A comparative study of models for predation and parasitism

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dy/dt: (-r' +a'x)y
= -r'y+a'xy (4a. lb).
Both LOTKA and VOLTERRA, assuming that all the coeffients involved were constant,
solved the above two equations simultaneously, from which emerged the familiar
'LOTKA-VoLTERRA oscillation' in a predator-prey interacting system. Both LOTKA and
VOLTERRA were aware that the assumption that the coefficients a and a' were cons-
tant was too simple, but VOLTERRA justified his assumption by stating that the
frequency of encounters between the prey and the predators
proportion to the densities.
must be in linear
For LOTKA, however, the justification of the constant
coefficients seemed to be purely for operational convenience, that is, to solve the
simultaneous equations. LOTKA carefully stated that factor a can, in a broad assump-
tion, be expanded as power series in x and y, i.e.
a:oz+~x +ry + ............
and a=oz can be an approximation if ~, r, etc. are sufficiently small for values of
both x and y not too large.
Some unreasonable aspects can be pointed out in the LOTKA-VoLTERRA equations
from a theoretical point of view.
First, LOTKA stated that the model is primarily
for parasitism, although he did not exclude predation explicitly. As I have pointed
out in w 3, however, the instantaneous hunting equation for parasitism would not take
the form of a differential equation as in eq. (4a. la). Therefore, LOTKA was mistaken
in this respect. Secondly, because VOLTERRA was thinking of a predation process,
the way he reasoned to get eq. (4a. lb) is not acceptable. First, it is obviously
incorrect to assume that predators die of starvation at the same rate when prey is
available as when no prey is available. In other words, the presence of the prey
population causes not only the rise of predator population by reproduction but also a
decrease in the death rate, because the predators are not as starved as when no prey
was given. This suggests that the first term in the right-hand side of eq. (4a. lb)
must also involve x, the prey density. It is acceptable, however, to assume that, in
eq.
(4a. la), the coefficient of increase for the prey population is equal to the
difference between r, the rate of net increase in the absence of enemies, and ay, the
rate of death due to predation when the predator population is added to the system
concerned. This is because it can be assumed that r is not influenced directly by
the presence of predators; its influence, if any, operating only through changes in
the prey numbers due to predation.
At this stage, let us rewrite eqs. (4a. la) and (4a. lb) in general forms for further
discussion, i. e.
dx/dt=g~(x) -f(x, y) (4a. 2a)
dy/dt=g2(x, y) (4a. 2b)
where gl, ge, and f are functional symbols. Note that the linear term for x in the
second equation is now excluded for the reason given above.
The second function, i.e.
f(x, y), on the right-hand side of eq. (4a. 2a) is of