A comparative study of models for predation and parasitism

7O
i). A geometric model for social interaction among parasites (this study)
In w 3, I introduced a function S, by which the effect of social interaction among
attacking species upon the instantaneous hunting efficiency is indicated.
Thus, if
social interaction is involved, the instantaneous hunting equation for predation is given
by eq. (3. 14) rather than eq. (3. 4). As already explained, however, the instantaneous
equation for parasitism does not take the form of a differential equation as in eq.
(3. 14), but, for indiscriminate parasites, it is expressed in terms of the number of
eggs laid per unit area, i.e. n, as in eq. (3. 1). Thus, the equation for indiscriminate
parasitism, equivalent to eq. (3. 14) for predation, will be written as:
n=S(Y, X)f(X) Yt (4i. 1),
and from eq. (3.22), we have an overall hunting equation for indiscriminate parasites
as below:
z=X{1-r (Y, X)f(X) Yt/X, V)} (4i. 2).
In order to study some fundamental characteristics of the function S, I shall again
use a geometric model similar to those used previously.
Suppose, firstly, that a given parasite individual encounters, within an area 8
around itself, other parasite individuals in the course of hunting.
these other parasites encountered within the area 8 is 0, 1, 2 .......
If the number of
or i, the instan-
taneous hunting efficiency of the given parasite, i.e. f(X), is changed by factors 2o,
21, ~2 ...... or 2~ respectively. It is conceivable, as a more general case, that 2 is
influenced not only by the number of parasites in the 8, but also by the number of
hosts. This is because, as already pointed out in w 3, the effect of, say, interference
might be strengthened or weakened if a lesser or greater number of hosts, respec-
tively, is available within the 8. Thus, it is more appropriate to indicate the number
of hosts too. Thus, 1~ is the index of the degree of social interaction when there
are i parasites and j hosts within the area 8; it should be noted that both i and j
take discrete values, 0, 1, 2 ..... , independently of each other.
Secondly, let m (j) be the probability of finding j hosts within the 6. Then the
average partial realization of the potential efficiency for a fixed value of i will be
oo
2~:~o(j). Similarly, let o(i) be the probability of finding i parasites within the 3.
j-0
Then the overall degree of changes in the instantaneous hunting efficiency for all j's
and i's, i.e. S(Y, X), will be
S(Y, X)= ~{ ~ 2~:~o(j)} o(i) (4i. 3).
i=o j=0
Now, ~o is the probability-distribution function of j (and can be determined when both
the average number of hosts within the area 8 and its variance are known).
So the
oo
value of ~2~jco(j) can be determined for a given value of X and for each value of i.
j-0
Therefore, if the value of X is fixed in the following argument, the expression
co
~,~o(j) can be indicated simply by 2,(X). Then, eq. (4i. 3) will be written as
j=o
co
S(Y, X) = ~ ,~,(X) o(i) (4i.4).
i~0