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Text S1. Analytical approximation for changes in species abundances with coevolution. The
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analyses presented in the Box in the main text can be used to both generalize the results from the
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simulations and expose the causal pathways underlying the role of coevolution in determining
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the changes in species abundances when climate change alters the intrinsic rate of increase of
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one species. Consider the general model
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N1,t+1 = N1,tF1(N1,t, 1(1,t, 2,t)N2,t, 1,t)
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N2,t+1 = N2,tF2(N2,t, 2(1,t, 2,t)N1,t, 2,t) .
(S1)
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This structure is an obvious generalization of the competition/mutualism model. It also has the
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same structure as the predator-prey model; if N1 is the prey and N2 is the predator, then 1(1, 2)
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= a(E, 1, 2), and 2(1, 2) = –a(E, 1, 2)c, where c is the predator conversion rate.
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From the Implicit Function Theorem,
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æ ¶G ö ¶G
¶X *
= -ç ÷
è ¶X ø ¶E
¶E
-1
(S2)
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where X*=(N1*,N2*,1*,2*) is the vector of the equilibrium densities and trait values for each
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species, and G is a vector of functions that all equal zero when population densities and traits are
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at their equilibrium values; specifically,
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G1(N1, 1(1, 2)N2, 1) = F1 – 1
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G2(N2, 2(1, 2)N1, 2) = F2 – 1
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G3(N1, 1(1, 2)N2, 1,[D11]N2) =
dF1 V1
du1 F1
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G4(N2, 2(1, 2)N1, 2,[D22]N1) =
dF2 V2
du 2 F2
1
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where the notation [Dif] denotes the partial derivative of f with respect to variable i; hence,
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[D11] = 1/1. Letting G/X = A, and assuming that E affects only species 1, the change in
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the sum of species abundances with respect to E is
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 X 1 *  X 2 *
cof ( A,1,1)  cof ( A,2,1) G

E
det( A)
E
(S3)
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where cof(A,i,j) = (–1)i+j det([A]i,j) and [A]i,j is matrix A with the ith row and jth column
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removed.
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The elements of A can be derived for G1 and G3 as
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a11 = [D1G1] = –1
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a12 = [D1G1]1
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a13 = [D2G1][D11]N2 + [D3G1] = 0
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a14d1 = [D2G1]N2[D21]
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a31 = [D1G3]
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a32 = [D2G3]1 + [D4G3][D11]
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a33 = [D2G3]N2[D11] + [D3G3] + [D4G3][D111]N2
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a34d1 = [D2G3]N2[D21] + [D4G4][D121]N2,
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with symmetric equations for G2 and G4. The term d1 represents the cross derivatives [D21] and
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[D121] that show when the interaction effect from species 2 on species 1, 1(1, 2), changes
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with the trait value of species 2. In the specific equations we use for competition and mutualism
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in which 1(1, 2) = 1(1 + d2), d is synonymous with d1.
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Expanding equation (S3) in terms of aij and excluding terms of order di2 yields
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cof(A,1,1)+ cof(A, 2,1) = (-1- a21 ) a33a44 - a23a32 a44 d2 + a23a31a44 d1
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det(A) = (1- a12 a21 ) a33a44 + d2 ( a23a32 a44 + a23a31a12 a44 ) + d1 ( a14 a41a33 + a14 a42 a21a33 )
2
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(S4)
Note that when d1 = d2 = 0,
¶ ( X1 *+X2 *)
(-1- a21 ) a33a44 ¶G = (1+ a21 ) ¶G
=¶E
(1- a12 a21 ) a33a44 ¶E (1- a12 a21 ) ¶E
(S5)
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which is identical to the result in the absence of coevolution. This shows that coevolution affects
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the response of equilibrium abundances to changes in E only when selection on the trait values of
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one species changes its interactive effects i(1, 2) on the other species.
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Letting C1 = (1 + a21)a33a44, C2 = (1 – a12a21)a33a44,
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¶X1 *+X2 *
C1 + a32 a23a44 d2 - a31a23a44 d1
¶G1
=
¶E
C2 + ( a23a32 a44 + a23a31a12 a44 ) d2 + ( a14 a41a33 + a14 a42 a21a33 ) d1 ¶E (S6)
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This expression gives a straightforward way of generalizing the simulation results for the
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specific forms of competition, mutualism, and predator-prey equations we used. For example,
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for competition
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¶X1 *+X2 *
¶G1
(1- a2 ) a1a2 N1N2 + a1a22 N12 d
=
2
2
¶E
(1- a1a 2 )a1a 2 N1 N 2 + a1a 2 (a1 N 2 + a 2 N1 )d ¶E
(S7)
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Because existence of a positive stable equilibrium requires |i| < 1, it follows that (X1*+
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X2*)/E increases with d, implying that non-conflicting coevolution will always increase the
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change in summed equilibrium abundances with changes in E.
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Broader generalities can be obtained directly from equation (S6) by noting constraints on
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the signs of the terms aij. The cases of competition and mutualism are similar, as follows. For
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traits to have optimal values, a33 < 0 and a44 < 0. If higher trait values vi reduce the interaction
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impact on species i from the other species (as in our models), then a23a32 > 0 and a14a41 > 0. For
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the exponential forms of equations that we used for simulations, i.e., F1(N1, 1N2, 1) =
3
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exp(f1(N1, 1N2, 1)), the terms a13 = a24 = 0. Assuming that these values are small, equation S6
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simplifies to
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¶X1 *+X2 *
C1 + a32 a23a44 d2
¶G1
=
¶E
C2 + a23a32 a44 d2 + a14 a41a33d1 ¶E
(S8)
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For competition and mutualism, C1 > C2, which leads immediately to the conclusion that
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increasing d = d1 = d2 will increase (X1*+ X2*)/E.
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For predator-prey interactions, equation (S8) also applies, although the signs of the terms
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differ from the cases of competition and mutualism. The term d1 represents the cross derivatives
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[D21] and [D121], and similarly d2 represents [D12] and [D122]. Because there is only a single
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function giving the predation rate, 1(1, 2) = a(1, 2), and 2(1, 2) = –a(1, 2)c, so d1 = –
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cd2. For traits to have optimal values, a33 < 0 and a44 < 0. In contrast to competition and
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mutualism, a23a32 ≤ 0 and a14a41 ≥ 0. Thus, positive values of d = d1 = –cd2, which will be the
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case for predator-prey relationships, will in general decrease (X1*+ X2*)/E. Although this
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analysis only addresses the case when the environmental change E affects the prey intrinsic rate
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of increase, a symmetric analysis shows similar results for the case of E affecting predators.
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