A comparative study of models for predation and parasitism

33
appear. This is perhaps because HOLLING used only one finger and also because n is
the number of discs removed per table. If, however, there are two fingers tapping
independently, the frequency of tapping, k in eq. (4c. 4), will be doubled provided
that there is no interference. In general, if there are Y fingers tapping per unit
area of the table we have, in place of eq. (4c. 1),
n =aXYG (4c. 5)
in which n is the number of discs removed per unit area rather than per table.
This generalization would not influence eq. (4c. 2), and so, eliminating t~, we get
n = aXYt/(1 + ahX) (4c. 6).
As the disc density has been fixed in the above model situation, eq. (4c. 6) is obviouslY
an instantaneous equation comparable to eq. (3.1), in which
f(X) =aX/ (l +ahX) (4c. 7).
Hence the overall hunting equation for predation, i.e. eq. (3.13), will be evaluated
by integrating
dx/dt = -ax Y/ (1 + ahx) (4c. 8).
Thus we have
z =x0 (1 - e -~(rt-7~)) (4c. 9).
Equation (4c. 9) is an overall form of HOLLINC'S disc equation, taking account of the
effect of diminishing returns, and is thus comparable with the LOTKA-VOLTERRA and
the NICHOLSON-BAILEY equations. It is at once clear that the equation is a generalized
NICHOLSON-BAILEY model, or that the latter is a special case of the former in which
factor h-0 (cf. eq. (3.8)). Equation (4c. 9) represents a surface in a Z-Xo-Yt coordinate
system, the shape of which is very much like that in Fig. 3a. A cross-section
parallel to the z-Yt plane shows the effect of diminishing returns similar to the
cross-section in the NmHOLSON-BAILEY competition surface (Fig. 2). The shape of
a cross-section parallel to the Z-Xo plane in HOLLING'S surface is, however, curvilinear,
unlike the NICHOLSON-BAILEY one which is rectilinear (see Fig. 2a). (It is difficult to
make the variable z in eq. (4c. 9) perfectly dependent : nevertheless the surface can
be drawn by assuming that Xo is the dependent variable and z and Yt independent
ones.)
If the disc model is applied to parasitism of the indiscriminate type, the righthand
side of eq. (4c. 7) should be substituted for f(x) in eq. (3. 22). Hence, if the
fingers tap, for example, entirely at random, the function r will be the zero-term of
a POISSON series, and so we have
z=X(1-e -a~/(l+ahz~) (4c. 10).
This also generates a surface similar to that in Fig. 3a.
My intention in generalizing HOLLING'S disc equation is in fact to point out three
mistakes commonly seen in the scattered publications in which the original disc equation
was applied directly to observed data (see e.g. HOLLING 1959). First, the
density of the predators is not always 1. If eq. (4c. 3) is fitted to observed data in
which Yr the estimates of the factor a and h in the equation inevitably involve