Appendix III. Derive the estimated distribution of production of new herbivore adults by plants.
To derive the estimated distribution of production of new herbivore adults by plants, it is useful to convert the axis of plant suitability to a logarithmic scale.  Let x = ln (ci), and let P(X:X-Z) equal the number of herbivore adults produced by plants with values of x less than X but greater than X-Z.  Then, P(x) dx, where P(x) = P’(x) / dx, is the number of new adults produced by plants at position x along the x-axis.  This can be written as:

 


               []                          ( III-A)


where D(x) equals the probability that any given egg will be placed on a plant on which . 
By analogy with equation (8) and incorporating equation (19),


                                      (III-B)

where, as in equation (21), sx = si / W, with W being the interval along the x-axis included within plant category i.  Note that “W” is the “width” of each category i, and is approximately 0.5 [=ln(1.7)] for the models in figures 4 and 5.  This is a way of scaling the discrete plant categories 1,2,3…i…n to a continuous logarithmic distribution of plants along the axis of plant vulnerability to the herbivore. 

Combining IIIA and IIIB::

(IIIBa)

Integrating IIIBa from x to (x - Z), the production of new adults is then:

          (III-C)

expanding and simplifying equation (III-C) yields:

(III-Cd)


To calculate the production of herbivore adults by plants with values of x ranging between xm and xm-Z, and remembering that xm = ln (cm), so that in equation IIICd, e(xm) in the numerator cancels out with cm in the denominator: 

                                                 (III-D)


The grand total of herbivore production ( = large value of Z) is then approximately:


                                                  (III-E)


and the proportion of the total produced by plants with values of x greater than xm - Z is:


                           (III-F)

This equals 82% when Z=2.3 = ln(10).  In other words, plants with values of ci within an order of magnitude of that of the marginal plant (cm) produce over 80% of the new herbivore adults.

An analogous analysis using the plant growth cutoff model instead of the logistic plant growth model, assuming equilibrium plant sizes are equal to sx for all plants with values ci < cm, yielded the proportion:
                                                                            (III-G)
This equals .90 (90%) when Z=2.3.  Regardless of the plant growth model, 80-90% of adult herbivores are produced by plants within 1 order of magnitude (2.3 log units) of cm along the axis of plant vulnerability to the herbivore.