A comparative study of models for predation and parasitism

41
(dy/dt)/y = C (1- e-~) (4d. 6)
where C and ,t are positive constants. I shall examine in the following: (1) whether
GAUSE'S proposal of eq.
(4d. 6) takes the same form as that of IVLEV'S eq.
(4d. 6) is acceptable, and (2) why the right-hand side of eq.
(4d. 2').
Although GAUSE was aiming at the formulation of the relationships between a
protozoan predator and its prey, his experimental justification of eq. (4d. 6) came
from an observation by ~MIRNOV and WLADIMIROW (cited by GAUSE 1934, pp. 139-140)
of a parasite, Mormoniella vitripennis, attacking its host, Phormia groenlandica. It
should be remembered that the relationship expressed in the form of a differential
equation as in eq. (4d. 6) will not be appropriate in the case of an entomophagous
parasite in which generations are discrete.
This is because, while the expression
C(1-e -~*) stands for the number of progeny (per unit area) produced per parasite
during t in the present generation to form the next generation, these progeny will
not reproduce in the present generation. Therefore, it is incorrect to equate the (dy/
dt)/y to C(1-e-~*). Before making further comments on eq. (4d. 6), however, I shall
investigate the second point, i.e. why were changes in the density of the parasites'
progeny, in relation to changes in host density as observed by SMIRNOV and WLADI-
MIROW, described by the formula C(1-e -~) which is of the same form as IVLEV'S ?
Let Y' be the density of progeny in the parasite population produced to form the
next generation, and z the density of hosts attacked in the present generation. Let
us assume hypothetically that Y' is more or less proportional to z, i.e. Y'=c'z where
c' is a proportionality constant. In the meantime, it has been shown that although it
is an instantaneous equation, eq.
(4d. 3) can nevertheless describe one cross-section,
parallel to the z-X plane, of an overall hunting surface for parasitism, e. g. eq.
(3.22). In other words, although eq. (4d. 3) should correctly be used to estimate the
value of n, the equation can, because of its considerable flexibility in fitting, also
describe the z-X relationship, provided that the coefficients a and b are appropriately
chosen for a given value of Yt, i.e. for a given cross-section of the hunting surface.
Under these circumstances, one may obtain, though only superficially,
Y'=c'b(1-e -ax) Yt,
or by transposing Yt to the left-hand side,
Y'/Yt =c'b (1 -e -ax) (4d. 7).
In the SMIRNOWWLADIMIROW observation, as presented in figure 40 of GAUSE'S
(1934) book, the factor Yt appears to be fixed at the same value for different values
of X throughout the observation, and this consistency satisfies the condition under
which eq. (4d. 7) has been derived.
Clearly, the expressions Y'/Yt and X in eq.
(4d. 7) are equivalent to (dy/dt)/y and x respectively in eq. (4d. 6), and the constants
c'b and a in eq. (4d. 6) can be written as C and 2 respectively as in eq. (4d. 6).
And this is why GAUSE's equation can be deduced from IVLEV'S.
The reasoning which led to eq. (4d. 7) explains why GAUSE adopted eq. (4d. 6),
and also why GAUSE and IvLgv proposed the same function, though GAUSE was