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! 2010 by The University of Chicago. All rights reserved. DOI: 10.1086/651591

Appendix from S. L. Nuismer et al., “When Is CorrelationCoevolution?”(Am. Nat., vol. 175, no. 5, p. 525)

Model Derivations and AnalysesDerivation of Recursions for Change in Mean Phenotype within a Single PopulationEach generation begins with the movement of individuals among locations. We assume that movement occursfollowing an island model, such that population mean phenotype of species i within a single location followingmovement is

′¯ ¯z p (1 ! m )z " m m , (A1)i i i i i

where mi is the expectation of taken across all populations.zi

Following movement, selection occurs. Assuming that additive genetic variance, Gi, is fixed and that traits arenormally distributed, the mean phenotype within a location following selection can be predicted using

!Wi′′ ′¯ ¯z p z " G , (A2)i i i ¯!zi

where is the population mean fitness of species i. If individuals encounter one another at random, the meanWi

fitness is given by

W p Wf f d d , (A3)! ! i z z z zi i j i j

where and are the phenotype distributions of species i and j within the population and individual fitness,f fz zi j

Wi, is given by equation (1) of the main text.Unfortunately, direct integration of equation (A3) is impossible. For this reason, we make the additional

assumptions described in the main text (i.e., and , where ) such that the fitness functiona ∼ O(") g ∼ O(") " K 1i

(eq. [1]) can be well approximated by a first-order Taylor series in " (Nuismer et al. 2005, 2007; Ridenhour andNuismer 2007). Substituting this approximate fitness function into equation (A3) makes the integrationstraightforward.

After selection, we assume that random genetic drift occurs such that the population mean of species i withina specific location at the end of a generation is given by

′′′ ′′¯ ¯z p z " z , (A4)i i i

where zi is a random variable with a mean of 0 and a variance of , where ni is the local effective populationG /ni i

size of species i. Equation (A4) is equation (3) of the main text.

Derivation of Statistical MomentsWe begin with equation (3) of the main text for the change in the mean phenotype of species i within a singledeme:

˜ ˜ ˜ ˜ ˜¯ ¯ ¯Dz p G b " (z ! z ) " z p G [2g (v ! z ) " 2s (z ! z ) " s ] " (z ! z ) " z ,i i i i i i i i i M j i D i i ii i

App. from S. L. Nuismer et al., “Correlation and Coevolution”

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where and is the mean in deme i immediately after migration.˜ ¯i ( j z p (1 ! m )z " m mi i i i i

A. recursion. Since , we have˜ ¯ ¯Dm E(z ) p E[(1 ! m )z " m m ] p (1 ! m )E(z ) " m m p mi 1 1 1 1 1 1 1 1 1 1

˜¯ ¯Dm p E(Dz ) p E[G b " (z ! z ) " z ]1 1 1 1 1 1 1

˜ ˜ ˜p G E[2g (v ! z ) " 2s (z ! z ) " s ]1 1 1 1 M 2 1 D1 1

¯p 2G [g (v ! m ) " s (m ! m ) " s ].1 1 1 1 M 2 1 D1 1

(The lack of a migration term verifies that migration has no effect on the global mean.) The analogous recursionfor can be found by exchanging subscripts 1 and 2.Dm2

B. recursion.Djz vi i

¯Dj p Cov (Dz , v )z v 1 11 1

˜ ¯p Cov (G b " z ! z " z , v )1 1 1 1 1 1

˜ ˜ ˜ ˜ ¯p Cov {G [2g (v ! z ) " 2s (z ! z ) " s ] " z ! z , v }1 1 1 1 M 2 1 D 1 1 11 1

2p 2G {g [j ! (1 ! m )j ] " s [(1 ! m )j ! (1 ! m )j ]} " (1 ! m )j ! j¯ ¯ ¯ ¯ ¯1 1 v 1 z v M 2 z v 1 z v 1 z v z v1 1 1 1 2 1 1 1 1 1 1 1

2p 2G {g [j ! (1 ! m )j ] " s [(1 ! m )j ! (1 ! m )j ]} ! m j¯ ¯ ¯ ¯1 1 v 1 z v M 2 z v 1 z v 1 z v .1 1 1 1 2 1 1 1 1 1

The analogous recursion for can be found by exchanging subscripts 1 and 2.Djz v2 2

C. , , recursion.Dj i ( jz vi j

¯Dj p Cov (Dz , v )z v 1 21 2

˜ ¯p Cov (G b " z ! z " z , v )1 1 1 1 1 2

˜ ˜ ˜ ˜ ¯p Cov {G [2g (v ! z ) " 2s (z ! z ) " s ] " z ! z , v }1 1 1 1 M 2 1 D 1 1 21 1

p 2G {g [j ! (1 ! m )j ] " s [(1 ! m )j ! (1 ! m )j ]} " (1 ! m )j ! j¯ ¯ ¯ ¯ ¯1 1 v v 1 z v M 2 z v 1 z v 1 z v z v1 2 1 2 1 2 2 1 2 1 2 1 2

p 2G {g [j ! (1 ! m )j ] " s [(1 ! m )j ! (1 ! m )j ]} ! m j .¯ ¯ ¯ ¯1 1 v v 1 z v M 2 z v 1 z v 1 z v1 2 1 2 1 2 2 1 2 1 2

The analogous recursion for can be found by exchanging subscripts 1 and 2.Djz v2 1

D. Recursion for variance.

2 2¯ ¯Dj p Var (z " Dz ) ! j¯ ¯z 1 1 z1 1

¯ ¯ ¯p 2 Cov (z , Dz ) " Var (Dz )1 1 1

˜ ˜¯ ¯ ¯p 2 Cov (z , G b " z ! z ) " Var (G b " z ! z ) " Var (z )1 1 1 1 1 1 1 1 1 1

G1˜ ˜ ˜¯ ¯ ¯ ¯p 2 Cov (z , G b " z ! z ) " Var (G b ) " Var (z ! z ) " 2 Cov (G b , z ! z ) "1 1 1 1 1 1 1 1 1 1 1 1 1 n1

G1 2˜ ˜ ˜¯ ¯ ¯ ¯p 2 Cov (z , G b " z ! z ) " Var (z ! z ) " 2 Cov (G b , z ! z ) " " O(" )1 1 1 1 1 1 1 1 1 1 1 n1

G1 2˜ ˜ ˜¯ ¯ ¯p 2 Cov (z , z ! z ) " Var (z ! z ) " 2 Cov (G b , z ) " " O(" )1 1 1 1 1 1 1 1 n1

G12 2 2˜ ˜ ˜ ˜¯ ¯p 2 Cov (z , z ) ! 2j " Var (z ) " j ! 2 Cov (z , z ) " 2 Cov G b , z " " O(" )¯ ¯1 1 z 1 z 1 1 1 1 11 1 ( ) n1

G12 2˜ ˜p Var (z ) ! j " 2 Cov (G b , z ) " " O(" ).¯1 z 1 1 11 n1

App. from S. L. Nuismer et al., “Correlation and Coevolution”

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Note that

2 2˜ ¯Var (z ) p Var [(1 ! m )z " m m ] p (1 ! m ) j ,¯1 1 1 1 1 1 z1

˜ ˜ ˜ ˜ ˜Cov (G b , z ) p G Cov [2g (v ! z ) " 2s (z ! z ) " 2s , z ]1 1 1 1 1 1 1 M 2 1 D 11 1

˜ ˜ ˜ ˜ ˜p 2G {g [Cov (v , z ) ! Var (z )] " s [Cov (z , z ) ! Var (z )]}1 1 1 1 1 M 2 1 11

2 2p 2G (1 ! m ){g [j ! (1 ! m )j ] " s [(1 ! m )j ! (1 ! m )j ]}.¯ ¯ ¯ ¯ ¯1 1 1 z v 1 z M 2 z z 1 z1 1 1 1 1 2 1

Putting it all together,

G12 2 2 2˜Dj p [(1 ! m ) ! 1]j " 2 Cov (G b , z ) " " O(" )¯ ¯z 1 z 1 1 11 1 n1

G12 2 2 2 2p 4G (1 ! m ){g [j ! (1 ! m )j ] " s [(1 ! m )j ! (1 ! m )j ]} ! (2m ! m )j " " O(" ).¯ ¯ ¯ ¯ ¯ ¯1 1 1 z v 1 z M 2 z z 1 z 1 1 z1 1 1 1 1 2 1 1 n1

The analogous recursion for can be found by exchanging subscripts 1 and 2.2Djz2

E. recursion.Dj¯ ¯z z1 2

¯ ¯ ¯ ¯Dj p Cov (z " Dz , z " Dz ) ! j¯ ¯ ¯ ¯z z 1 1 2 2 z z1 2 1 2

˜ ˜¯ ¯ ¯ ¯p Cov (G b " z ! z " z , z ) " Cov (z , G b " z ! z " z )1 1 1 1 1 2 1 2 2 2 2 2

˜ ˜¯ ¯" Cov (G b " z ! z " z , G b " z ! z " z )1 1 1 1 1 2 2 2 2 2

˜ ˜¯ ¯ ¯ ¯p Cov (G b " z ! z , z ) " Cov (z , G b " z ! z )1 1 1 1 2 1 2 2 2 2

˜ ˜¯ ¯" Cov (G b " z ! z , G b " z ! z )1 1 1 1 2 2 2 2

2˜ ˜ ˜ ˜p Cov (G b , z ) " Cov (G b , z ) " Cov (z , z ) ! j " O(" ).¯ ¯1 1 2 2 2 1 1 2 z z1 2

Using

˜ ˜ ¯ ¯Cov (z , z ) p Cov [(1 ! m )z " m m , (1 ! m )z " m m ]1 2 1 1 1 1 2 2 2 2

p (1 ! m )(1 ! m )j ,¯ ¯1 2 z z1 2

˜ ˜ ˜ ˜ ˜Cov (G b , z ) p G Cov [2g (v ! z ) " 2s (z ! z ) " 2s , z ]1 1 2 1 1 1 1 M 2 1 D 21 1

˜ ˜ ˜ ˜ ˜ ˜p 2G {g [Cov (v , z ) ! Cov (z , z )] " s [Var (z ) ! Cov (z , z )]}1 1 1 2 1 2 M 2 1 21

2p 2G (1 ! m ){g [j ! (1 ! m )j ] " s [(1 ! m )j ! (1 ! m )j ]}¯ ¯ ¯ ¯ ¯ ¯1 2 1 z v 1 z z M 2 z 1 z z2 1 1 2 1 2 1 2

results in

2Dj p 2G (1 ! m ){g [j ! (1 ! m )j ] " s [(1 ! m )j ! (1 ! m )j ]}¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯z z 1 2 1 z v 1 z z M 2 z 1 z z1 2 2 1 1 2 1 2 1 2

2" 2G (1 ! m ){g [j ! (1 ! m )j ] " s [(1 ! m )j ! (1 ! m )j ]}¯ ¯ ¯ ¯ ¯ ¯2 1 2 z v 2 z z M 1 z 2 z z1 2 1 2 2 1 1 2

2! (m " m ! m m )j " O(" ).¯ ¯1 2 1 2 z z1 2

App. from S. L. Nuismer et al., “Correlation and Coevolution”

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Equilibrium and Stability for Interactions Mediated by Phenotype MatchingEquilibrium Analysis

Using the recursions derived in “Derivation of Statistical Moments,” we find that interactions mediated byphenotype matching have a single equilibrium, where the variance among replicate populations is given by

2u R " u {!4R R " (m " 2R )[m " m " 2(R " R )]}j M i M M j T i j T Ti i j j i j2 2j p " O(" ), (A5)zi 2[m " m " 2(R " R )][!4R R " (m " 2R )(m " 2R )]i j T T M M i T j Ti j i j i j

and the covariance is given by

u R (m " 2R ) " u R (m " 2R )2 M 1 T 1 M 2 T 21 1 2 2j p " O(" ), (A6)¯ ¯z z1 2 [m " m " 2(R " R )][!4R R " (m " 2R )(m " 2R )]1 2 T T M M 1 T 2 T1 2 1 2 1 2

where , , , and .u p G /n R p G s R p Gg R p R " Ri i i M i M A i i T M Ai i i i i i

Equations (A5) and (A6) define the equilibrium correlation between the traits of the interacting species

r p¯ ¯z z1 2

2[u R (m " 2R ) " u R (m " 2R )]2 M 1 T 1 M 2 T1 1 2 2

2 2"(u R " u {!4R R " (m " 2R )[m " m " 2(R " R )]})(u R " u {!4R R " (m " 2R )[m " m " 2(R " R )]})2 M 1 M M 2 T 1 2 T T 1 M 2 M M 1 T 1 2 T T1 1 2 2 1 2 2 1 2 1 1 2

2" O(" ). (A7)

Note that the correlation , as shown, exists only when the system is stable (and thus must meet the stabilityr¯ ¯z z1 2

conditions presented in “Stability Analysis”).

Stability Analysis

The eigenvalues of the Jacobian matrix generated using recursion equations from D and E and equilibrium (eqq.[A5], [A6]) are

l p ![2(R " R ) " m " m ], (A8)1 T T 1 21 2

2"l p ![2(R " R ) " m " m ] " [2(R ! R ) " m ! m ] " 16R R , (A9)2 T T 1 2 T T 1 2 M M1 2 1 2 1 2

2"l p ![2(R " R ) " m " m ] ! [2(R ! R ) " m ! m ] " 16R R . (A10)3 T T 1 2 T T 1 2 M M1 2 1 2 1 2

Because we have assumed weak selection and gene flow, the stability of the system of discrete recursionequations given by D and E can be analyzed as a continuous time system. Consequently, for the system to bestable, . Further analysis of the largest eigenvalue (l2) reveals that the following2(R " R ) " m " m 1 0T T 1 21 2

condition must be met for stability:

1R R ≤ (m " 2R )(m " 2R ),M M 1 T 2 T1 2 1 24

4R R ≤ (m " 2R " 2R )(m " 2R " 2R ),M M 1 M A 2 M A1 2 1 1 2 2

2R (m " 2R ) " 2R (m " 2R ) " (m " 2R )(m " 2R ) ≥ 0, (A11)M 2 A M 1 A 1 A 2 A1 2 2 1 1 2

2R 2RM M1 2" ≥ !1.m " 2R m " 2R1 A 2 A1 2

If and only if the system is stable, it is possible for the dynamics to show damped oscillations when [2(R !T1

App. from S. L. Nuismer et al., “Correlation and Coevolution”

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. This implies that oscillations are not possible in mutualistic or competitive2R ) " m ! m ] " 16R R ! 0T 1 2 M M2 1 2

interactions (i.e., this occurs only in antagonistic interactions).

Figure A1: Contour plots of the correlation between the trait means of the interacting species as a function ofthe strength of biotic selection for species 1 (Y-axis) and species 2 (X-axis). Contour plots were generated by aleast squares regression fit of a quadratic model of the general form 2 2r p b " c y " c y " c y y " c y " c y1 1 2 2 3 1 2 4 1 5 2

to the simulated data, where the variable yi is the strength of biotic selection acting on species i.

App. from S. L. Nuismer et al., “Correlation and Coevolution”

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Figure A2: Contour plots of the correlation between the trait means of the interacting species as a function ofthe strength of abiotic selection for species 1 (Y-axis) and species 2 (X-axis). Contour plots were generated by aleast squares regression fit of a quadratic model of the general form 2 2r p b " c g " c g " c g g " c g " c g1 1 2 2 3 1 2 4 1 5 2

to the simulated data, where the variable gi is the strength of abiotic selection acting on species i.