A comparative study of models for predation and parasitism

42
thinking of a function equivalent to g2 in eq. (4a. 2b) while IVLEV was thinking of
one equivalent to the function f in eq.
(4a. 2a). This reasoning suggests, however,
that GAUSE'S propgsal of his equation, the justification of which was based erroneously
on an observation of parasites with discrete generations, is not acceptable, for two
reasons, if applied to predators with continuous generations. First, as pointed out by
NICHOLSON and BAILEY (1935, p. 552, first paragraph), the number of progeny in a
predator population, unlike that in a parasite population, will not in general be propor-
tional to the number of prey eaten by the parental predators.
be avoided if it could be verified that the exponential function as in eq.
This objection might
(4d. 6) still
holds for protozoan predators in which the value and the ecological significance of
the coefficient ~ are different from those of the coefficient a in eq. (4d. 7), and there-
fore that the similarity between the two equations is only coincidental.
My second objection, however, seems unavoidable. Clearly, dy/dt is a linear func-
tion of y in eq. (4d. 6), suggesting that the predator population exhibits a geometric
increase for any given value of x. This contradicts GAUSE'S own suggestion of a
logistic law with respect to the natural increase of the prey population in the absence
of predators; in particular with respect to the function gl (x) in the generalized LOTKA-
VOLTERRA eq. (4a. 2a). GAUSE'S inconsistency in this respect, i.e. adopting the logistic
law for gl and neglecting it for gz, appears to be attributed to the misconception in
his statement (GAuSE 1934, p. 53, last paragraph): "In the general form the rate of
increase in the number of individuals of the predatory species resulting from the
devouring of the prey dN2/dt Eequivalent to dy/dt in my notation] can be represented
by means of a certain geometrical increase which is realized in proportion to the
unutilized opportunity of growth.
This unutilized opportunity is a function of the
number of prey at a given moment." It appears that GAUSE overlooked in the last
sentence above that the 'unutilized opportunity of growth' in the predator population
is a function not only of the density of prey but also of the density of the predators
at the same time.
To summarize, in section IV of chapter III in his book, GAUSE (1934) tried to
improve the LOTKA-VOLTERRA formulation of the prey-predator system. His sugges-
tions, however, were reasonable only with respect to the function gl (x) in eq. (4a. 2a)
and not with g2(x, y) in eq. (4a. 2b). As to the function f(x, y) in eq. (4a. 2a),
there was no suggestion by GAUSE, and the function f was left uncriticized as a
linear function of x, which is unreasonable. For these reasons, GAUSE'S mathematical
investigations into the prey-predator interaction system are unsatisfactory.
e). ROYAMA'S model of random searching and probability of random encounters
In the previous sections, it has been shown that the instantaneous hunting func-
tion, i. e. f, is, unlike the simple assumption by LOTKA and VOLTERRA and by
NICHOLSON and BAILEY, not a linear function of prey or host density. This finding
influences the notion of 'the area of discovery' that was originally used by NICHOLSON