Matthijs Vos, Antonie M. Verschoor, Bob W. Kooi, Felix L. Wäckers, Donald L. DeAngelis, and Wolf M. Mooij. 2004. Inducible defenses and trophic structure. Ecology 85:2783–2794.

Appendix A. Partial derivatives with respect to k for the positive and negative branches of Eq. 13.

Demonstrating that for the positive branch

,

for all x, and for the negative branch

,

for all x, is easier if we rescale k in Eq. 13 as

.

 

Now P_tot can be written as:

,

 

where

.

 

This (x) ³0 and (x) does not depend on k, since H(x) does not. The partial derivative of the negative branch is:

,

 

which has to be negative, that is

,

 

which is true since  ² > 0 and > . The latter is a necessary condition for the bitrophic system to exist. For the positive branch we need

,

which is true since > .