A comparative study of models for predation and parasitism

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HOLLING (1966). In order to maintain consistency throughout this study, an effort
will be made, as far as possible, to use the same symbols denoting the same factors,
parameters, etc. For example, x stands for the density of a prey (host) species as
against y for the predator (parasite) density, and t for a time-interval during which
the prey (host) species are exposed to predation (parasitism). Symbols used extensively
are listed and defined in Appendix 4. The consistency of using the same symbols
for the same meaning in different models makes it difficult to use those of the
original authors.
Each subsection begins with the presentation of the model concerned, more or
less in the manner that the original author presented it, so that the way he reasoned
can be studied easily.
a). The LOTKA-VOLTERRA model
LOTKA (1925) and VOLTERRA (1926) independently proposed equations which
are essentially the same. Both authors' methods are largely analytical (i. e. by mathematical
analysis), though considering to some extent analogies from kinetics. VOLTE-
RRA was thinking of predation whereas it was explicitly stated by LOTKA that his
equations were for parasitism.
Their first assumption is the geometric increase of a population; in the case of
the prey population, its instantaneous rate of increase per individual, i.e. (dx/dt)/x, is
assumed to be constant in the absence of predators. Thus we have dx/dt-rx where
r is a coefficient of increase (or of net birth -= birth minus death due to factors other
than predation). Similarly, for the predator population, we have dy/dt=-r'y where
-r' is a coefficient of decrease in the absence of the prey population, as predators
will die if no food is available. However, if the two populations are put together,
the prey population will now diminish as much as it is preyed upon. That is to say,
in the presence of predators, the coefficient of increase must be equal to the difference
between the net birth in the absence of predators and the death due to predation.
It is assumed secondly that the number preyed upon is proportional to the number
of encounters between prey and predator individuals, and so the rate of loss due to
predation is equal to the rate at which an individual prey is encountered by predators,
i.e. ax where a is a proportionality factor of encounters. Then r, under these circumstances,
should be replaced by the expression (r-ay). Similarly, the predator population
can now increase because food is available, and its rate of increase per predator
must be equal to the difference between the death rate and the birth rate due to the
intake of food. So, under the assumption that the birth rate is proportional to the
amount of food eaten, which in the above assumption is proportional to the number
of encounters with prey, the coefficient of the net increase in the predator population
is equal to the expression (-r'+a'x), where a' is a positive constant. Thus we have,
dx/dt = (r- ay) x
=rx-ayx
(4a. la)