A comparative study of models for predation and parasitism

40
finding a disc. So in HOLLING's disc experiment, the value n/Yt should never exceed
k/(l+hk). It should be noticed, however, that eq. (4c. 6) describing HOLLING'S disc
situation also holds for sweep-feeding predation as in the NICHOLSON-BAILEY model.
In this case, there is no limitation to the prey density, and so the maximum ration
is set at n/Yt~l/h.
The effect of the handling time, h, can easily be incorporated into the toss-a-ring
equation by eliminating t~ from eqs.
kYt(l_e -~R2X)
n=l + hk(l_e- ~R2x )
(4c. 2) and (4d. 4). Thus we have
(4d. 5).
In the above equation, the maximum value of n/Yt is k/(l+hk) which is reached
asymptotically as X becomes infinity. This elaboration, however, may be of doubtful
value, for the following reasons. First, the purpose of the toss-a-ring model is only
to explain why IVLEV'S equation takes that form, and also to find where the missing
variables, the predator density and the time factor, should appropriately be placed.
Secondly, if the model is to reflect a certain type of predation process, it may be
compared only to that of some shore birds probing with their beaks to find food,
but it is not a common method of hunting for the predators of interest. Therefore,
it would be wise to leave IVLEV's equation in the form, i.e. eq. (4d. 3) :
n = b (1 - e -~x) Yt,
as purely descriptive without attaching any serious meaning to it. Equation (4d. 3)
has an excellent power of describing instantaneous hunting curves of the shape of
the one in Fig. 1, which has been widely observed with various predators.
As long as it is remembered that eq. (4d. 3) is an empirical form of an instant-
aneous hunting equation, it may be used safely, according to the situation to which
the equation is applied, either as in the present form or as the basis of deducing a
theoretical trend in the higher order of synthesis. For example, I have used IVLEV'S
equation for my model of the clutch size variation in birds (RoYAMA 1969) and the
instantaneous form of HOLLING'S equation for explaining the hunting behaviour of
the great tit (Parus major L.) (ROYAMA 1970). In both cases, I could neglect depletion
of the prey density caused by predation by the birds. If, however, eq. (4d. 3) is to
be applied to a situation in which the prey density is being depleted, further synthesis
is needed to deduce an overall hunting equation. This has in fact been done in w 3,
as in eq. (3. 12) derived from eq. (3.10). If eq. (4d. 3) is used to describe the
instantaneous relationship in parasitism of the indiscriminate type, eq.
than eq.
(3.12) should be used.
(3. 24) rather
A brief mention will be made of an equation that GAUSE (1934) had proposed
before IVLEV. GAUSE, in his study of protozoan predators, found that the rate of
increase per individual in the predator population was not a linear function of prey
density, and he proposed an exponential function to replace the equation, i.e. eq.
(4. lb), proposed by LOTKA and VOLTERRA for that relationship. Thus