# S1 Appendix.

```Supporting Information
for the paper:
Where two are fighting, the third wins: Stronger selection facilitates greater
polymorphism in traits conferring competition-dispersal tradeoffs
S1
Appendix: Calculating the thresholds
Here, we calculate (i) s0 , the critical value of s above which q = 1 is not dominant, and (ii) s1 ,
the critical value of s above which the steady state population in our model becomes dimorphic
(Fig. 3). To calculate s0 , note that if s = s0 , D(1) = 0. Therefore, it follows from Eq. (10) that
∞
X
M =0
PM (1)
1 − M s0 F 0 (1)
=0.
(M + 1)2
P∞
Since Q−1 (λ) ≡ M =0 PM /(M +1) = λ1 (1−e−λ ), it follows from Eq. (5) that Q−1 (αΓ(q1 )) =
1/(αq1 ), which implies
1
αQ−2 (αΓ(1))
s0 = 0
,
(S1)
F (1) 1 − αQ−2 (αΓ(1))
where
∞
X
Q−2 (λ) ≡
PM /(M + 1)2 .
(S2)
M =0
Fig. S1 demonstrates that s0 decreases with α, approaches infinity as α → 1, and approaches
zero as α → ∞.
At the second threshold, where s = s1 , a mutant at q = 1 has an equal per-capita growth-rate
as a qc individual, namely f (1, qc ) = 0. In addition, since D(qc ) = 0, it follows that, for
F (q) = −q,
αQ−2 (αΓ(qc ))
s1 =
.
(S3)
1 − αqc Q−2 (αΓ(qc ))
Finding qc is more complicated, but we calculate a higher bound for s, ŝ1 , by assuming that the
branch at q = 1 invades if and only if it can sustain by using empty patches that are not seeded
by any qc -seeder. Then, f (1, qc ) = 0 implies αP0 (qc ) = 1, which yield exp(−αΓ(qc )) = 1/α.
Thus,
ln(α)
Γ(qc ) =
,
(S4)
α
and from Eq. (5) it follows that
ln(α)
qc =
,
(S5)
α−1
and therefore,
αQ−2 (ln(α))
.
(S6)
ŝ1 =
ln(α)
1 − αα−1
Q−2 (ln(α))
Fig. S1 demonstrates that ŝ1 approaches s1 when α is large, and also that both s1 and ŝ1
approaches infinity either as α = 1 or as α → ∞.
S1
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