A comparative study of models for predation and parasitism

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This problem will not be discussed any further in this paper.
The solution of simultaneous eqs. (4a. la) and (4a. lb) under the assumption of discrete
generations was not considered by the original authors. The solution, as I will
show in the next section, is in fact possible and is related to the NICHOLSON-BAILEY
model.
b).
The NICHOLSON-BAILEY model
This model is known as the 'Competition model'. It is, primarily, constructed for
the purpose of demonstrating NICHOLSON'S hypothesis that animal populations are in
the state of balance fluctuating around a steady density of each species concerned,
and that this steady density (or steady state) is brought about by competition among
the members of the parasite species (NICHOLSON 1933). NICHOLSON with the collabo-
ration of a mathematician, BAILEY (NICHOLSON and BAILEY 1935), intended to con-
struct a model on the assumption of a very simple, idealized situation, concerning a
theoretical relationship in densities between host and parasite species. By altering
conditions in this simple model, they drew numerous conclusions about the mode of
existence of steady states.
Whether or not NICHOLSON'S basic philosophy that animal populations are in the
state of balance is a useful one, is not of concern here.
It is more important to
determine whether the basic premises in the NICHOLSON-BAILEY theory can produce
a reasonable model for parasitism, so that a comparison between the model and
observation can provide any useful direction.
original authors.
The following is the reasoning by the
Let x0 be the number of objects (hosts) originally present in a unit area, and let
x be the number left undiscovered after an area s has been traversed (by all parasites
concerned). Then the number of previously undiscovered objects discovered in a
fraction of area traversed, i.e. ds, is xds. This must be equal to the decrease, -dx,
of the number of undiscovered objects per unit area, i.e.
-dx--xds (4b. 1),
and since x=xo when s=0, integrating eq. (4b. 1) for the range (0, s), and hence
(x0, x), we obtain
X ~Xoe-*
from which we have
z/xo: 1- e-' (4b. 2)
where z=xo-x. Factor s is called by the authors the 'area traversed', which is the
area that is searched effectively by all parasites involved and includes possible overlaps.
In passing, the average area traversed by each individual parasite is called the 'area of
discovery'. As against the area traversed, the net total area searched by all parasites
concerned is called the 'area covered', which excludes areas already searched. Thus,
the right-hand side of eq.
(4b. 2) shows that the proportion of the 'area covered'
increases only asymptotically as the 'area traversed' increases, and that therefore the