A comparative study of models for predation and parasitism

73
the curve must be decreasing for large values of Y. Although the proof is
curtailed here (because it can easily be confirmed by calculating the second
order derivative), it should be mentioned that whether the rate of decrease is
accelerated or decelerated depends on the rate of decrease in 2~(X) with increasing
i; the curve is decreasing with an increasing rate if the value of 2
decreases comparatively fast as i increases, but the rate of decrease in the curve
may become lower if the value of 2 decreases only slowly with increasing i.
Some examples of curves generated by eq. (4i. 9) are shown in Fig. 12. These hypothetical
curves cannot be compared directly with the observed curves and scattergram
in Fig. 11, because the values of f(X), t, and ~ are not known in these observations.
tOO
0,50
Xo(x} = 1.0
e
)h(x) = 0,5 I
(i~
1l
0,10
(1} o= 0
(2) = 0,25
0,0. ~
13) = 0.50
[z, ) = 0,75
X
:0
l,f,-
0
=,
~ 1.0C
i ' ' ' I I
0,1
I I I L I I I i| I I I I I I I II I I
1.0 10,0
o~5o
t
{1)
(2)
0 , 2 0 , 5 , B,0,2,,,5 \ \,~,
1.00 1,50 0,80 0.50 0.25 0,21 0.18 014 011 0.09 0.08 0.07
1oo ,.,2o loo .... o.o oJo 0,50 o: 030 21; 0.27 ::: 0.25 ::; 0,23
0,IC
(1}
..... 0:, . . . . . . . . ,'0 . . . . . . . . ,~0 ' '
MEAN No. OF PARASITES PER EFFECTIVE AREA OF INTERACTION
Fig. 12a. Hypothetical relationships between the values of ii/{f(X)t/X} and 6Y,
(mean number of parasites per effective area) calculated from eqs. (4i. 6) and
(4i. 8), plotted in the natural logarithmic scale on both axes. The values of
2i (X) shown in the figure decreases as i increases, indicating that social
interference only is considered here.
Fig. 12b. The same as in Fig. 12a, but the value of 2i(X) increases from i=0
to 1, indicating social facilitation, and then decreases towards higher values of i.