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Appendix S2: Ratio of Transition Probabilities for Weak Selection J. Zukewich1∗ , V. Kurella1 , M. Doebeli1,2 , C. Hauert1 1
Department of Mathematics, University of British Columbia, Vancouver, BC, Canada. Department of Zoology, University of British Columbia, Vancouver, BC, Canada. ∗ E-mail: Corresponding
[email protected] 2
For each update rule and structure, the transition probabilities Ti+ , Ti− are different. In each case we determine Ti− /Ti+ in the limit of weak selection (w � 1). In this limit Ti− /Ti+ ≈ 1 + wθi , where the coefficient θi captures the effect of population structure and update rules.
Well-Mixed Population (BD) Under the BD rule the number of C’s increases by one when a C reproduces and a D then dies: Ti+ =
N −i ifi ifi + (N − i)gi N
(S2.1)
(N − i)gi i ifi + (N − i)gi N
(S2.2)
Similarly, the number of C’s decreases if a D reproduces and a C dies: Ti− =
Hence Ti− /Ti+ = gi /fi , where fi = 1 − w + w[(i − 1)πCC ) + (N − i)πCD ] and gi = 1 − w + w[iπDC + (N − i − 1)πDD ] are the fitness of C and D given that there are i C-individuals and N − i D-individuals [1]. By Taylor expanding and neglecting higher order terms we get: Ti− /Ti+ ≈ 1 − w(fi − gi ),
(S2.3)
θiBD = −αi + N (πDD − πCD ) + πCC − πDD .
(S2.4)
and hence where α = πCC − πCD − πDC + πDD is used for convenience.
Well-Mixed Population (DB) Under the DB rule the number of C’s increases by one when a D dies and a C then reproduces: Ti+ =
N −i ifi . N ifi + (N − i − 1)gi
(S2.5)
i (N − i)gi . N (i − 1)fi + (N − i)gi
(S2.6)
Similarly, the number of Cs decreases if a C dies and a D reproduces: Ti− = Hence
Ti− /Ti+ = or, up to first order, Ti− /Ti+
gi (i − 1)fi + (N − i)gi , fi ifi + (N − i − 1)gi
= 1 − w(fi − gi )
and θiDB =
�
N θBD . N −1 i
N N −1
�
(S2.7)
(S2.8)
(S2.9)
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Structured Population (BD) The transition probabilities under BD for structured populations are given in Eqs. (9)-(10) based on the fitness of a focal cooperator (fkC ) and a focal defector (gkC ) with kC C-neighbours: fkC = 1 − w + w(kC πCC + (k − kC )πCD ),
gkC = 1 − w + w(kC πDC + (k − kC )πDD ).
(S2.10) (S2.11)
In the limit of weak selection the separation of time scales results in the quasi-steady state condition (Eq. S1.5) that can be used to simplify Eqs. (9)-(10): φTi+ = 1 + w[πCC + πCD (k − 1) − 1 + (πCC − πCD )(k − 2)pC ], pCD φTi− = 1 + w[πDD + πDC (k − 1) − 1 + (πDD − πDC )(k − 2)pD ], pCD
(S2.12) (S2.13)
where φ indicates the total fitness of all individuals in the population. Using Ti− /Ti+ ≈ 1 + wθi , we find: θiBD = −α + k(πDD − πCD ) − (k − 2)α
i , N
(S2.14)
with α = πCC − πCD − πDC + πDD .
Structured Population (DB) Under DB the transition probabilities are given in Eqs. (12)-(13) where f˜j and g˜j denote the fitness of C and D neighbours of a focal j individual: f˜j = 1 − w + w{[δjC + (k − 1)qC|C ]πCC + [δjD + (k − 1)qD|C ]πCD },
g˜j = 1 − w + w{[δjC + (k − 1)qC|D ]πDC + [δjD + (k − 1)qD|D ]πDD }, where δj� = 1 if j = � and δj� = 0 otherwise. Using the quasi steady-state condition (??), Eqs. (12)-(13) simplify to: � � wξCD Ti+ = pCD 1 − [k − 1 − (k − 2)pC ] , k � � wξ DC Ti− = pCD 1 − [k − 1 − (k − 2)pD ] , k
(S2.15) (S2.16)
(S2.17) (S2.18)
with Using
Ti− /Ti+
ξij = (k − 2)pi (πji − πjj − πii + πij ) + k(πjj − πij ) − πii + πij .
≈ 1 + wθi , we find: � � 1 2 i θiDB = k (πDD − πCD ) + k(πCD − πCC ) − α − α(k − 2)(k + 1) . k N
(S2.19)
(S2.20)
Structured Population (Mixed Update) Under the mixed update, DB is used with probability δ and BD with probability 1 − δ. The probabilities to increase or decrease the number of C players by one C are: Ti+δ =Ti+BD (1 − δ) + Ti+DB δ,
Ti−δ Therefore, θiδ = θiBD (1 − δ) + θiDB δ.
=Ti−BD (1
− δ) +
Ti−DB δ.
(S2.21) (S2.22)
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References 1. Nowak M, Sasaki A, Taylor C, Fudenberg D (2004) Emergence of cooperation and evolutionary stability in finite populations. Nature 428: 646–650.
