A comparative study of models for predation and parasitism

and BAILEY but redefined in w
43
as /~=(1/Y) In{xo/(Xo-Z)} for predation, and /i=
(l/Y) ln{X/(X-z)} for parasitism. If the instantaneous function f(x)is a non-linear
function, z is also a non-linear function of x0 or X, and hence /~ cannot be constant.
Although the models proposed by HOLLING and IVLEV, reviewed in w 4c and d, are
adequate to show that /~ will not be constant theoretically, these models are not quite
sufficient to show how /~ can logically vary. One of the key points for this analysis
seems to lie in understanding the coefficient a (not /~).
The coefficient a appeared consistently in the instantaneous hunting function of
models reviewed in preceding sections, with the same geometric meaning, namely the
size of an area immediately around either a prey or a predator individual within
which the predator would take action to catch the prey. In all the models, it was
assumed that the coefficient a was constant, or independent of variables x, y, and t.
Some objections arose among ecologists (see below) against the validity of the assump-
tion that the coefficient was constant,
yet. The main purpose of the present
nature of this coefficient.
The justification of the assumption
and the debate has not quite been settled
section is to give a much clearer idea of the
that a is constant is partly concerned with
the validity of the assumption of random searching--whether or not such an assumption
is reasonable in predation or parasitism theories. The idea of random searching and
random encounter is not new. It has been used very widely among theorists, and
while the concept of randomness is clearly defined in the field of mathematics,
confusion has resulted from loose application of the concept to an ecological situation.
First, one must separate the concepts of random searching and random encounters,
as the one does not necessarily follow the other. Perhaps the most satisfactory defini-
tion of the term 'random searching' is that the path of each individual predator is
affected neither by the location of prey individuals in the hunting area nor by the
paths of other members of the predator population. 'Random encounter', as opposed
to 'random searching', can be defined as meaning that every prey individual in the
area concerned has an equal probability of being encountered by predators searching
for a limited time interval. Now, the simple theory of the kinetics of gas molecules,
used by LOTKA, VOLTERRA, and NICHOLSON and BAILEY, assumes random encounters
between molecules, and so the theory can be a model for random searching to a
limited extent. This is because, as pointed out earlier, the analogy can be reasonably
applied to hunting behaviour only when an encounter is made by bodily contact;
otherwise, random encounters are approximately guaranteed only if either (1) the
number of predators searching independently is high, (2) the time for searching is
unlimited, or (3) the density of prey is relatively low.
This can be illustrated by assuming that a few molecules are marked as predators,
and all other molecules, unmarked, are prey. Then the movement of these few marked
molecules for a certain limited time interval must be limited to a small fraction of
the total area concerned, so that the unmarked molecules located in the immediate