A comparative study of models for predation and parasitism

9
of this fact, it is curious that they ignored it.
One possible justification may be
related to two statements. The first one (the last paragraph of the introductory part
of their paper) stated:
"These oscillations can be stabilized [the NICHOLSON-BAILEY equations generate
host-parasite oscillations with ever-increasing
amplitude] by reducing the area
of discovery as parasite density increases, but changes in area of discovery in
relation to host density do not promote stability".
The second statement (the second and third paragraphs on p. 1135 of their paper)
is summarized below:
The NICHOLSON-BAILEY equation did not exactly fit the observed relationship
between the winter moth, Operophtera brumata (L.), and its parasite Cratiech-
neumon culex (MuELLER) in two aspects: (1) the calculated peak of the parasite's
density lagged two generations after the peak of the host's density whereas the
observed lag was only one generation; (2) while in the NICHOLSON-BAILEy model
more than two parasite species could not coexist [because of the competitive
exclusion of one species by another], there were several parasite species coexisting
in the field. But, when eq. (4h. 2) was used instead of the NICHOLSON-BAILEY,
it was found that both of the above difficulties disappeared.
With respect to the first statement, it is certainly agreeable that reduction of "the
area of discovery", i. e. d, in relation to increase in host density will not promote
stability. It should be noticed, however, that such changes in the d tend to accelerate
instability (see TINBERGEN and KLOMP 1960). This implies that the effect of changes
in the d in relation to host density, which must be involved in actual host-parasite
interaction systems, has to be counteracted by other factors, or conditions, more
strongly than in a hypothetical situation in which changes in host density have no
influence on the value of d. Since the HASSELL-VARLEY equation assumes that the d
is independent of host density, this bias has to be cancelled out by another bias, and
this latter bias is in fact involved in the assumption expressed by eq. (4h. 1). Hence,
the fact that the observed relationship between O. brumata and C. culex agreed with
the theoretical relationship expressed in eq. (4h. 2) suggests that this model is another
example of c21 under C2 in w 2.
It was pointed out in w
that HOLLING'S disc equation involved some bias, be-
cause it assumed that the discovery of a prey was regarded as the capture of it.
Nevertheless, the model enhanced the importance of the factor h, the handling time.
By the same token, although these models involve certain contradictions, the HASSELL-
VARLEY model, as well as WATT'S model, implies strongly the significance of social
interference among parasites as one important regulatory mechanism in host-parasite
interaction systems. As the effect of social interference seems very important, I shall
investigate it more in detail in the following subsection.