A comparative study of models for predation and parasitism

66
(4g. 8). Then f(X) for a standard X is the efficiency of the species concerned.
further discussion, see the appendix to w 4i).
It should be pointed out here that, on the whole, the review of models by WATT
(1959) is invalid, firstly because his mathematics is often wrong, and secondly because
he was confused between
(For
instantaneous and overall functions, between parasitism
and predation, and between the Z-Xo and z-Y relationships. It should also be noticed
that the criticism against the assumption f(x)=ax invalidating the NICHOLSON-BAILEY
predation equation does not invalidate THOMPSON'S parasitism equation, since the latter
does not assume f(x)=ax.
h).
The HASSELL-VARLEY model of social interference in parasites
Although this model is called by the authors (HASSELL and VARLEY 1969) 'a
new model' based on the NICHOLSON-BAILEY competition equation (see w it is in
fact a special case of the generalized THOMPSON'S model for indiscriminate parasites,
eq. (4g. 8), in which the instantaneous hunting function is a modified NICHOLSON-
BAILEY linear function. As already pointed out, THOMPSON'S equation for parasitism
takes the same form as the NICHOLSON-BAILEY 'competition equation' for predation
if the instantaneous hunting function f(X) is assumed to be a linear function of X,
i.e. f(X) =aX, in which the coefficient a is the 'effective area of recognition per
unit time'. Under these circumstances, the value of /~, as defined by eq. (4b. 8), becomes
at, a constant.
It has been shown in w 4e, however, that the /~ cannot be constant, but is at least
a function of X, host density. HASSELL and VARLEY, however, found that in some
published data the value of ~ was not independent of Y, parasite density. These data
are shown graphically in Fig. 11; this is a reproduction of figure 1 in HASSELL and
VARLEY (1969) with a slightly different arrangement. These data show that the
value of In ii tends to decrease as the value of In Y increases. The interpretation of
these relationships by HASSELL and VARLEY is that the parasites interfered with each
other more strongly as their density increased, and hence the reduction in "the area
of discovery", i.e. /f. The authors stated that "the striking feature of the relationships
in (Fig. 11) is that they are linear over several orders of magnitude", and that "the
data for Chelonus texanus CRESS. [curve (c)] cover a narrow range of parasite densities
but seem to imply a curvilinear relationship".
Thus, it was concluded that the relationships were described by the following
formula
In ii=ln Q-m In Y (4h. 1)
or
= O Y-~ (4h. 1'),
in which the factor Q is called "the quest constant" and m "the mutual interference
constant".
If the above relationships are incorporated into the NICHOLSON-BAILEY model