A comparative study of models for predation and parasitism

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in the binomial series, the variance is npq as against the mean which is np, and so
the variance-mean ratio is npq/np=q (where n is the number of eggs laid, p the
probability of a given host receiving one egg, and q=l-p).
small, the variance-mean ratio, which is q=l-p,
If p is not sufficiently
must be smaller than unity. This
refutes the common belief that if the ratio is less than unity the parasites concerned
are of the discriminate type. The interpretation of frequency distribution of parasite
progeny on hosts should therefore be made with the utmost caution.
Another relevant point here is the use of the negative binomial series developed
by BLISS and FISHER (1953). The expectation of the 0-term (the proportion of hosts
receiving no parasite eggs) in the negative binomial series is given by
Pr{0} = (1 +m/k) --k (4g. 5)
where m is the mean number of eggs per host, which in my notation is n/X. Sub-
stituting n/X for m in eq. (4g. 5) we have
Pr {0} = (1+ n/kX) -k (4g. 6).
Here k is a positive constant, and if k-~oo the negative binomial distribution coincides
with the PoISsON distribution, and if k-~0, it becomes the logarithmic series (BLISS
and FISHER 1953). When we choose an appropriate value of k, the negative binomial
series describes various types of non-random distribution (excluding an even distribu-
tion). Substituting the right-hand side of eq. (4g. 6) for Pr{O} in eq. (3.20) we have
z =X {1 - (1 +n/kX) - k} (4g. 7).
GRIFFITHS and HOLLING (1969) proposed the use of eq. (4g. 7) for parasitism in
which the parasite egg distribution is not random (or, more precisely, the variance-
mean ratio is larger than unity).
The negative binomial series, however, does not
distinguish the type of underlying mechanisms involved, and so it is purely a descrip-
tive formula. Although the negative binomial distribution is identical with what is
known as the POLYA-EGGENBERGER distribution (IT(5 1963) which has a specific model
structure, I found it difficult to relate this model structure to the process of para-
sitism. In passing, although GRIFFITHS and HOLLING (lOt, cit.) suggested the use of
eq. (4g. 7) also for predation, it is not legitimate to do so.
As already discussed in w 3, predation and parasitism models will not differ from
each other in the form of the instantaneous hunting function, i.e. eq. (3. 1) holds
for both cases as long as the prey or host density can be considered to be constant.
In parasitism, the overall equation in general form is eq. (3.22), in which f(X) can
be anything as long as it is not influenced by the pattern of encounters between
parasites and hosts. Therefore, THOMPSON did not have to assume a particular form
for f(X) but only needed to say that n was the number of eggs laid.
It does not
matter whether n is an observed value or an assumed function of X, as long as one
can justify the assumption of the POISSON or similar distribution. Consequently, if
f(X) Yt in eq. (3. 1) is used as the theoretical expectation of n, the following simul-
taneous equations will hold,
,~n =f(X) Y, Jt