdual s behavour accordng to Assumpton 1,, Assumpton N, than the behavour that emerges s more lkely to be behavour A than t s to be behavour B.

47 How to qut the Prce equaton

48 Prce (as smple as t gets) N q N z N q z N q N q z Q Q Q 1 2

49 Prce (as smple as t gets) z q z q Q N N N

50 Prce (as smple as t gets) Q Cov z, q

51 Prce (as smple as t gets) z q z q Q N N N

52 Correct would be Q Sample cov z,q f the numbers are data

53 or Q Covz q E, If z and q random varables for all.

54 Model: draw twce, both tmes P (red) = p P (whte) = 1 - p Parent generaton Offsprng generaton Ind. 1 Ind. 2

55 Ind. 1 Parent generaton Offsprng generaton Model: draw twce, both tmes P (red) = p P (whte) = 1 - p Ind. 2 q q z z Q Cov z,q Q 1 2 zq z q

56 Ind. 1 Parent generaton Offsprng generaton Model: draw twce, both tmes P (red) = p P (whte) = 1 - p Ind. 2 q q z z Q Cov z,q Q 1 2 zq z q

57 Ind. 1 Parent generaton Offsprng generaton Model: draw twce, both tmes P (red) = p P (whte) = 1 - p Ind. 2 q q z z Q Cov z,q zq z q Q

58 Ind. 1 Parent generaton Offsprng generaton Model: draw twce, both tmes P (red) = p P (whte) = 1 - p Ind. 2 q q z z Q Cov z,q zq z q Q

59 Propertes of the model Parent generaton Possble offsprng generatons I II III IV Model: draw twce, both tmes P (red) = p Ind. 1 P (whte) = 1 - p Ind. 2 p 2 (1-p) 2 p(1-p) p(1-p) Randomly draw a parent (hypothetcally) X ts genotype Y ts number of offsprng Cov X, Y p 1 2 Q "Cov z,q "

60 What would a statstcan do? Ind. 1 Parent generaton Possble offsprng generatons I II III IV Model: draw twce, both tmes P (red) = p P (whte) = 1 - p Ind. 2 p 2 (1-p) 2 p(1-p) p(1-p) 1) Estmate p 2) Test f p > 0 Q "Cov z,q "

61 Prce 2.0 N Q "Covz,q " z z q' q z

62 Prce 2.0 N Q "Covz,q " z z q' q z Meoss term

63 Prce 2.0 N Q "Covz,q " z z q' q z

64 Prce 2.0 N Q "Covz,q " z z q' q z 8 71

65 Prce 2.0 Parent generaton Possble offsprng generatons I II CCLVI Ind. 1 Ind Ind. 3 Ind. 4 q q 05. q 05. q 0 z z z z q' 1 1 q' 0 2 q' 1/ 3 3 q' 0 4

66 Prce 2.0 N Q "Covz,q " z z q' q z

67 What would a modeler do? Show that the assumpton of far meoss mples that E z q' q z 0

68 What would a statstcan do? Test the hypothess of far meoss, usng the realzaton of z q' q z

69 Projecton of ntuton onto the Prce equaton N Q "Covz,q " z z q' q z

70 Parent generaton Prce 3.0 Possble offsprng generatons Model: 1) match randomly 2) play I II 3) draw each ndvdual wth Ind. 1 Ind. 2 probabltes proportonal to payoffs... Ind. n Cov X, Y 1 n Ind. 1 Ind Ind. n Cov X, Y 1 n

71 Dynamcal suffcency

72 Cov X, Y 1 n Cov X, Y 1 n

73 A lst of numbers Left hand sde = Rght hand sde