A comparative study of models for predation and parasitism

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since
z =Xo (1 - e-ay~
lira (1 - e-,'t)/r': t.
r~-~O
Clearly, y0 corresponds to my previous notation Y, and so we obtain the NICHOLSON-
BAILEY eq. (3. 8).
Now it is very clear that the NICHOLSON-BAILEY model is only a special case of
the new solution of the LOTKA-VOLTERRA model, i.e. eq. (4b. 6), in which r, r', and
a' are all zero. The above conclusion is contradictory to a statement by NICHOLSON
and BAILEY (1935, second paragraph, p. 551) :
"... , we have not been able to derive our theory from LOTKA'S fundamental
equations. Competition does not appear explicitly in any of his equations, and few,
if any, indicate the existence of this factor."
It should be mentoned that NICHOLSON and BAILEY appeared to refer to 'LOTKA'S
fundamental equations' as those in chapter VI of LOTKA'S book, but that those which
are relevant to the NICHOLSON-BAILEY treatise, i.e. eqs.
(4a. la) and (4a. lb) in the
present paper, appear in chapter VIII. However, LOTKA called the equations in chapter
VIII a 'special case' and those in chapter VI, a 'general case'. Since a general case
involves a special case, the NICHOLSON-BAILEY criticism quoted above must be meant
to apply also to eqs. (4a. la) and (4a. lb), and such a criticism cannot be taken seriously.
Contrary to the NICHOLSON-BAILEY view, the LOTKA-VOLTERRA equations are
comparatively more general and detailed than the NICHOLSON-BAILEY one. Obviously,
the only necessary condition which makes the LOTKA-VOLTERRA equations match the
condition of discrete generations is that a'-0.
And r and r' are, unlike the simpler
assumption by NICHOLSON-BAILEY, not generally zero. That is to say, the whole of
the NICHOLSON-BAILEY model is covered by the LOTKA-VOLTERRA one, and so we
do not need the former. However, some specific assumptions tentatively adopted by
LOTKA are not satisfactory from an ecologist's point of view. What is needed is the
generalized LOTKA-VOLTERRA eqs. (4a. 2a) and (4a. 2b), which have both necessary and
sufficient conditions for computation of the final densities of both populations, if
appropriate functions for f, gl, and gz are found. As animals with discrete generations
are assumed here, we need separate equations to evaluate the initial densities in the
following generation. If, however, generations are not discrete, eqs. (4a. 2a) and (4a. 2b)
are sufficiently comprehensive.
Although TINBERGEN and KLOMP (1960) introduced into the NICHOLSON-BAILEY
model the effect of mortality in both populations (the authors considered parasitism
rather than predation as they did not think that a distinction was needed), it was
assumed that their mortality factors acted only after the period of attack had ended,
but not during the attack. This was perhaps because the arithmetic method used by
TINBERGEN and KLOMP was not quite capable of incorporating the effect of mortality
during the attack period. An analysis of the process in which death takes place during