A comparative study of models for predation and parasitism

34
the effect of Y. In order to show the point, let us assume that a and h in eq. (4c. 6)
are the true estimates of the factors originally defined, whereas those in eq. (4c. 3)
are superficial and are written a' and h' respectively. Then the fit of eq. (4c. 3) to
data, which should be correctly fitted by eq. (4c. 6), means that
a'Xt/ (1 + a'h'X) = aXYt/ (1 + ahX)
and transposing we get
h'=h/Y+ (a'-aY)/aa'XY
or
a'=aY/ (l +aX(h- h'Y) ).
Thus, estimates a' and h' not only change as the predator density changes but also
they are a decreasing function of the prey density, unless a'=aY and h'=h/Y. Fur-
ther, they are not independent of the units in which the densities of both populations
are measured. Hence, such estimates have no universal value.
Secondly, the depletion of prey or, in the case of parasitism, superparasitism,
often occurred in the observations. If eq. (4c. 3) is applied to such data, it is the
same as fitting the instantaneous equation to one of the cross-sections parallel to the
Z-Xo plane of the surface generated by eq. (4c. 9) in the case of predation, or eq.
(4c. 10) in the case of parasitism. Obviously, the curve of a cross-section parallel to
the Z-Xo plane, as in Fig. 3a, changes as Y or t changes, and therefore the estimates
for factors a and h should change accordingly. Again such estimates have no univer-
sal value.
The third point is concerned with a precaution that should be taken when prey
density is changed by changing the size of the experimental universe, a method often
adopted to obtain an extremely high prey density with relatively few individuals (e. g.
MORRIS 1963; HAYNES and SISOJEVIC 1966).
If this is done, however, it should
be noticed that the predator density changes too. To assess z in this method, we
measure a cross-section of the surface which goes diagonally across the xo-Yt plane.
As far as I know, all the published literature in which HOLLING'S disc equation
is applied, involves at least one of the above three misapplications. Yet, the goodness
of fit is, in many cases, remarkably high. This is because the equation involves two
factors that are normally estimated from the observed relationships between z and
x0. That is to say, the nature of the estimation ensures that the resultant curve fits
the observation well. Hence, a good fit does not constitute verification of the theory.
In my opinion, the significance of HOLLING'S simulation experiment with discs
and tapping is to show the importance of the factor h, and this simple model is
sufficient to show that "the area of discovery", i.e. /i, cannot be independent of prey
or host density, as clearly deduced from eqs. (4c. 9) or (4c. 10). However, an application
of the original model as in eq. (4c. 3) to a situation in which the effect of diminishing
returns is obviously involved, is incorrect.