A comparative study of models for predation and parasitism

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where ci and ce are integral constants. Hence, the following relationship should hold:
f ~f,dN~-cl= f
r~dP-c~.
That is, the functions rl and r2 are not independent of each other.
(2), an ecological condition that the function rl should reflect: since rl is a partial
derivative ON~/ONo, it should reflect the limited capacity of each predator according
to its maximum number of attacks, i.e.K.
(3), an ecological condition that the function r2 should reflect: since r2 is a partial
derivative ONA/~P, it should reflect the effect of both types of competition, i.e. the
effect of diminishing returns and the effect of social interference.
In the derivation of eq. (4f. 7), WATT evaluated the factor A arbitrarily, neg-
lecting the first condition and part of the third condition, i.e. the effect of diminishing
returns.
Although it seems as though WATT was considering the effect of compe-
tition, i.e. condition (3), his consideration in terms of the factor A was concerned
only with the effect of social interference. This is obvious because eq. (4f. 4) does
not involve No, whereas the effect of diminishing returns should be a function of
No. The neglect of the effect of diminishing returns in WATT'8 mathematics, and
hence the failure to determine the function r2 in relation to r~, resulted in the contra-
diction when eq. (4f. 7) was regarded as an overall hunting equation.
My alternative interpretation of eq. (4f. 7) is therefore that it is an instantaneous
hunting equation equivalent to eq. (3. 1); i.e. Na is not z, but n. It automatically
follows that No is X, and this means that the density of the attacked species is kept
constant during the attack period. Then P is Y. Now, K was defined by WATT as
'the maximum number of attacks made per P during an attack period' (though the
expression 'per P' is not clear, this is perhaps 'per individual predator'). Then K
corresponds to the expression bt used in my toss-a-ring experiment, i.e. b was the
frequency of tosses per unit time so that the maximum possible number of attacks
for time t per individual ring was bt (see w 4d).
mine as above, except A, eq. (4f. 3) is rewritten as
On/OX= A Y(Ybt - n)
So, changing WATT'S notations to
and eliminating A from the above equation, using the relationship A--aY-a (cf. eq.
(4f. 5) in which P~ Y, and a-=a), we have
n=b(1-e -aYI-pX) Yt (4f. 8).
If we set ~=1, then eq. (4f. 8) becomes
n=b(1-e -~) Yt,
and this equation is identical to eq. (4d. 3), i.e. IVLEV'S equation with the addition
of the attack period t and the density, Y, of the attacking species.
While WATT stated that the structure of eq. (4f. 3) was intuitively obvious, it
appears that intuition was a poor guide in this case, and the structure has become
intelligible in the light of the toss-a-ring experiment.
That is, eq. (4f. 7), which is
equivalent to eq. (4f. 8), represents a generalized instantaneous equation of the toss-