(B.1)
(B.2)
.(B.3)
The densities of R, N, and P at the three-species equilibrium of model II with omnivory are
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(B.1)
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(B.2)
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. |
(B.3)
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Substitution of Eq. B.1 into Eq. B.2 and setting Eq. B.2 to zero yields the maximum conversion efficiency eRP1, beyond which N goes extinct, as a function of K:
 . |
(B.4)
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Similarly, substitution of Eq. B.1 into Eq. B.3 and setting Eq. B.3 to zero yields the minimum conversion efficiency eRP2, beyond which P can exist, as a function of K:
 . |
(B.5)
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eRP1 and eRP2 are monotonously decreasing functions of K that intersect once, as shown by their partial derivatives with respect to K:
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(B.6)
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 . |
(B.7)
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A three-species equilibrium with omnivory is feasible whenever eRP is high enough to allow P to exist but low enough to not drive N extinct, i.e., eRP1 > eRP2 (in that case eRP1 and eRP2 describe the upper and lower bounds of the coexistence region in Fig. 3). This inequality amounts to
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(B.8)
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At a three-species equilibrium,
the right parenthesis in Eq. B.8 must be negative, because a necessary condition
for a three-species equilibrium is that the intermediate consumer has a positive
growth rate in the absence of the top consumer (i.e., 
). Thus, a three-species equilibrium with omnivory requires that the
left parenthesis (which corresponds to the numerator in Eq. B.1) is positive.
The same condition ensures that the slope of the upper bound of the coexistence
region in Fig. 3 (Eq. B.6) is less negative than the slope of the lower bound
(Eq. B.7). Thus, whenever a three-species equilibrium exists at some K,
three-species coexistence is possible also at infinitely high K for conversion
efficiencies bounded by the asymptotic values of eRP1 and
eRP2. The latter are given by:
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(B.9)
|
 . |
(B.10)
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eRP1 is a decreasing function of K but is always positive. Thus, Eq. B.9 gives the upper threshold of the conversion efficiency eRP for which three-species coexistence with omnivory is possible at infinite K (provided that Inequality B.8 is fulfilled).