Sebastian Diehl. 2003. The evolution and maintenance of omnivory: dynamic constraints and the role of food quality. Ecology 84:2557–2567.

Appendix A. The derivations of the stability of the three-species equilibrium of a two-consumer–one-resource system with unidirectional consumer interference and of the criteria for mutual invasion of the two consumers. A pdf version is also available.

	(A.1)
	(A.2)
.	(A.3)

Specific resource growth and consumer functional responses are assumed to obey the following inequalities:

.

(A.4)

The last two inequalities imply that consumer N suffers from interference by consumer P, but that interference comes at no cost to consumer P.

Stability of the three-species equilibrium

The above system may have an equilibrium with all three species present, which, however, is always unstable. The signs of the elements J_ij of the Jacobian matrix J are

J =

(A.5)

One of the Routh-Hurwitz conditions necessary for stability is (Guerney and Nisbet 1998): . Given the signs of the elements of J, this condition cannot be fulfilled and the three-species equilibrium is always unstable. Two consumers that compete exploitatively for a single biotic resource may persist indefinitly on a periodic, oscillating trajectory (Armstrong and McGehee 1980). The same holds, if interference competition is included (S. Diehl, unpublished results). In the following analysis, such unstable persistence is ignored, and only locally stable point equilibria are considered.

Invasion criteria

In the absence of consumer P, an RN-system equilibrates at . From Eq. A.2 follows that, at this equilibrium,

.

(A.6)

Correspondingly, in the absence of consumer N, an RP-system equilibrates at , with

.

(A.7)

Consumer N can invade an RP-system at equilibrium if , i.e., . Substitution of Eq. A.6 yields

.

(A.8)

Because of interference from P , Inequality A.8 can only be fulfilled if

.

(A.9)

Similarly, consumer P can invade an RN-system at equilibrium if , i.e., . Substitution of Eq. A.7 yields

,

(A.10)

which requires

.

(A.11)

Clearly, Inequalities A.9 and A.11 cannot be simultaneously fulfilled and mutual invasibility is impossible.

If consumer P is the superior resource competitor (i.e., ), Inequality A.9 and A.11 imply that consumer N cannot invade an RP-system at equilibrium, whereas consumer P always invades an RN-system at equilibrium. A more interesting case arises if consumer N is the superior resource competitor (i.e., ). In that case, consumer P cannot invade an RN-system at equilibrium, but consumer N does not necessarily invade an RP-system at equilibrium. If the costs of interference suffered by N are sufficiently high (e.g., at high P_P*), N cannot invade and alternative equilibria with either N or P at equilibrium with the resource are possible.

Literature cited

Armstrong, R. A., and R. McGehee. 1980. Competitive exclusion. American Naturalist 115:151–170.

Gurney, W. S. C., and Nisbet, R. M. 1998. Ecological dynamics. Oxford University Press, Oxford, UK.