A comparative study of models for predation and parasitism

72
Under which the value of In ~, for a given fixed value of X, is increasing, decreasing,
or remaining constant, the first order partial derivative Oln ii/Oln Y will be calculated
below:
Oln ii .. dY' ~-Y,~+I (X) (~Y')~ 1 ) (4i. 11)
Oln Y= o" ~ d Y ----;~Y'\~
L z2,(x) ~ ~i )
in which the derivative d Y'/dY is, from eq. (4i. 6):
d Y'/d Y= { (1 - e -~r) -/~ Ye -~r } / (1 - e -st) 2>0.
From the above evaluation of the partial derivative, the following conclusions will be
drawn:
(1). When Y->O, the partial derivative converges to zero, so that the curve is parallel
to the In Y axis at the level of
In ii=ln{f (X)t/X} +ln 20(X)
(see eq. (4i. 10)).
~ oo
(2). When Y is sufficiently small, so that ~ 2i+1 (X) (~Y') ~/i [ and 2E 2i (X) (~Y') ~/i !
i=1 i~l
are negligible as compared with ,h(X)and ;o(X) respectively, then
Oln ii/Oln Y~--~Y(dY'/dY) {2~(X)/2o(X)-1}.
Therefore: (a) if 2~ (X) >20 (X) , i.e. social facilitation, the curve is increasing
as Y increases, but (b) if At (X) ~ At(X) (SY')'/i [,
i=0 i=0
the partial derivative in eq. (4i. 11) is positive, and so the curve is increasing,
but (b) if the effect of interference outweighs that of facilitation, the curve is
decreasing.
(4). When Y becomes sufficiently large, both lower and higher terms in the series
{2,(X)(~Y')~/i!} will become negligible as compared with mid-terms, i.e. for
certain numbers k and k', we have
co
k t
2~(X) (~Y')~/i ! ~ ~ At(X) (~Y')'/i !
i=O
i=k
and the same applies to the series {A,+~(X)(~Y')*/i!}. Now it is unlikely that
the degree of social facilitation increases indefinitely as i increases; the effect
of interference must sooner or later become apparent. Hence, beyond a certain
number for i, e. g. k, the inequality A, (X) >Ak+~ (X) will always hold. Under
these circumstances, the partial derivative becomes always negative, and hence