Appendix IV. Approximate distribution of herbivore production, foraging preference model.
To derive equations describing equilibrium values of the logistic plant growth model incorporating foraging preference parameter a, I started with text equation (24), the foraging preference model:


(IV-A)


converted to the logarithmic “x” axis; x = ln(ci) and xm = ln(cm)

(IV-B)

collecting terms and rearranging:

(IV-C)

(IV-D)

integrating along the x-axis and rearranging (IV-D) yields:

(IV-E)

(IV-F)

(IV-Fa)
To evaluate herbivore production by plants with values of x [x = ln(ci)] between Xm and –K (K = xmin = ln(minimum value of ci)) is done by first evaluating for x= xm then subtracting the value for x = xm - K:


[x = xm] (IV-G)


[x = xm - Z] (IV-H)

(IV-J)

subtracting IV-J from IV-G:

(IV-K)

(IV-L)

Collecting terms gives the number of herbivores pruduced by plants within Z log units of xm on the axis of plant suitability:

(IV-M)
To calculate the proportion of the total herbivore production represented by (IV-M) above, we need to know the total herbivore production. This is done by evaluating equation IV-Fa at x = -K and subtracting from IV-G above.


(IV-N)

 

[K>6; exp(-K) approaches 0] (IV-O)

The proportion of the total herbivore production will then be:

(IV-P)