A comparative study of models for predation and parasitism

37
big the container Was or if the prey density was kept constant; though, judging from
his statement (chapter 2, p. 18 of his book) that "Food consumption was studied...
by estimation of the food left over out of the quantity given .... ", the prey density
was apparently not kept constant. Obviously, in the light of my analysis leading to
eq. (4d. 3), IVLEV's variable u must have been influenced by the size of container and
consequently the density of fish: more precisely, the size of container influence-~ the
density of fish, and consequently the coefficient b, if the density of food species was
kept constant to meet the condition required for describing the instantaneous relationship.
If the food species diminished gradually during the course of predation, the
estimates of both coefficients a and b obtained by fitting eq. (4d. 3), even though it
takes factors Y and t into account, would have been different between observations
with different predator density, simply because it amounts to fitting an instantaneous
equation to one particular cross-section of the hunting surface. Hence, these coefficients
a and b estimated in IVLEV'S experiments are specific to these experiments and
have no universal meaning. This is the same criticism I raised in regard to HOLLINO'S
model. In order to eliminate the awkwardness pointed out above, it is necessary to
deduce an overall hunting equation. But before doing so, I shall examine in more
detail the reason why IVLEV'S instantaneous equation takes such specific form: although
eq. (4d. 3) has no apparent contradiction as an instantaneous hunting equation, the
justification for starting our inference from the differential eq. (4d. 1) is yet to be
rationalized.
IVLEV obtained his idea concerning eq. (4d. 1) from three existing equations
developed in physical chemistry and physiology. The first was an equation for unimolecular
reaction. The velocity equation for a simple unimolecular reaction takes
the form
dx/dt = -ax
where x is the density of molecules at time t, and a the reaction coefficient. So,
letting x0 be the initial density of the molecules, we have
z =x0 (1 - e -~)
where z--xo-x. As already seen, this is the NICHOLSON-BAILEY equation in which
Y=I. Of course, there is a resemblance between the velocity equation and IVLEV's
equation in their mathematical form, but the meanings are entirely different since
the derivative dx/dt in the velocity equation is the rate of change in time, whereas
dn/dX in IVLEV'S is the rate of change with density. These two attributes are,
needless to say, totally irrelevent to each other. Therefore, the quotation of the
velocity equation by IVLEV is absolutely irrelevant in his context.
IVLEV also quoted the "WEBER-FECHNER law" (but without citing the literature
source). As far as I know, this is a law in neuro-physiology representing the relationship
between the strength of a stimulus and the reaction of a nerve. However,
there appears to be no possible resemblance between this law and IVLEV'S equation
in their mathematical forms.