A comparative study of models for predation and parasitism

47
mately true. Consequently, HOLLING'S disc equation should be a good approximation
for low prey densities.
On the other hand, if X is very large, then
lira a(X)=lim 2V~/v'X.
X~o~ X~oo
Thus, when X increases, a (X) decreases in inverse proportion to the square root of
X. This is because the probability of any one P being found in the recognition
radius of a O increases as X increases, and so in its extreme situation, the movement
of the Q is largely governed by the location of P's. Under these circumstances, the
movement of the Q can no longer be independent of the location of P's. More
precisely, if the Q is most attracted to the closest one of the P's found in the recog-
nition area, then O's path consists largely of the distance to the closest P. As already
demonstrated by MORISITA (1054) and CLARK and EVANS (1954), the distance between
an arbitrarily selected point and the closest one of a number of randomly distributed
points in a two-dimensional plane is inversely proportional to the square-root of the
density of the random points. The same conclusion applies when the recognition
radius R is large relative to X. This suggests that HOLLING'S disc equation would
show its bias, even if X is low, when R is very large. Conversely, if R is comparatively
small, the disc equation is again a good approximation even if X is high. That is to
say, the equation holds whenever the R is reduced to an immediate area around the
prey so that a capture is made practically by bodily contact, and this confirms my
previous conclusion.
It should be pointed out, however, that the inverse proportionality of a(X) to
the square-root of X holds only when the hunting area is a two-dimensional plane.
If the hunting area is one-dimensional, such as thin branches of a tree, a (X) is in-
versely proportional to X;and if it is a three-dimensional space, a(X) is inversely
proportional to the cubic-root of X (for the proof, see CLARK 1956).
The above investigation suggests that if HOLLING'S disc eq. (4c. 6) is applied to
my present model, the coefficient a thus estimated will show a decreasing trend as X
increases. MORRIS (1063) applied the disc equation to the instantaneous hunting curve
observed with Podisus maculiventris (Hemiptera) eating larvae of Hyphantria cunea
(Lepidoptera). (MORRIS kept the density of the prey constant during the observation
to meet the condition for an instantaneous hunting curve.) It was found that, when
the factor h was assumed to be constant, the factor a decreased as the prey density
was increased. MORRIS thought that this was due to satiation by the predator. While
this interpretation is reasonable, it is equally tenable that the trend observed was due
to a decrease in the factor a as in my a(X). Hence, it is not known, unless the
experiment is designed accordingly, to what extent the trend is attributable to satiation
and to what extent to the geometric property of the hunting behaviour.
MILLER (1960) pointed out that both HOLLING'S and WATT'S equations (to be
reviewed in the next subsection) tended to deviate from the observed trend in various