A comparative study of models for predation and parasitism

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the situation again, though it is not quite necessary, the area around each prey is
assumed to be a circle of radius R. Then, as the predator moves from A to B, it
sees those prey individuals with hatched circles (Fig. 4a) ;or if, alternatively, the
predator, rather than the prey, is given a circle of radius R as in Fig. 4b, then those
prey within the hatched belt along the predator's path will be recognized.
To calculate the number of prey found by the predator along its path of search,
Fig. 4b will be used. First, if the predator can see a prey anywhere in the circle,
the size of the effective area in which prey are found between A and B must be the
size of the hatched area plus the circle at A. If, however, the distance between A
and B is sufficiently large as compared with radius R, the area covered with the
circle around A can be neglected as compared with the size of the hatched area.
Second, and alternatively, if the path between A and B is considered to be a given
fraction of a path of search, point A is the last point reached in the preceding section
of search, and so the circle at A is the area already searched. Thus, it is sufficient
to know the size of the hatched area in order to calculate the number of prey found
between A and B. The size of the hatched area is clearly the product of the width
2R and the length Vt, so that the number of prey found in the area is 2RVXt, where
X is the density of prey fixed during t in each observation.
If there are Y predator individuals searching at the same time, their paths being
entirely independent of each other, the total number of prey found by these Y predators
for time t, i. e. n, will be
n-:2RVXYt (4b. 3).
Equation (4b. 3) is clearly equivalent to eq. (3.1). That is, expression 2RVX is the
instantaneous function f(X) in eq. (3.1), i.e.
f(x) :2 RVX.
Consequently, if R and V are assumed to be independent of X, i.e. changes or variation
in both R and V are independent of X, we can replace the complex factor 2RV
by a single factor, say, a, which can conveniently be treated as a constant. Thus we
have in this model
f(X) -aX (4b. 4)
or
n =aXYt. (4b. 5).
Clearly, the complex factor aYt(=-2RVYt) is the area traversed by all the predators
for time t, and is therefore equal to the NICHOLSON-BAILEY factor s. Also, if t is the
whole length of time that each predator spends hunting in the generation concerned,
the expression at is the whole area effectively searched by each individual predator
hunting for the generation. So, this factor at is the 'area of discovery' and is assumed
in NICHOLSON-BAILEY'S argument to be constant for a given species. (It should be
mentioned here that NICHOLSON-BAILEY did not find any reason to separate parasitism
from predation, and so the above factor was in fact called the area of discovery of
a parasite species for its life time which usually ends at the end of a generation.)