A comparative study of models for predation and parasitism

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particles.
As the average number of particles within the ring is 7rR2X and the
number of tosses in time ts is kt,, where k is the frequency of tosses per unit time,
the total number caught by the ring will be
n~ =~R~kXt~.
Although the right-hand side of the above formula is the same as in eq. (4c. 4)
describing HOLLING'S experiment, notation n, in the left-hand side is not n as in eq.
(4c. 4). It was assumed in HOLLING'S model that all the discs tapped were taken,
and it was a justifiable assumption since only one disc was discovered at a time as
there was no overlap of the discs. In the present model, however, more than two
particles may occur within the ring at one toss. Thus, if the predator is allowed to
catch only one prey individual within its recognition zone at a time, the total number
captured (i. e. n) will be the number of successful tosses rather than ns, the total
number discovered.
In the present model, the distribution of the prey individuals and the tosses of
the ring are ideally at random, and so, provided that R is not too large as compared
with the table, the frequency of tosses in which a given number of particles falls
must follow a POISSON series with its mean equal to the average number of particles
in the ring at one toss, i.e. rcR~X. Hence, as the frequency with which no particle
occurs within the ring in the period t, will be kt,e -~R*X the total frequency of
successful tosses in t, will be
n =kt, (l-e- ~R~X ).
In the above model, only one ring was assumed to be tossed at the frequency of k
per unit time per table.
If there are Y rings per unit area tossed independently,
each at the average frequency of k per unit time, the total frequency is kYts, and
so the above equation becomes
n=kYt~(1-e -'~R~X ) (4d. 4).
If kYt~ in the above equation is set equal to b, and rcR 2 to a, we have IVLEV'S eq.
(4d. 2"). Thus, we find that IVLEV'S equation, in terms of the toss-a-ring simulation,
is a generalized version of HOLLING'S experiment, since eq. (4d. 4) is a generalization
of eq. (4c. 5). That is, recognition zones can overlap, and so there is no restriction
to the range of variation in the prey density X (remembering that HOLLING'S eq.
(4c. 5) holds in the disc experiment only for X